SCALAR CURVATURE AND THE EXISTENCE OF GEOMETRIC STRUCTURES ON 3-MANIFOLDS, II

MICHAEL T. ANDERSON

Contents

0. Introduction. 1 1. Background Setting and Results. 3 2. The Linearized Equations. 10 3. Level Set Measures and Masses. 22 4. Uniformity Estimates. 27 5. Mass Gaps and Essential Spheres. 36 6. The Collapse Situation. 47 7. Non-Degeneration Situations. 51 Appendix A. 57 Appendix B. 58 References 60

Abstract. This paper studies the degeneration, i.e curvature blow-up, of sequences of metrics approaching the Sigma constant σ(M) on 3-manifolds M with σ(M) ≤ 0. The degeneration is related to the sphere decomposition of M in case M is σ-tame.

0. Introduction.

This paper is a continuation of [1], and is mainly concerned with the Sphere conjecture of [1,§0]. We recall the basic issues and set-up.

Let M be a closed oriented 3-manifold and σ(M) the Sigma constant of M, i.e. the supremum of the scalar curvatures of unit volume Yamabe metrics on M. Throughout the paper, it is assumed that

(0.1)  σ(M) ≤ 0;

this is the case if for instance M has a K(π, 1) factor in its sphere decomposition, c.f. [11]. Consider the L^{2} norm of the negative part of the scalar curvature as a functional on the space of metrics on M, i.e.

(0.2)  S_{−}^{2}(g) = (v^{1/3} ∫_{M}(s^{−})^{2}dV_{g})^{1/2},

where s^{−} = min(s, 0), s = s_{g} is the scalar curvature of the metric g, v is the volume of (M, g). The volume power in (0.2) is chosen so that S_{−}^{2} is scale invariant. By [2, Prop.3.1],

(0.3)  inf S_{−}^{2} = |σ(M)|,

under the assumption (0.1), so that a minimizing sequence for S_{−}^{2} has similar characteristics to a maximizing sequence of Yamabe metrics on M.

Partially supported by NSF Grants DMS 9802722 and 0072591.

2000 Mathematics Subject Classification. Primary 58J60, 57M50; Secondary 58E11, 53C20.

Conjectures I and II of [1] concern the limiting behavior of certain minimizing sequences {g_{i}} for S_{−}^{2} on irreducible 3-manifolds. These conjectures imply the geometrization conjecture for 3manifolds satisfying (0.1). In [1, Thm.0.2], the validity of these two conjectures was proved for tame 3-manifolds, i.e. 3-manifolds M which admit a minimizing sequence of unit volume metrics {g_{i}} for S_{−}^{2} such that, for some K < ∞,

(0.4)  Z^{2}(g_{i}) = ∫_{M} |z_{g_{i}}|^{2}dV_{g_{i}} ≤ K, as i → ∞,

where z = r − \frac{s}{3}g is the trace-free Ricci curvature. c.f. also [5] for an outline of the proof.

The assumption (0.4) is a rather strong topological assumption and it is not clear which 3manifolds actually satisfy (0.4). For instance if N_{i}, i = 1, 2, are hyperbolic 3-manifolds then M = N_{1}#N_{2} does not satisfy (0.4). The fundamental issue concerning Conjectures I and II above is in fact whether the converse holds, i.e. if M is irreducible, is then M necessarily tame.

In this paper, building on the work in [1], substantial progress is made on this problem. To explain this, let

(0.5)  σ̃(g) = S_{−}^{2}(g) − |σ(M)|,

so that σ̃(g) ≥ 0, and a sequence {g_{i}} is minimizing for S_{−}^{2} if and only if σ̃(g_{i}) → 0. Definition 0.1. A 3-manifold M satisfying (0.1) is σ-tame, if there exist constants k < ∞ and K < ∞, and some unit volume minimizing sequence {g_{i}} for S_{−}^{2}, such that

(0.6)  σ̃(g_{i})^{k} · Z^{2}(g_{i}) ≤ K, as i → ∞.

Thus, M is σ-tame if there exists a sequence of unit volume metrics {g_{i}} with σ̃(g_{i}) → 0 such that the (trace-free) curvature of g_{i} blows up in L^{2} at most as fast as some polynomial in σ^{−1}. This is of course a topological condition on M, which is clearly much weaker than the tameness condition (0.4), (which corresponds to the case k = 0).

In fact we believe that any closed 3-manifold M with σ(M) ≤ 0 is σ-tame, regardless of whether M is irreducible or reducible; this will be pursued in subsequent work since the methods dealing with this question are not related to the main issues here. In [2,3], numerous examples or, more precisely, models for potentially minimizing sequences {g_{i}} for S_{−}^{2} were constructed. All of these models satisfy (0.6) with k = 1. Using these models it is not difficult to show that if Conjectures I and II are valid, then any 3-manifold M, σ(M) ≤ 0, is σ-tame with k = 1. Conversely, a much simpler proof of the Main Theorem below can be given for σ-tame 3-manifolds with k = 1, (based on the work in [2, Thm.C]).

The purpose of this paper is to prove the following result: Main Theorem. If M is σ-tame, then Conjectures I and II of [1] are valid.

As noted above, this implies the geometrization conjecture for σ-tame irreducible 3-manifolds M with σ(M) ≤ 0, as well as the evaluation of their Sigma constant σ(M). We refer to [1,5] for background discussion on these Conjectures. Exact statements of the Main Theorem in the cases σ(M) < 0 and σ(M) = 0 are given in Theorems 7.6 and 7.7 respectively.

The overall strategy of proof for the Main Theorem is briefly as follows; further details are given at the beginning of each section, c.f. also Remarks 1.3, 1.6 and 2.2. Basically, a modified version of the Sphere conjecture in [1] is proved for σ-tame 3-manifolds. Thus, suppose M is σ-tame but not tame, so that the curvature of any minimizing sequence {g_{i}} for S_{−}^{2} blows up in L^{2}, i.e. (0.4) fails. By far the most crucial issue of the paper is to locate “natural” 2-spheres S^{2} from the geometry of the sequence {g_{i}}, as i → ∞. Under two relatively weak hypotheses, namely a degeneration hypothesis and a non-collapse hypothesis, it is shown that such 2-spheres arise as a certain locus where the curvature of {g_{i}} blows-up, i.e. becomes very large, for σ̃(g_{i}) small. This core issue of the paper requires most of the technical work.

Given the existence of such 2-spheres, one then needs to prove that they are essential in M, i.e. do not bound 3-balls. This is done by means of the cut and paste or comparison methods already developed and used in [1, §2,§4] just for this purpose. The upshot is that M is then necessarily a reducible 3-manifold.

Recall that Conjectures I and II apply only to irreducible 3-manifolds. Hence, if M is irreducible, one of the two aforementioned hypotheses must fail. In the last section of the paper, §7, this is shown to imply that there exists a minimizing sequence which does not degenerate at all, so that M is in fact tame. This proves the Main Theorem, as discussed preceding (0.4). In sum, if M is σ-tame, but not tame, then M is necessarily reducible.

The crucial issue is thus to locate geometrically natural 2-spheres from the geometry of the degeneration of a suitable minimizing sequence {g_{i}}. The approach taken in the previous papers [1,2] was to find such spheres near regions where the curvature is blowing up at a maximal rate, since these are geometrically the most obvious places. Blow-up limits of the metric sequence based at such points are solutions of a certain elliptic system of equations, the Z_{c}^{2} equations, c.f. §1, studied in detail in [2]. The Sphere conjecture states that such limit solutions or metrics have an asymptotically flat end, c.f. also Remark 1.6; such an end thus carries a natural 2-sphere.

While we believe the Sphere conjecture above is true, in this paper a somewhat different approach is taken. Namely, 2-spheres in M are detected as a locus where the curvature of {g_{i}} is blowing up, but blowing up at a, comparatively speaking, almost minimal rate; (similar issues regarding the rate of curvature blow-up play a central role in [3]). The metric blow-up limits based at such points are flat, and the near limiting metric behavior is governed by solutions of linearized equations closely analogous to the linearized Z_{c}^{2} equations. These linearized equations are derived in §2 and their implications for the near limiting geometry are analysed further in §3-§4. The main point, developed in §5, is the identification of 2-spheres in almost flat regions of rescalings (M, g′_{i}) of (M, g_{i}), which enclose regions where the curvature is much larger. Such 2-spheres serve as a substitute for the spheres at infinity in the Sphere conjecture. Remark 3.6 explains briefly one important reason why this approach is advantageous to the analysis of general Z_{c}^{2} solutions.

Thus, the most significant part of the paper is the work in §5, c.f. in particular Theorems 5.4, 5.9 and 5.12, given the preparatory work in §2-§4 leading up to §5. More detailed descriptions of the contents and the strategy of the arguments are given at the beginning of each section as well as in the Remarks.

1. Background Setting and Results.

In this section, we set the stage and discuss background material needed for the work to follow; most of this is taken from [2,4]. Remarks 1.3 and 1.6 give some preliminary perspective on the overall strategy of the paper.

For a given ε > 0, consider the scale-invariant functional I_{ε}^{−} = v^{1/3}εZ^{2} + S_{−}^{2} on the space of metrics on the closed, oriented 3-manifold M, σ(M) ≤ 0, i.e.

(1.1)  I_{ε}^{−} = εv^{1/3} ∫_{M} |z|^{2}dV + (v^{1/3} ∫_{M}(s^{−})^{2}dV )^{1/2}.

Although seemingly complicated at first sight, it is explained in detail in [2, §1], (c.f. also [1]), why this is, in a natural sense, an optimal functional to consider from the point of view of geometrization of M via direct methods in the calculus of variations.

It is proved in [2,Thm.3.9] that, for any ε > 0, there exists a domain Ω_{ε} and a complete C^{2,α}∩L^{3,p} smooth Riemannian metric g_{ε} on Ω_{ε} with vol_{g_{ε}}Ω_{ε} = 1, which realizes the infimum of I_{ε}^{−}. The domain Ω_{ε} has a finite number q = q(ε, M) of components and is weakly embedded in M in the sense that any domain with smooth compact closure in Ω_{ε} embeds in M. Further, there exists an exhaustion {K_{j}} of Ω_{ε} by compact subsets such that M \ K_{j} is a graph manifold. The pair (Ω_{ε}, g_{ε}) is called a minimizing pair for I_{ε}^{−}. In brief, (Ω_{ε}, g_{ε}) gives a geometric decomposition of M w.r.t. the functional I_{ε}^{−}.

As ε → 0, the metrics g_{ε} form a minimizing sequence or family for S_{−}^{2}, in that

(1.2)  S_{−}^{2}(g_{ε}) → |σ(M)| and εZ^{2}(g_{ε}) → 0, as ε → 0.

The essential issue is to understand the behavior of sequences of minimizing pairs (Ω_{ε}, g_{ε}), where ε = ε_{i} is some sequence with ε_{i} → 0.

The domain Ω_{ε} is empty if and only if M is a graph manifold which does not admit a flat metric; in fact, if Ω_{ε} is non-empty, then no component of Ω_{ε} is a graph manifold. Similarly, it is apriori possible that a sequence (Ω_{ε_{i}}, g_{ε_{i}}) is constant, i.e. independent of ε_{i}, as ε_{i} → 0. In this case of course (0.4) holds, so the Main Theorem is already proved; thus [1, Thm.0.2] implies that (Ω_{ε}, g_{ε}) = (H, g_{o}), where g_{o} is a constant curvature metric with scalar curvature σ(M) < 0, with ∂H incompressible in M, or Ω_{ε} = M is flat if σ(M) = 0. In these situations, the geometric decomposition of M w.r.t. I_{ε}^{−} above already gives the geometrization of M, for any ε > 0, c.f. [1, §2] for further details.

Thus, in effect, only the situation where (Ω_{ε}, g_{ε}) is not empty and not constant in ε needs to be considered. All of the work in §2-§7 is to prove that this situation can occur only if M is reducible. Hence, if M is irreducible, then the point is to prove that either Ω_{ε} is empty, (so M is a graph manifold), or (Ω_{ε}, g_{ε}) is constant in ε as above, c.f. Theorems 7.2, 7.6, and 7.7.

Abusing terminology slightly, a metric of constant negative curvature will be called hyperbolic, even if the sectional curvature is not −1. The topology of Ω_{ε} may, apriori, change with ε.

The metrics g_{ε} satisfy the system of Euler-Lagrange equations for I_{ε}^{−}, c.f. [2, Thm.3.3]:

(1.3)  L^{∗}w = ε∇Z^{2} + ϕ · g,
(1.4)  2Δ(w − \frac{εs}{12}) + \frac{1}{4}sw = \frac{1}{2}ε|z|^{2} − 3c.

Here ∇Z^{2} is the gradient of Z^{2} = ∫ |z|^{2}dV, c.f. (B.4), Δ = trD^{2} where D^{2} is the Hessian, and

(1.5)  L^{∗}u = D^{2}u − Δu · g − u · r,

where r is the Ricci curvature, (all w.r.t. (Ω_{ε}, g_{ε})). The subscript ε has been dropped from the notation in (1.3)-(1.4). The function ϕ is given by ϕ = −\frac{1}{4}sw + c, and the constant c is given by

(1.6)  c = \frac{1}{12σ} ∫ (s^{−})^{2} + \frac{ε}{6} ∫ |z|^{2},

where σ = (v^{1/3} ∫ (s^{−})^{2})^{1/2}. Observe that σ is scale invariant and converges to |σ(M)| = −σ(M) as ε → 0, by (1.2). Most importantly, the function w is given by

(1.7)  w = −\frac{s^{−}}{σ} ≥ 0,

so that the L^{2} norm of w over Ω_{ε} equals 1. The equation (1.4) is the trace of (1.3). The metric g_{ε} is C^{2,α} ∩ L^{3,p}, for any p < ∞, while the scalar curvature s is Lipschitz. In any region where s ≠ 0, the metric is C^{∞}, in fact real-analytic. The function −w was denoted by τ in [1].

Again the equations (1.3)-(1.4) are a complicated nonlinear system of elliptic PDE (in the metric g), but are explained and analysed in detail in [2, §1,3]. Essentially all of the work in this paper depends crucially on the exact form of the terms in these equations; the analysis developed is thus specific to these equations and may not be relevant to general variational problems on the space of metrics.

Let T = T_{ε} = sup_{Ω_{ε}} w. Then one has the estimate, (c.f. [2, (3.43)]),

(1.8)  1 ≤ T ≤ (1 + 2\frac{εZ^{2}}{σ})^{1/2}.

Thus, if σ(M) < 0, then

(1.9)  T_{ε} → 1, as ε → 0,

and so in particular

(1.10)  w_{ε} → 1, a.e.

in the sense that for any μ > 0, the measure of the set where |1 − w_{ε}| ≥ μ converges to 0, as ε → 0. Moreover, in general for σ(M) ≤ 0, one has

(1.11)  ∫_{Ω_{ε}} |dw_{ε}|^{2}dV_{g_{ε}} → 0, as ε → 0,

c.f. [3, (3.39),(3.46)]. However, if σ(M) = 0 and M is σ-tame, then (1.9) does not hold. Although it can be shown that T_{ε} remains bounded as ε → 0, this will not be used here. Thus to unify these two situations, regardless of the behavior of T, the potential w will always be renormalized as

(1.12)  u = \frac{w}{T} ,

so that, by fiat,

(1.13)  sup_{Ω_{ε}} u = 1.

The points x_{ε} ∈ Ω_{ε} where u(x_{ε}) → 1 play the central role throughout the paper.

The equations (1.3)-(1.4) then read

(1.14)  L^{∗}u = ε̄∇Z^{2} + ϕ̄ · g,
(1.15)  2Δ(u − \frac{ε̄s}{12}) + \frac{1}{4}su = \frac{1}{2} ε̄|z|^{2} − 3c̄,

where ε̄ = ε/T, c̄ = c/T, ϕ̄ = ϕ/T. The equations (1.14)-(1.15) will be used throughout the paper.

The non-negative function u in (1.12) is of course just a renormalization of the non-positive part of the scalar curvature of g_{ε}. It is viewed as a potential function on (Ω_{ε}, g_{ε}), since it satisfies the potential type equation (1.15). The main point of view of the paper is to understand the degenerating geometry of (Ω_{ε}, g_{ε}) as ε → 0 from analysis on the scalar curvature, or more precisely u, as a solution of the equations (1.14)-(1.15).

While all the terms in (1.14)-(1.15) are important, the two most important are the potential u and the constant term c̄. Most of the paper is concerned with the distribution of the values of u over Ω_{ε}, and its relation to the behavior of the curvature blow-up of g_{ε}. It will be seen in §3, c.f. Theorem 3.5, that c in (1.4) or c̄ in (1.15) gives some important global control on the value distribution of u. Observe that c̄ is the only global term in the equations (1.14)-(1.15). Lemma 1.7 below relates the two terms of c in (1.4) to the condition that M is σ-tame.

Let ρ(x) = ρ_{ε}(x) be the L^{2} curvature radius of (Ω_{ε}, g_{ε}) at x; ρ(x) is the largest radius such that, for all geodesic balls B_{y}(s) ⊂ B_{x}(ρ(x)) ⊂ (Ω_{ε}, g_{ε}),

(1.16)  \frac{s^{4}}{volB_{y}(s)} ∫_{B_{y}(s)} |r|^{2} ≤ c_{o},

where c_{o} is a fixed constant, c.f. [2,(2.1)], [4,§3]. The radius ρ(x) measures the degree of curvature concentration near x; ρ(x) is small if and only if the L^{2} average of the curvature of g_{ε} is large near x. The function ρ is a Lipschitz function, with |∇ρ| ≤ 1.

We recall the following result from [2,Rmk.7.3], (compare also with [3,Thm.3.3]), which will be used repeatedly throughout the paper.

Proposition 1.1. There are constants υ_{o} > 0 and ρ_{o} > ∞, independent of ε, such that if x ∈ U^{υ_{o}} = {x ∈ Ω_{ε} : u(x) ≥ 1 − υ_{o}}, then

(1.17)  ρ(x) ≥ ρ_{o} · min(t_{υ_{o}}(x), 1),

where t_{υ_{o}}(x) = dist(x, L_{υ_{o}}), and L_{υ_{o}} = {x ∈ Ω_{ε} : u(x) = 1 − υ_{o}} is the 1 − υ_{o} level of u. Further, analogous higher order estimates also hold, i.e, for any k ≥ 0,

(1.18)  |∇^{k}r|(x) ≤ c(k) · max(t_{υ_{o}}(x)^{−2−k}, 1), |∇^{k}du|(x) ≤ c(k) · max(t_{υ_{o}}(x)^{−1−k}, 1).

The constant 1 on the right in (1.17)-(1.18) arises from the scalar curvature of (Ω_{ε}, g_{ε}). The estimate (1.17) can be re-expressed scale-invariantly as

(1.19)  ρ(x) ≥ ρ_{o} · min(t_{υ_{o}}(x), ρ_{s}(x)),

where ρ_{s} is the L^{2} scalar curvature radius, defined as in (1.16) with s in place of r. Similarly, (1.18) holds when 1 is replaced by the appropriate power of ρ_{s}(x).

As throughout the previous papers, we use the L^{2} Cheeger-Gromov theory, c.f. [7,8], [4, §3] to understand the limiting behavior of the metrics (Ω_{ε}, g_{ε}) as ε → 0. Thus, suppose x_{ε} ∈ Ω_{ε} are base points such that

(1.20)  u(x_{ε}) ≥ u_{o},

for some u_{o} > 0, and

(1.21)  ρ(x_{ε}) ∼ 1,

i.e. ρ(x_{ε}) is bounded away from 0 and ∞ as ε = ε_{i} → 0. Then there two possible behaviors.

Convergence (Non-Collapse). Suppose vol_{g_{ε}}B(ρ(x_{ε})) ≥ ν_{o}, for some ν_{o} > 0. Let U_{j} ≡ U_{j,ε} ⊂ Ω_{ε} be the component of {q_{ε} ∈ Ω_{ε} : 2^{−j} ≤ ρ(q_{ε}) ≤ 2^{j}} containing x_{ε}, and choose a sequence j = j(ε) → ∞ as ε → 0. Then for any sequence ε = ε_{i} → 0, there is a subsequence such that (U_{j}, g_{ε}, x_{ε}) converges, (modulo diffeomorphisms), in the C^{α} topology, α < \frac{1}{2}, and weak L^{2,2} topology, uniformly on compact subsets, to a maximal limit (U, g_{o}, x). In particular, any compact domain in U embeds in U_{j}, for j sufficiently large, (in the subsequence).

Collapse. Suppose vol_{g_{ε}}B(ρ(x_{ε})) → 0 as ε = ε_{i} → 0. Then, for any given R < ∞ and ε sufficiently small, the domains U_{j}(R) = U_{j} ∩ B_{x_{ε}}(R) above are either Seifert-fibered spaces over a surface V, or are torus bundles over an interval. In both cases, the diameter of the fiber F_{y_{ε}} over y_{ε} ∈ U_{j}(R) satisfies diam_{g_{ε}}F_{y_{ε}} → 0 as ε → 0 and the inclusion map of the fiber in U_{j}(R) induces an injection on π_{1}. In particular, there are finite covers Ũ_{j}(R) of U_{j}(R), of degree d_{j} → ∞ as ε → 0, such that (Ũ_{j}(R), g_{ε}) does not collapse, and so has a C^{α} convergent subsequence as above. Letting R = R_{j} → ∞, j = j(ε) as ε → 0 gives rise to a maximal limit (Ũ, g_{o}, x). The limit has a free isometric S^{1} or S^{1} × S^{1} = T^{2} action, arising from the unwrapping of the collapse.

There is of course some freedom in the choice of coverings unwrapping the collapse. It will be assumed throughout the paper that such coverings are chosen so that the diameter of the limiting S^{1} orbit, or diameter and area of the limiting T^{2} orbit, equal 1 at the limit base point x.

Analogous to (1.16) define the volume radius ω(x) of (Ω_{ε}, g_{ε}) by

(1.22)  ω(x) = sup\{r : \frac{volB_{y}(s)}{s^{3}} ≥ ω_{o}, ∀B_{y}(s) ⊂ B_{x}(r)\},

where ω_{o} is a fixed (small) positive constant. Both ρ and ω scale as distances. The two possibilities convergence/collapse above correspond to the situations ω(x_{ε}) ≥ v_{o} · ρ(x_{ε}) for some v_{o} > 0, or ω(x_{ε}) << ρ(x_{ε}) respectively. A similar collapse result holds if B_{x_{ε}}(ρ(x_{ε})) is sufficiently collapsed, for any fixed ε, i.e. if ω(x_{ε}) ∼ μ_{o} · ρ(x_{ε}), for μ_{o} small. In this case, there are finite covers as above, of order about μ_{o}^{−1}, such that in the covering space, ω(x_{ε}) ∼ ρ(x_{ε}).

Although the collapse case corresponds to a degeneration of the sequence of metrics, this situation is in fact somewhat easier to handle than the convergence situation, since the collapse may be (locally) unwrapped to obtain convergence to a limit which has special properties, namely a free isometric S^{1} action. This extra structure is of importance and leads to simplifications not present in the convergence situation in general.

In both cases, one has in fact C^{∞} convergence to the limit within regions satisfying (1.20)-(1.21), due to regularity estimates as in (1.18), c.f. [2, Rmk.4.3]. It is elementary but important to note that the Euler-Lagrange equations (1.14)-(1.15) are invariant under passing to covering spaces, as is, (modulo bounded factors), the curvature radius ρ.

The metric boundary ∂U of U is the limiting locus where the curvature of (U_{j}, g_{ε}) blows-up. Thus p ∈ ∂U if and only if there is a Cauchy sequence p_{k} ∈ U, with p_{k} → p in the metric completion Ū of U and such that if q_{ε}(k) ∈ U_{j} converges to p_{k} as ε → 0 then ρ_{ε}(q_{ε}(k)) ≤ μ(k), where μ(k) → 0 as k → ∞. Thus, ∂U is typically, although not necessarily, singular; it is possible that U may extend through (parts of) ∂U to a larger smooth manifold.

Since each g_{ε} satisfies the Euler-Lagrange equations (1.14)-(1.15), the limit metric g_{o} (of a subsequence) satisfies the same equation with ε set to 0, i.e.

(1.23)  L^{∗}(u) + (\frac{1}{4}su + \frac{1}{12} \frac{σ(M)}{T})g = 0,
(1.24)  2Δu + \frac{1}{4}(su − \frac{σ(M)}{T}) = 0.

These are the Euler-Lagrange equations for S_{−}^{2} at the value σ(M). When σ(M) = 0, the equations (1.23)-(1.24) are the static vacuum Einstein equations with potential u:

(1.25)  L^{∗}u = 0, Δu = 0.

(Here one uses the fact from [2,(3.45)] that s → 0 wherever u ≥ 0).

The following result characterizes these solutions at points x_{ε} satisfying u(x_{ε}) → 1 as ε → 0. Proposition 1.2. Let x_{ε} be base points in (Ω_{ε}, g_{ε}) such that u(x_{ε}) → 1, and suppose (1.21) holds. Then any limit (U, g_{o}, x) constructed above is hyperbolic, with constant scalar curvature σ(M) if σ(M) < 0, or flat if σ(M) = 0. Proof: If σ(M) = 0, then the maximum principle applied to trace equation in (1.25) implies that u ≡ 1 in U, and hence the full equation L^{∗}u = 0 implies g_{o} is flat, c.f. (1.5).

The argument is essentially the same when σ(M) < 0. Thus, (1.9) implies that in the limit one has s = −u · |σ(M)| and so, setting ω = u − 1 ≤ 0, the trace equation (1.24) can be written as

(1.26)  2Δω − \frac{1}{4}|σ(M)|(u + 1)ω = 0.

Since ω(x) = 0 = max ω, the strong maximum principle, c.f. [9, Thm. 3.5], implies that ω ≡ 0 in U and hence from (1.23), the metric g_{o} is of constant negative curvature.

The situation where

(1.27)  ρ(x_{ε}) ≥ ρ_{o} > 0,

for all x_{ε} ∈ (Ω_{ε}, g_{ε}) as ε → 0, i.e. when the metrics g_{ε} do not degenerate anywhere in Ω_{ε}, is analysed in detail in [1]; in particular (1.27) holds as ε → 0 if and only if M is tame, i.e. (0.4) holds. As mentioned in §0, the Main Theorem has been proved in this case.

Observe that Proposition 1.1 shows that the metrics g_{ε} do not degenerate in the region U^{υ_{o}}, provided one stays a fixed distance away from the boundary L^{υ_{o}}. However, while the estimates (1.9)-(1.11) give some useful information on the structure of U^{υ_{o}}, for instance vol_{g_{ε}}U^{υ_{o}} → 1 as ε → 0 in case σ(M) < 0, it is possible apriori that the level L^{υ_{o}} is δ-dense in (Ω_{ε}, g_{ε}), with δ → 0 as ε → 0.

By far the most important issue is to understand the situation where

(1.28)  ρ(x_{ε}) → 0 as ε → 0,

for at least some x_{ε} in the region where u ≥ u_{o}, for some u_{o} > 0, so that the curvature blows up at or very near x_{ε}. We will not attempt to understand the global geometry of the degeneration in all of U^{u_{o}}, but it will be important to understand the local behavior of the degeneration, say in unit balls B_{x_{ε}}(1) ⊂ (Ω_{ε}, g_{ε}). Remark 1.3. To give some perspective, here is a simple model situation. Suppose there are base points x_{ε}, u(x_{ε}) → 1 and satisfying (1.28), but which are isolated in the sense that there exists d_{o} > 0 such that within B_{x_{ε}}(d_{o}), ρ(y_{ε}) ≥ dist_{g_{ε}}(y_{ε}, x_{ε}). Thus, if there is no collapse in B(d_{o}) = B_{x_{ε}}(d_{o}), and u → 1 in B(d_{o}/2), the metrics converge in B(d_{o}/2) to a constant curvature metric g_{o} outside x = lim x_{ε}, but the curvature blows-up at x_{ε}. The limit metric g_{o} extends smoothly through x, giving a constant curvature metric on a 3-ball B^{3}. In this situation, one has a natural 2-sphere in B(d_{o}/2) that surrounds x_{ε}, and it is easy to see, from the arguments in [1,§3], that this 2-sphere is essential, (at least on the ”inside”). Unfortunately, it seems very difficult to prove that base points x_{ε} with this good behavior necessarily exist.

In the situation where (1.28) holds, we always consider the blow-up metrics

(1.29)  g′_{ε} = ρ(x_{ε})^{−2} · g_{ε},

so that ρ′(x_{ε}) = 1, where ρ′ is the L^{2} curvature radius w.r.t. g′_{ε}. (The rescaling (1.28) is also used when ρ(x_{ε}) → ∞; this situation will arise only at the end of the paper, in §7.3).

In the rescaling (1.29), the equations (1.14)-(1.15) become

(1.30)  L^{∗}u = ᾱ∇Z^{2} + ϕ̄ · g,
(1.31)  2Δ(u − \frac{ᾱs}{12}) + \frac{1}{4}su = \frac{1}{2} ᾱ|z|^{2} − 3c̄′.

All quantities in (1.30)-(1.31) are w.r.t. the g′_{ε} metric. The constant ᾱ = ᾱ(ε, x_{ε}) is given by ᾱ = ε̄/ρ^{2}, ρ = ρ(x_{ε}), while c′ = ρ^{2}c. As emphasized in [2], the function u is scale-invariant, i.e. it is considered as a function, and not as the normalized scalar curvature in the scale g′_{ε}.

The following result from [2, Cor.5.2] implies there is an apriori maximal rate or scale at which the curvature can blow up on (Ω_{ε}, g_{ε}) as ε → 0. Proposition 1.4. There is a constant κ > 0, independent of M and (Ω_{ε}, g_{ε}), such that

(1.32)  ρ^{2}(x_{ε}) ≥ κ\frac{ε}{T} = κε̄,

for all x_{ε} ∈ Ω_{ε}. Equivalently, ᾱ ≤ κ^{−1}.

The convergence/collapse possibilities, as well as Proposition 1.2, hold for the sequence g′_{ε} exactly as in the case (1.21) holds; as before, the convergence is modulo diffeomorphisms but this will usually be ignored in the notation. The possible structure of blow-up limits, i.e. limits (N, g′, x) of the Riemannian manifolds (Ω_{ε}, g′_{ε}, x_{ε}) has been analysed in detail in [2]; specifically from [2, §4,§5], one has: Proposition 1.5. If u(x_{ε}) ≥ u_{o}, for some constant u_{o} > 0, then any blow-up limit (N, g′, x) of the metrics (Ω_{ε}, g_{ε}, x_{ε}) is of the following form:

(i) a complete non-flat solution of the Z_{c}^{2} equations, with uniformly bounded curvature, (ii) an incomplete non-flat solution of the static vacuum Einstein equations, (iii) an incomplete flat solution of the static vacuum Einstein equations.

The static vacuum equations are given by (1.25) while the Z_{c}^{2} equations with potential u are

(1.33)  L^{∗}u = ᾱ∇Z^{2},

(1.34)  2Δ(u − \frac{ᾱ}{12}s) = \frac{1}{2} ᾱ|z|^{2},

where ᾱ is a positive constant, (the limit of ε̄/ρ^{2}). These equations are obtained from (1.30)-(1.31) by replacing ε̄ = ε/T by ᾱ, replacing ϕ̄ and c̄ by 0 and setting su = 0. Thus, on a Z_{c}^{2} solution, the support of s and of u are disjoint in their interiors. Note that when ᾱ = 0, the equations (1.33)- (1.34) become the static vacuum equations (1.25). Further, to leading order, the Euler-Lagrange equations (1.14)-(1.15) are identical to the Z_{c}^{2} equations (1.33)-(1.34).

The three cases above are characterized as follows: Case (i) occurs exactly when ᾱ > 0, Case (ii) occurs exactly when ᾱ = 0 and u is non-constant on the limit, while Case (iii) occurs exactly when ᾱ = 0 and u = const. on the limit.

The limit (N, g′, x) is a Z_{c}^{2} solution whenever x_{ε} is a point of locally maximal curvature blow-up, i.e. ρ(x_{ε}) ≤ ρ(y_{ε}), for all y_{ε} ∈ B_{x_{ε}}(δ) ⊂ (Ω_{ε}, g_{ε}), for a fixed δ > 0. By [2, Thm.B], whenever M is not tame, i.e. ρ → 0 somewhere on (Ω_{ε}, g_{ε}) as ε → 0, there exist base points x_{ε} whose blow-up limit is a Z_{c}^{2} solution. The limit is a possibly flat static vacuum solution when the curvature blows up at a rate much slower than the maximal rate. Observe that if the limit (N, g′, x) based at x_{ε} is flat, then at any base points y_{ε} converging to points in ∂N, the curvature blows up much faster than at x_{ε}, i.e. ρ(y_{ε}) << ρ(x_{ε}).

Remark 1.6. Recall the Sphere conjecture from [1]: if (N, g′, y) is a complete Z_{c}^{2} solution arising as a blow-up limit of (Ω_{ε}, g_{ε}, y_{ε}), then (N, g′) has an asymptotically flat end. The Sphere conjecture implies Conjectures I-II.

Any asymptotically flat end of (N, g′) of course carries natural 2-spheres, namely the spheres S^{2} approximating the spheres S^{2}(R) of large radius in ℝ^{3}. Since the convergence to the limit is smooth, such spheres also lie in (Ω_{ε}, g′_{ε}), and hence also in M. Theorem C of [2] proves the Sphere conjecture in case the potential function u on (N, g′) is bounded away from 0 outside a compact set, and the level sets of u are compact. The essential remaining difficulty in proving the conjecture lies in dealing with the situation where the levels of u are non-compact.

In this paper, in place of attempting to locate natural 2-spheres in ends of a Z_{c}^{2} solution as in the Sphere conjecture above, we instead locate 2-spheres directly in flat blow-up limits, as in Case (iii) of Proposition 1.5. Of course these two issues may, (and in fact should be), closely related since if a Z_{c}^{2} solution has an asymptotically flat end, then by “blowing it down”, i.e. rescaling the metric by factors converging to 0, one obtains a limit which is flat, with an isolated singular point; the 2-sphere near ∞ thus becomes a standard 2-sphere, of size on the order of 1, surrounding the origin in ℝ^{3}. This blow-down of the limit Z_{c}^{2} solution can be obtained directly as a flat “blow-up” limit of (Ω_{ε}, g_{ε}) at base points x_{ε} nearby but distinct from y_{ε}. Note that such a 2-sphere in (Ω_{ε}, g_{ε}) is very small; it has size on the order of ρ(x_{ε}) → 0 and is being rescaled to approximately unit size in g′_{ε} = ρ(x_{ε})^{−2} · g_{ε}; compare with Remark 1.3.

The main concern of the paper is to locate such 2-spheres by carefully analysing the structure of level sets of the potential u, i.e. studying the value distribution theory of u on (Ω_{ε}, g_{ε}). In effect, the main point is to show that there exist base points x_{ε} satisfying u(x_{ε}) → 1 and (1.28) such that in rescalings (1.29) converging to a flat limit, the level sets of u have at least one uniformly compact and separated component C, c.f. Remark 2.2. It is then not difficult to show that such components are surrounded in a natural way by 2-spheres.

We conclude this section with the following elementary consequence of the σ-tame condition.

Lemma 1.7. Suppose M is σ-tame, so that there exist constants k < ∞ and K < ∞ such that, on the space of unit volume metrics on M,

(1.35)  lim inf_{σ̃→0} σ^{k}Z^{2} ≤ K,

where σ̃ is as in (0.5). Then there is a sequence ε = ε_{i} → 0, such that on any minimizing pair (Ω_{ε}, g_{ε}), ε = ε_{i},

(1.36)  εZ^{2} ≤ (1 + K)ε^{1/(k+1)}, and σ̃ ≤ (1 + K)ε^{1/(k+1)}.

Proof: For any given ε the metric g_{ε} is a minimizer of I_{ε}^{−}, so that

(1.37)  εZ^{2}(g_{ε}) + σ̃(g_{ε}) ≤ εZ^{2}(g′) + σ̃(g′),

for any unit volume metric g′ on M. By (1.35), there is a sequence of metrics g_{i} on M with σ̃_{i} = σ̃(g_{i}) → 0 and

Z^{2}(g_{i}) ≤ Kσ_{i}^{−k}.

Substituting this in (1.37) gives

εZ^{2}(g_{ε}) + σ̃(g_{ε}) ≤ εKσ_{i}^{−k} + σ̃_{i}.

Now this estimate holds for any ε. Choose ε = ε_{i} by setting

ε_{i} = σ_{i}^{k+1},

so that ε_{i} → 0 as i → ∞. This gives

ε_{i}Z^{2}(g_{ε_{i}}) + σ̃(g_{ε_{i}}) ≤ Kε_{i} · ε_{i}^{−k/(k+1)} + ε_{i}^{1/(k+1)} ≤ ε_{i}^{1/(k+1)}(1 + K),

which gives the result.

The estimate (1.36) is the only consequence of the σ-tame condition that will be used in the paper. Observe that (1.36) gives an estimate for the constant term c̄ in (1.15); this will be used to obtain the global control in Theorem 3.5 and Theorem 5.4.

Finally, recall the following formulas for the change of curvature under conformal deformation of the metric, c.f. [6, 1.159ff]. Thus, if g̃ = f^{2} · g, then the Ricci curvature r̃ of g̃ is given by

(1.38)  r̃ = r − f^{−1}D^{2}f + 2(d ln f)^{2} − f^{−1}Δf · g.

Similarly, if g̃ = ψ^{4} · g, then

(1.39)  ψ^{5}s̃ = −8Δψ + sψ.

It should also be understood that throughout the paper, we always pass to subsequences where convergence issues are concerned, without explicitly mentioning this at each use. A sequence is said to sub-converge if a subsequence converges.

2. The Linearized Equations.

In this section, we study in detail the linearization of the metrics g_{ε} and their rescalings g′_{ε} in (1.29) about their constant curvature limits given by Proposition 1.2. This forms the foundation for the work to follow in later sections. Throughout the paper, we study the behavior of (Ω_{ε}, g_{ε}, x_{ε}) at base points x_{ε} satisfying

(2.1)  u_{ε}(x_{ε}) → 1,

on some arbitrary sequence ε = ε_{i} → 0. From §4 on, the sequence ε_{i} will no longer be arbitrary, but will be required to satisfy (1.36).

A general introductory discussion on such linearizations is given in §2.1, while in §2.2 the three possible forms of the linearized equations and the associated form of the metrics are analysed in detail. These are then summarized and unified in §2.3, c.f. Theorem 2.11.

2.1. The metrics g_{ε} satisfy the Euler-Lagrange equations (1.14)-(1.15). After a little algebra on the trace equation, these may be rewritten in the form

(2.2)  L^{∗}u = ε̄∇Z^{2} + ϕ̄ · g,

(2.3)  2Δ(u − \frac{ε̄s}{12}) = ψ_{ε} + \frac{1}{2} ε̄|z|^{2},

where

(2.4)  ψ_{ε} = \frac{1}{4}σT (u + 1)(u − 1) + d̄_{ε},

and d̄_{ε} is a constant, depending on the global geometry of (Ω_{ε}, g_{ε}), given by

(2.5)  d̄_{ε} = \frac{1}{4}σT (1 − T^{−2}) − \frac{1}{2} ε̄Z^{2}.

All quantities in (2.2)-(2.5) are w.r.t. (Ω_{ε}, g_{ε}) and ε̄Z^{2} ≡ ε̄Z^{2}(g_{ε}). Observe that (1.8) implies that d̄_{ε} ≤ 0, so that both summands of ψ_{ε} are non-positive. The term σT → |σ(M)| when σ(M) < 0, while σT → 0 when σ(M) = 0, again by (1.8). Thus, ψ_{ε} is also uniformly bounded below as ε → 0, i.e. there exists K < ∞ such that

(2.6)  −K ≤ ψ_{ε} ≤ 0.

Further, in regions where u → 1, one has

(2.7)  ψ_{ε} → 0.

Apriori, the L^{2} curvature radius ρ(x_{ε}) = ρ_{ε}(x_{ε}) may converge to 0, remain bounded away from 0 and ∞, or diverge to ∞, (as usual in subsequences), as ε → 0. Regardless of which of these behaviors occurs, we will always work in the scale g′_{ε} given by

(2.8)  g′_{ε} = ρ(x_{ε})^{−2} · g_{ε},

so that ρ′(x_{ε}) = 1. The metrics g′_{ε} then satisfy the rescaled Euler-Lagrange equations (1.30)-(1.31).

As discussed in §1, one may then pass, (in subsequences), to a maximal connected (blow-up) limit (F, g′_{o}, x_{o}) of the metrics (Ω_{ε}, g′_{ε}) based at {x_{ε}}, passing to sufficiently large finite covers if the sequence collapses at x_{ε}. As noted following Proposition 1.4, the assumption (2.1) and Proposition 1.2, (in the scale (2.8)), imply that the maximal limit (F, g′_{o}, x_{o}) at x_{o} = lim x_{ε} is a constant curvature manifold, i.e. either flat or hyperbolic, with limit potential u ≡ 1, provided ρ(x_{ε}) ≤ R_{o}, for some R_{o} < ∞. If ρ(x_{ε}) → ∞, so that g_{ε} is being ’blown-down’, this remains true but does not follow directly from Proposition 1.2; since the proof is out of place here, it is given in Appendix A. (The very special situation when ρ(x_{ε}) → ∞ will only arise in §7.3 and so may be ignored until then). Note also that F so defined is not apriori maximal w.r.t. a smooth flat or hyperbolic structure, i.e. F may possibly extend to a larger smooth flat or hyperbolic manifold, (c.f. the discussion preceding (1.23)).

It is of course apriori possible that the metrics (Ω_{ε}, g_{ε}) do not degenerate at all, i.e. ρ(x_{ε}) ≥ ρ_{o} > 0 whenever u(x_{ε}) is sufficiently close to 1. The question of whether the metrics g_{ε} degenerate in this region plays an important role in §4, but is not used until then. The main case of interest is when the metrics degenerate, so that ρ(x_{ε}) → 0.

To study the linearization of the metrics g′_{ε} at the limit constant curvature metric (F, g′_{o}), write

(2.9)  g′_{ε} = g′_{o} + δ · h + o(δ),

where the parameter δ = δ(ε) → 0 as ε = ε_{i} → 0 and h is a symmetric bilinear form, formally corresponding to \frac{dg′_{ε}}{dδ} |_{δ=0}. The choices of the parameter δ and the form h of course depend on each other. While the sequence {g′_{ε}}, ε = ε_{i}, converges smoothly to the limit g′_{o}, it may not converge smoothly with respect to ε, i.e. dg′_{ε}/dε may not exist at ε = 0, so that one cannot necessarily choose δ = ε. (Of course, this is related to the fact that the choice of base points x_{ε} is far from unique).

The natural value for the parameter δ is the local size of the curvature near the base point x_{ε}, which gives a measure as to how far away g′_{ε} is from the limit metric g′_{o}. For reasons which will only become clear later, we use for this the trace-free Ricci curvature z in place of the full Ricci curvature r. Thus, set

(2.10)  δ = δ_{z} = (∮_{B′_{xε}( \frac{1}{2} )} |z′_{ε}|^{2}dV_{g′_{ε}})^{1/2} = (ρ^{4}(x_{ε}) ∮_{B_{xε}( \frac{1}{2} ρ(x_{ε}))} |z_{ε}|^{2}dV_{g_{ε}})^{1/2},

where ∮ denotes the average value, i.e. the integral divided by the volume of the domain. Observe that the right side of (2.10) is scale invariant, so that δ is scale invariant. The quantity δ depends only on the isometry class of g_{ε} and so is independent of diffeomorphisms ’reparametrizing’ g_{ε}. Clearly δ = δ(x_{ε}) depends strongly on the choice of the base point. The factor \frac{1}{2} in (2.10) could be replaced by any other fixed constant < 1. The quantities δ_{r} and δ_{s} are defined in the same way as δ, by replacing z by the Ricci curvature r or scalar curvature s respectively.

Note that from the definition of ρ and δ,

δ ≤ 1

always. Further, since the convergence to the limit is smooth in (B_{x}(1), g′_{o}) and the limit g′_{o} is of constant curvature since (2.1) holds, one has

(2.11)  δ(x_{ε}) → 0, as ε → 0.

On the other hand, only x_{ε} such that δ(x_{ε}) > 0 for ε sufficiently small need to be considered. For if δ(x_{ε}) = 0 on some sequence ε = ε_{i} → 0, then g_{ε} is of constant curvature, (hyperbolic or flat), and u = const, on B_{x_{ε}}(ρ(x_{ε})). Since g_{ε} is real-analytic in the region where u > 0, g_{ε} is of constant curvature on the full component C_{ε} of Ω_{ε} containing x_{ε}. In this case, the metrics (C_{ε}, g_{ε}) = (C, g_{o}) are constant, i.e. independent of ε, for all ε, and no further analysis is necessary, c.f. also §1.

There are three basic cases for the behavior of δ at base points x_{ε}, as ε = ε_{i} → 0:

(i) δ >> ρ^{2},
(2.12)  (ii) δ ∼ ρ^{2},
(iii) δ << ρ^{2}.

By definition (2.10), δ/ρ^{2} is just the L^{2} average of z over B(\frac{1}{2}ρ). Observe also that s_{g′_{ε}} = ρ^{2}s_{g_{ε}} = O(ρ^{2}), so that in all cases

(2.13)  δ_{s} ≤ c · ρ^{2}.

It is obvious from (2.11) that ρ(x_{ε}) → 0 in Cases (i) and (ii). In Case (iii), either of the behaviors ρ(x_{ε}) → 0 or ρ(x_{ε}) ≥ ρ_{o} > 0 is possible. Thus, in Cases (i) and (ii), the maximal blow-up limit (F, g′_{o}, x) based at {x_{ε}} with x = lim x_{ε}, is flat, while in Case (iii), it is either flat or hyperbolic.

For δ as in (2.10), the local L^{2} norm of h in (2.9) is on the order of 1 in the Cases (i) and (ii) above. More precisely, define

(2.14)  h_{ε} = \frac{g′_{ε} − g′_{o}}{δ} .

Then

(2.15)  ∮_{B_{xε}( \frac{1}{2} )} |h_{ε}|^{2}dV_{g′_{ε}} ∼ 1 as δ → 0.

This is because the L^{2} norm of the curvature r′_{ε} of g′_{ε} is bounded by δ in (B′_{x_{ε}}(\frac{1}{2}), g′_{ε}) and the components of g′_{ε} − g′_{o} in a harmonic coordinate chart on these balls are bounded in the L^{2,2} topology by the L^{2} norm of the curvature. Moreover, since g′_{ε} satisfies the elliptic system (1.30)- (1.31), the covariant derivatives ∇^{k}z_{g′_{ε}}/δ, k ≥ 0, are uniformly bounded in L^{2}, while α∇^{k}z_{g′_{ε}}/δ → 0 as δ → 0, both on compact subsets of (F, g′_{o}). The proof of this (technical) result is given in Appendix B.

The estimate (2.15) may not hold in Case (iii), since one might have δ_{z} << δ_{s}, so that s′_{g_{ε}}/δ may be unbounded as δ → 0. This case must then be treated somewhat more carefully.

In analogy to (2.9), let

(2.16)  u_{ε} = 1 + ν · δ + o(δ),

where ν is (formally) du_{ε}/dδ at δ = 0. More precisely, as in (2.14), set

(2.17)  ν_{δ} = \frac{u_{ε} − 1}{δ},

and let ν = lim ν_{δ} if the limit exists. Special care must be taken here, since apriori ν might be infinite.

We complete this subsection with the following characterization of the boundary ∂F.

Proposition 2.1. For any maximal limit (F, g′_{o}, x) with x_{ε} → x satisfying (2.1), the boundary ∂F is the Gromov-Hausdorff limit of the level set L^{υ_{o}} from (1.17).

Proof: Recall that ∂F is formed by limits of sequences where ρ′ → 0. By the scale-invariant estimate (1.19), ρ′ → 0 implies t′_{υ_{o}} → 0, where t′_{υ_{o}} = dist_{g′_{ε}}(L^{υ_{o}}, ·). Hence ∂F is contained in the Gromov-Hausdorff limit of L^{υ_{o}}, c.f. [10,Ch.3], or [14,Ch.10] for discussion of Gromov-Hausdorff limits.

To prove the converse, it suffices to prove that the converse of Proposition 1.1 also holds, i.e. there is a constant R_{o} < ∞, independent of ε, such that

(2.18)  ρ(x_{ε}) ≤ R_{o} · t_{υ_{o}}(x_{ε}).

To see this, the estimate (2.18) is scale-invariant, and so it suffices to prove it in the scale g′_{ε} where ρ′(x_{ε}) = 1. The metrics (Ω_{ε}, g′_{ε}, x_{ε}) converge smoothly to the maximal limit (F, g′_{o}, x), and the potential u converges smoothly on compact subsets of F to the limit function u_{o}. Proposition 1.2, or more precisely its proof, shows that u_{o} ≡ 1. It follows that the level set L^{υ_{o}} cannot converge to any interior region of F, and hence t′_{υ_{o}}(x_{ε}) is at least 1 in the limit, which gives (2.18).

Note that for any flat manifold (F, g′_{o}), the curvature radius ρ′ satisfies

(2.19)  ρ′ = t′,

where t′(x) = dist_{F} (x, ∂F); in (1.16) the balls B_{x}(ρ′(x)) are required to be contained in F. Similarly, if the limit is hyperbolic, i.e. of constant negative curvature, then ρ′ ≤ t′. Note that equality, i.e. (2.19), holds when (s′)^{2} ≤ 3c_{o}, where s′ is the (constant) scalar curvature of g′_{o} and c_{o} is as in (1.16).

The structure of ∂F plays an important role throughout the paper. The metric closure F̄ = F ∪ ∂F is a length space, c.f. [10, Ch.1], and is viewed as a singular flat (or hyperbolic) manifold. In situations where F is embedded in ℝ^{3}, (or a quotient of ℝ^{3}), F̄ is then just the closure of a domain in ℝ^{3}, (or the quotient).

Some of the more technical issues in §3 and §4 arise because unit balls in F̄ are not necessarily compact. Consider for instance F of the form of a product F = ℝ × V, where V is the universal cover of ℝ^{2} \ {0}; V may naturally be identified with the upper half plane. In this setting, the topological boundary of ℝ × V is ℝ^{2}, while the metric boundary is a line, i.e. ℝ × {0}. Hence, ∂F, or even better ∂V = {0}, acts as a ”wormhole”, in that points of V that are far apart, when measured by the length of curves a bounded distance away from ∂V, may be close when the curves are allowed to be arbitrarily close to or within ∂V. On the other hand, it will be seen later following Proposition 4.8 that this is essentially the only way that ∂F can act as a wormhole.

Remark 2.2. Much of the work of the paper through §5 can be characterized as follows: find a flat blow-up limit (F, g′, y) of (Ω_{ε}, g_{ε}) such that ∂F is compact with F non-compact, or at least there is a ball B_{x}(R) of finite radius R with center x ∈ F̄, and a constant d_{o} > 0 such that

(2.20)  B_{x}(R) ∩ ∂F ≠ ∅, but A_{x}(R, (1 + d_{o})R) ∩ ∂F = ∅,

where A_{x}(r, s) is the geodesic annulus about x of radii r, s. Thus ∂F has a compact region, separated by a definite amount from any other part of ∂F. The relation (2.20) is established in Theorem 5.4. It is in such gap regions that geometrically natural 2-spheres in (Ω_{ε}, g_{ε}) will be located.

2.2. The main results of this subsection, (Propositions 2.3-2.4 and Corollary 2.8), describe the form of the linearizations of g′_{ε} and u at their limits (g′_{o}, 1). For clarity, the results are separated into three cases according to the three possibilities in (2.12). These three distinct situations will then be unified in §2.3, c.f. Theorem 2.11. An analysis similar to the analysis here was previously carried out in [1, §4], and also in [2, §7] in the context of Z_{c}^{2} solutions; this may serve as an introduction to the discussion below.

The first, and most important case, is the following. Proposition 2.3. Suppose the base points x_{ε} satisfy (2.1) and suppose that

(2.21)  δ >> ρ^{2}(x_{ε}), as ε = ε_{i} → 0.

Then there is a sequence of affine functions a_{δ} such that ν_{δ} − a_{δ} subconverges to a smooth limit function ν on F. The linearizations h = lim_{ε→0} h_{ε} and ν are a non-trivial solution of the linearized static vacuum Einstein equations

(2.22)  r′ = \frac{dr}{dδ} |_{δ=0} = D^{2}ν, Δν = 0,

at the flat metric g′_{o}. Thus, the limit function ν is a non-affine, locally bounded and smooth harmonic function on (F, g′_{o}).

To 1^{st} order in δ, the metric g′_{ε} is conformally flat. In fact, for ε small, and on any compact domain K ⊂ F, the metric g′_{ε} has the expansion, modulo diffeomorphisms,

(2.23)  g′_{ε} = (1 − 2νδ)(g′_{o} + δχ) + o(δ),

where δ → 0 as ε → 0, |o(δ)| << δ, and χ is an infinitesimal holonomy deformation, i.e. g′_{o} + δχ is a curve (in δ) of flat metrics on F. Proof: The metrics g′_{ε} satisfy the Euler-Lagrange equations (2.2)-(2.3) in this scale, i.e.

(2.24)  L^{∗}u = ᾱ∇Z^{2} + (−\frac{1}{4}su + c̄_{ε}) · g
2Δ(1 + \frac{εσ}{12} )u = ψ′_{ε} + \frac{ᾱ}{2}|z|^{2}.

Here and below, the prime and subscript ε are usually omitted from the notation, and u > 0. The term ψ′_{ε} is given by ψ′_{ε} = ρ^{2} · ψ_{ε}, ρ = ρ(x_{ε}). To obtain the linearization, divide the equations (2.24) by δ, and consider the limit as δ → 0, (as always in subsequences).

By Proposition 1.5 and the discussion following it, ᾱ = ε/ρ^{2}(x_{ε}) → 0 as ε → 0. Consider first the linearization of the ᾱ|z|^{2} term, and write this in the form:

\frac{ᾱ|z|^{2}}{δ} = ᾱ|z|\frac{|z|}{δ}.

Proposition 1.1 implies that |z| → 0 in L^{∞} on any compact subset of F, while ᾱ|z|/δ is bounded, by the discussion following (2.15), c.f. also Appendix B. It follows that, as δ → 0,

(2.25)  \frac{ᾱ|z|^{2}}{δ} = o(1),

on compact subsets of F. The same reasoning, again from the discussion following (2.15), implies

(2.26)  (ᾱ∇Z^{2})/δ = o(1).

Next consider the remaining terms on the right in (2.24). By definition

\frac{ψ′_{ε}}{δ} = \frac{ρ^{2}}{δ} ψ_{ε}.

From (2.7), ψ_{ε} → 0 as ε → 0. Since, by assumption, δ >> ρ^{2}, this term tends to 0 uniformly on compact subsets of F. The same analysis holds for the terms su and c̄_{ε}; note that s = g_{g′_{ε}} = ρ^{2}s_{g_{ε}}, and similarly for the term c̄_{ε} in this scale.

Hence, these arguments imply that

(L^{∗}u)/δ = o(1) and (Δu/δ) = o(1),

in L^{∞} on compact subsets of F. In particular, by (1.5),

(2.27)  u\frac{r}{δ} = D^{2}ν_{δ} + o(1), Δν_{δ} = 0 + o(1).

The equations (2.27), with the o(1) and δ terms removed, are the static vacuum Einstein equations (1.25). Since u → 1, it follows that any limit (h, ν) of (h_{δ}, ν_{δ}) is a solution of the linearization of the static vacuum equations (2.22) at the flat limit g′_{o}, with u ≡ 1. To prove the existence of such limits, the regularity discussion following (2.15), (c.f. Appendix B), implies that r/δ converges, in a subsequence, to a limit r′ and further h_{ε} converges, modulo diffeomorphisms, to a limit h on (F, g′_{o}).

Thus, (2.27) implies that the functions ν_{δ} are bounded, modulo addition of functions in the kernel of D^{2}. Since Ker D^{2} consists of affine functions on F, there are affine functions a_{δ} such that ν_{δ} − a_{δ} converges to a limit harmonic function ν on (F, g′_{o}). This proves the first statement.

To prove the second statement, write

(2.28)  \frac{u − 1}{δ} = ν + a_{δ} + o(1),

and set υ = u − a_{δ} · δ = 1 + νδ + o(δ). Clearly a_{δ} · δ → 0 as δ → 0. Now consider the conformally equivalent metric

(2.29)  g̃_{ε} = υ^{2}g′_{ε},

A simple computation using (2.27) and (1.38) shows that

(2.30)  r̃ = 2(d ln υ)^{2} − \frac{2}{υ} Δυ · g − \frac{ᾱ}{υ}∇Z^{2},

where all metric quantities on the right in (2.30) are w.r.t. g = g′_{ε}. Hence,

\frac{r̃}{δ} = \frac{2(d ln υ)^{2}}{δ} − \frac{2Δυ}{υδ} · g − \frac{ᾱ∇Z^{2}}{υδ} .

By the arguments in (2.25)-(2.27), the last two terms on the right go to 0 as δ → 0. Similarly, since d ln υ → 0 as δ → 0, and (d ln υ)/δ = \frac{1}{υ} dν_{δ} + o(1) → dν as δ → 0, it follows that

(2.31)  \frac{r̃}{δ} → 0, as δ → 0.

Thus, to first order in δ, g̃_{ε} is flat and so g′_{ε} is conformally flat. More precisely, one may write

(2.32)  g̃_{ε} = g̃_{o} + δχ + o(δ) = g′_{o} + δχ + o(δ),

where χ is an infinitesimal flat variation of the metric g′_{o}. The variation χ is a sum of a trivial term, due to variations of g̃_{ε} by diffeomorphisms, and a possibly non-trivial term due to variations in the holonomy, i.e. periods, of (F, g′_{o}). Ignoring the contribution due to diffeomorphisms, it follows that

g′_{ε} = υ^{−2}(g′_{o} + δχ) + o(δ) = (1 − 2ν · δ)(g′_{o} + δχ) + o(δ),

which proves the result.

As noted above in the proof, the affine indeterminacy of ν arises from the fact that Ker D^{2} consists of affine functions.

Next we turn to the Case (ii) of (2.12). Proposition 2.4. Suppose the base points x_{ε} satisfy (2.1) and suppose that

(2.33)  δ ∼ ρ^{2}(x_{ε}), as ε → 0.

Then there exists a sequence of affine functions a_{δ} such that the functions ν_{δ} −a_{δ} converge to a limit function ν. The linearizations h_{ε} from (2.14) converge to a solution of the linearized S_{−}^{2} equations,

(2.34)  z′ = D^{2}ν, Δν = 0,

on (F, g_{o}). This solution is non-trivial in the sense that D^{2}ν ≠ 0. On compact subsets of F, the metric g′_{ε} has the expansion, modulo diffeomorphisms,

(2.35)  g′_{ε} = (1 − 2ν · δ)(g′_{−δ} + δχ) + o(δ),

where g′_{−δ} is the space form of constant curvature (σ(M)r_{o}/6)δ, r_{o} = lim_{ε→0}(ρ^{2}/δ)(x_{ε}) and χ is an infinitesimal holonomy deformation of (F, g′_{−δ}). Proof: Following the proof of Proposition 2.3, the estimates (2.25)-(2.26) remain valid here. Further, since ρ^{2}/δ is bounded and ψ_{ε} → 0 on F by (2.7), it follows from (2.24) as before that

(2.36)  Δν_{δ} = o(1).

Similarly, as before from the full equation in (2.24), one obtains

(2.37)  L^{∗}(\frac{u}{δ}) = \frac{σT}{3} \frac{ρ^{2}}{δ} · g′_{ε} + o(1),

using the fact that u → 1 and −s ∼ ρ^{2}σT in this scale. When σ(M) < 0, one does not obtain here a solution to the linearized static vacuum equations, since

(2.38)  \frac{s_{g′_{ε}}}{δ} = −\frac{ρ^{2}}{δ} uσT = \frac{ρ^{2}}{δ} σ(M) + o(1).

The equations (2.36) and (2.37) imply, in the limit δ → 0, the equations

(2.39)  z′ = D^{2}ν, Δν = 0.

These equations are the linearization of the equations for critical points of S_{−}^{2}, i.e. the linearized S_{−}^{2} equations.

Exactly the same arguments as in Proposition 2.3 shows that the limit h exists, and there exist affine functions a_{δ} such that ν_{δ} − a_{δ} converges to a harmonic function ν on (F, g′_{o}). Similarly, for g̃_{ε} as in (2.29), the arguments in (2.30)-(2.31) give here that

(2.40)  r̃_{ε}/δ = \frac{σ(M)}{3} · g′_{ε} + o(1),

and so

(2.41)  z̃_{ε}/δ = 0 + o(1).

Thus, to first order in δ, g̃_{ε} is of constant curvature, and hence g_{ε} to first order is again conformally flat. More precisely, the formula (1.39) for conformal deformation and the estimate (2.36) give

(2.42)  \frac{s̃}{δ} = \frac{s_{g′_{ε}}}{δ} υ^{−2} + o(1) = −\frac{ρ^{2}}{δ} σT (1 + o(1)).

Hence, to first order in δ, g̃_{ε} is locally isometric to the space form of constant curvature −δ(\frac{ρ^{2}}{δ} σT)/6. As always, by passing to subsequences, one may assume that ρ^{2}/δ → r_{o} > 0, while by (1.8) −σT → σ(M). Thus, to first order in δ, g̃_{ε} is locally isometric to the space form g′_{−δ} of constant curvature δr_{o}σ(M)/6. The remainder of the proof, in particular (2.35), then follows as in Proposition 2.3.

Next, we turn to the situation where δ << ρ^{2}, which is somewhat more complicated. Remark 2.5. Actually, the situation below where δ << ρ^{2} or the situation (2.33) where δ ∼ ρ^{2}, although logically needed for the work through §5, are needed only in a comparatively minor way, (in terms of the overall picture). Hence on a first reading, it is advisable to skip at this point to the summary Theorem 2.11, and then onto §3, returning to this case as needed.

(ii). Similarly, given Proposition A, the work to follow in §2.2 - §2.3 also holds when ρ(x_{ε}) → ∞ as ε → 0; this will be used however only in §7.3.

Some preliminary estimates are needed in this case before deriving the analogue of Propositions 2.3 and 2.4. First, return to the equations (2.24), at the blow-up scale g′_{ε}. The estimate (2.26) also holds here, i.e. on compact subsets of F,

(2.43)  \frac{ᾱ}{δ}∇Z^{2} → 0, as ε → 0,

c.f. again Appendix B for the details of the proof. Taking the trace-free part of (2.24) then gives

(2.44)  \frac{z}{δ} = D_{o}^{2}ν_{δ} + o(1),

where ν_{δ} = (u − 1)/δ as before and D_{o}^{2} is the trace-free Hessian.

Observe that the quantity z/δ in (2.44) is both bounded away from 0 and bounded away from ∞, in the blow-up scale. Thus it has a limit, in a subsequence, and the limit is not identically 0.

We return to the estimate (2.44), the trace-free analogue of (2.27) or (2.37) in Corollary 2.8 below, but first need to understand in detail the behavior of the trace equation in (2.24) in this setting.

Referring to ψ_{ε} in (2.4), let ψ_{δ} = ψ_{ε}/δ, so that ψ′_{δ} = ρ^{2} · ψ_{δ} in the g′_{ε} scale (2.8). Hence from (2.43), or more directly from the definition of δ, one has

(2.45)  2Δν_{δ} = ρ^{2}(x_{ε})(ψ_{δ} + o(1)) ≤ o(1).

Thus, in any limit, the potential is superharmonic. Of course, since the behavior of u is being examined near its maximal value, this is not unexpected.

The following result gives control on the size of the right side of (2.45) from below. Lemma 2.6. Let x_{ε} ∈ (Ω_{ε}, g_{ε}) be any base points satisfying (2.1) and suppose one of the following conditions holds: there are positive constants χ_{o} or ρ_{o} such that

(2.46)  |u − 1|(x_{ε}) ≥ χ_{o} · ε̄Z^{2}, or
(2.47)  ρ(x_{ε}) ≥ ρ_{o}.

Then there exists a constant K < ∞, depending only on χ_{o} or ρ_{o}, such that

(2.48)  ξ(x_{ε}) ≡ ρ^{2}(x_{ε})ψ_{δ}(x_{ε}) ≥ −K.

Proof: The proof is by contradiction, and so suppose

ξ(x_{ε}) = ρ^{2}(x_{ε})ψ_{δ}(x_{ε}) → −∞, as ε → 0.

Renormalize the equations (2.45) and (2.48) by dividing them by |ξ(x_{ε})|, so that

(2.49)  2Δν̄_{δ} = ξ̄+ o(1),

where ν̄_{δ} = ν_{δ}/|ξ(x_{ε})|, ξ̄ = ξ/|ξ(x_{ε})|, so that

(2.50)  ξ̄(x_{ε}) = −1.

It then follows from (2.44) and the uniform bound on (z_{g′_{ε}}/δ) that

(2.51)  D_{o}^{2}ν̄_{δ} = o(1),

on compact subsets of F. Consider first functions ϕ such that

(2.52)  D_{o}^{2}ϕ = 0,

on the limit (F, g′_{o}, x), g′_{o} = lim g′_{ε}. If ρ(x_{ε}) → 0, the limit F is flat, while if ρ(x_{ε}) ≥ ρ_{o} > 0, then the limit is either flat or hyperbolic. We treat first the case where (F, g′_{o}) is flat.

On a flat manifold, the only functions ϕ satisfying (2.52) locally are, modulo affine functions,

(2.53)  ϕ = b_{o} · t_{o}^{2},

where b_{o} is a constant and t_{o}(x) = dist(x, p_{o}), for some base point p_{o} ∈ F. It follows from (2.51)- (2.53) that ν̄_{δ} converges to a limit function ν̄_{∞} modulo addition of affine functions and functions ϕ of the form (2.53). By (2.51), the limit function ν̄_{∞} necessarily satisfies (2.52) and hence

Δν̄_{∞} = 6b_{o}.

The normalization (2.50) gives b_{o} = −1/12 and further ξ̄ = ξ̄_{ε} converges to the constant function −1 as ε → 0. Hence, provided y_{ε} → y ∈ F, from (2.4) we have

ξ̄(y_{ε}) = \frac{1}{|ψ(x_{ε})|}[\frac{1}{4}σT (u + 1)ν_{δ} + d̄_{δ}](y_{ε}) → −1,

where d̄_{δ} = d̄_{ε}/δ and d̄_{ε} is as in (2.5). Via (2.49), this implies that on compact subsets of F,

(2.54)  2Δν̄_{δ} = 2Δ\frac{ν_{δ}}{ρ^{2}(x_{ε})|ψ(x_{ε})|} = \frac{1}{|ψ(x_{ε})|}[\frac{1}{4}σT (u + 1)ν_{δ} + d̄_{δ}] → −1.

Now suppose first (2.46) holds, and consider the right side of (2.54). The term d̂_{δ} ≡ d̄_{δ}/|ψ(x_{ε})| is constant, and u → 1 uniformly. Hence the oscillation of ν̂_{δ} ≡ ν_{δ}/|ψ(x_{ε})| converges to 0 as δ → 0, i.e.

(2.55)  sup_{B} ν̂_{δ} − inf_{B} ν̂_{δ} → 0,

where B is any compact subset of F. The hypothesis (2.46) and the definition of d̄_{δ} imply that |ν_{δ}|(x_{ε}) = −ν_{δ}(x_{ε}) ≥ −χ_{o} · d̄_{δ} = χ_{o} · |d̄_{δ}| and so |ν̂_{δ}|(x_{ε}) ≥ χ_{o} · |d̂_{δ}|. Via (2.55), this now implies that

(2.56)  sup_{B} ν̂_{δ}/ inf_{B} ν̂_{δ} → 1,

and so of course sup_{B} ν̄_{δ}/ inf_{B} ν̄_{δ} → 1. In turn, this means that Δν̄_{δ} → 0 weakly, which contradicts (2.54). An essentially identical argument holds if (2.47) holds, since in this case, |ν̄| ≤ ρ_{o}^{−1}|ν̂|.

Next suppose the limit (F, g′_{o}) is hyperbolic, i.e. of constant negative curvature, so that in particular (2.47) automatically holds. The argument in this case is essentially the same as the flat case, with some minor modifications. Thus, since u(x_{ε}) → 1, the scalar curvature of g′_{ε} is approximately −σT ρ^{2}, ρ = ρ(x_{ε}), so that g′_{ε} approximates a metric of constant sectional curvature −κ^{2}, κ = ρ(σT/6)^{1/2}.

The arguments (2.49)-(2.52) hold as before. An elementary computation shows that the only solutions of (2.52) on the limit, (mod constants), are

(2.57)  ϕ = b_{o} · cosh κt_{o},

with t_{o} again the distance from some point on F. A simple computation then gives

(2.58)  Δϕ = 3κ^{2}ϕ.

On the other hand, arguing exactly as in (2.54), one has

2Δ\frac{ν̂_{δ}}{ρ^{2}} ∼ \frac{σT}{2} ν̂_{δ} + d̂_{δ},

or equivalently

(2.59)  Δν̄_{δ} ∼ \frac{σT}{4} ρ^{2}ν̄_{δ} + \frac{1}{2}d̂_{δ}.

Since ν̄_{δ} converges to a solution of (2.52), (mod constants), ν̄_{δ} approaches a solution of (2.58). However, since κ^{2} = ρ^{2}(σT/6), this contradicts (2.59), (since κ > 0 in the limit).

Lemma 2.6 leads to the following definition: Definition 2.7. Base points x_{ε} ∈ (Ω_{ε}, g_{ε}) are called allowable if u(x_{ε}) → 1 and |Δν_{δ}|(x_{ε}) remains uniformly bounded in the scale g′_{ε} = ρ(x_{ε})^{−2} · g_{ε}, as ε = ε_{i} → 0.

Thus, the results above imply that x_{ε} is allowable if either δ ≥ c · ρ^{2} at x_{ε}, as ε → 0, for some c > 0, or if δ << ρ^{2}, then either (2.46) or (2.47) holds. Observe that if x_{ε} is allowable, then both terms in ξ(x_{ε}), i.e. ρ^{2}d̄_{δ} and ρ^{2}ν_{δ}(x_{ε}), are bounded as δ → 0.

Note that x_{ε} allowable does not imply that the linearized scalar curvature, i.e. s_{g′_{ε}}/δ remains uniformly bounded at x_{ε} as ε → 0. On the other hand, by the statement following (2.44) and Lemma 2.6, the full Hessian |D^{2}ν_{δ}|(x_{ε}) does remain bounded on allowable sequences.

The discussion above proves the following partial analogue of Propositions 2.3 and 2.4. Corollary 2.8. Suppose the base points x_{ε} are allowable and suppose that

(2.60)  δ << ρ^{2}(x_{ε}), as ε → 0.

Then there exists a sequence of affine functions a_{δ} such that the functions ν_{δ} − a_{δ} converge to a non-affine limit function ν on the maximal limit (F, g′_{o}, x). The linearization of the metrics g′_{ε} at ε = 0 gives rise to a solution of the equations,

(2.61)  z′ = D_{o}^{2}ν, 2Δν = ξ ≤ 0,

where ξ = lim ρ^{2}(x_{ε})(\frac{1}{2}σT ν_{δ} + d̄_{δ}) on (F, g′_{o}). Proof: The first equation in (2.61) follows from the limit of (2.44). For the second equation in (2.61), as noted above the constant term ρ^{2}d̄_{δ} in ξ is bounded as ε → 0. Further, the coefficient ρ^{2}σT of ν_{δ} in ξ is also bounded, since it approximates the scalar curvature of g′_{ε}. Hence, elliptic regularity for the trace equation in (2.45) implies that ξ = ξ_{ε} is locally bounded on F and so (2.61) follows as the limit of (2.45). The proof that the functions ν_{δ} converge to ν modulo affine functions follows for the same reasons as before in Proposition 2.3.

The situation in Corollary 2.8 of course differs somewhat from that in Propositions 2.3 and 2.4, where the limit potential ν is harmonic. It is not to be expected that ξ = 0 in (2.61) in general in situations where δ << ρ^{2}. The form of the metric g′_{ε}, i.e. the analogue of (2.23) or (2.35) in this situation, is discussed below in Theorem 2.11. Remark 2.9. If the limit (F, g_{o}) in Corollary 2.8 is hyperbolic, i.e. of constant negative curvature, then the functions ν_{δ} converge modulo constants to the limit ν; this follows since hyperbolic manifolds do not carry, even locally, any non-constant affine functions.

Remark 2.10. Suppose the base points x_{ε} are non-allowable, in that both (2.46) and (2.47) fail. The implication that (2.55) implies (2.56) no longer holds, since one may have |ν̂_{δ}| << |d̂_{δ}|. Hence ξ = ψ′_{δ} may become unbounded as δ → 0. Observe that in this case the dominant term in ξ is ρ^{2}d̄_{δ}. Whether this dominant term ρ^{2}d̄_{δ} stays bounded or not depends on the relation of ρ^{2}/δ, which depends only on the local geometry at x_{ε}, with the global term ε̄Z^{2}. There are other, much more fundamental reasons to work only with allowable base points which arise in §4 and §5 below.

2.3. In this section, we complete and unify the three forms of the linearizations, according to the three cases of (2.12). To do this, given allowable base points x_{ε} ∈ Ω_{ε}, let g_{−κ} be the space form of constant curvature

(2.62)  −κ^{2} = −\frac{T σ_{g_{ε}}}{6} ρ^{2}(x_{ε}).

If δ(x_{ε}) >> ρ^{2}(x_{ε}) as ε → 0, so κ^{2} << δ, set

(2.63)  g′_{−κ} ≡ g′_{o},

where g′_{o} is the flat metric on F. If δ(x_{ε}) ∼ ρ^{2}(x_{ε}) or δ(x_{ε}) << ρ^{2}(x_{ε}), set

(2.64)  g′_{−κ} ≡ g_{−κ}.

Theorem 2.11. Let x_{ε} be any sequence of allowable base points. Then there is a sequence of affine functions a_{δ}, either 0 or diverging to −∞, such that ν_{δ} − a_{δ} sub-converges to a non-affine function ν, defined on the maximal constant curvature limit (F, g′_{o}, x), x = lim x_{ε} of (Ω_{ε}, g′_{ε}, x_{ε}). The linearization h = lim_{δ→0} h_{ε} from (2.14) exists and gives rise to a non-trivial solution of the equations

(2.65)  z′ = D_{o}^{2}ν, Δν = λν + a_{∞} ≤ 0,

where λ is a constant, either 0 or positive, and a_{∞} is a non-positive affine function, possibly constant. For ε sufficiently small, the metric g′_{ε} has the form, modulo diffeomorphisms,

(2.66)  g′_{ε} = (1 − 2νδ)(g′_{−κ} + δχ) + o(δ),

where χ is an infinitesimal holonomy deformation of (F, g′_{−κ}). Proof: The first equation in (2.65) has already been verified in (2.22), (2.34) and (2.61). As in (2.29), consider the conformally equivalent metric g̃_{ε} = υ^{2}g′_{ε}, for υ as following (2.28). The trace-free part of (1.38) implies

(2.67)  \frac{z̃_{ε}}{δ} = \frac{z_{ε}}{δ} − \frac{D_{o}^{2}υ_{δ}}{υ} + 2\frac{(dυ_{δ}dυ)_{o}}{υ^{2}} ,

where the subscript o denotes trace-free part and υ_{δ} = (υ − 1)/δ. Since υ_{δ} converges to the limit ν, dυ_{δ} is bounded, while dυ → 0 uniformly on compact subsets of F. Hence, by the first equation in (2.65), one obtains

(2.68)  \frac{z̃_{ε}}{δ} → 0, as ε → 0,

on F, so that g̃_{ε} is of constant curvature, and hence g′_{ε} is conformally flat, both to first order in δ. For the trace equation in (2.65), by (2.45) one has

(2.69)  Δν_{δ} = \frac{ρ^{2}}{8} σT (u + 1)ν_{δ} + \frac{ρ^{2}d̄_{δ}}{2} + o(1),

where d̄_{δ} = d̄_{ε}/δ. Recall that u → 1 uniformly on compact subsets of the limit F. Lemma 2.6 implies that both terms on the right side of (2.69) are uniformly bounded as ε → 0, while (2.5ff) implies both are non-positive. As in (2.28), ν_{δ} = ν + a_{δ} + o(1), where a_{δ} is either 0 or a divergent sequence of affine functions, so that (2.69) has the form

(2.70)  Δν_{δ} = \frac{ρ^{2}}{4} σT (ν + a_{δ}) + \frac{ρ^{2}d̄_{δ}}{2} + o(1).

Suppose first that there exists c_{o} > 0 such that, as ε → 0,

(2.71)  ρ^{2}σT ≥ c_{o},

Then the term a_{δ} must be uniformly bounded as ε → 0, and so it may be ignored, i.e. absorbed into ν. In this case, (2.70) limits to

(2.72)  Δν = lim(\frac{ρ^{2}}{4} σT)ν + lim(\frac{ρ^{2}d̄_{δ}}{2} ),

giving (2.65). On the other hand, if ρ^{2}σT → 0, then ρ^{2}σT ν → 0 also, so the limit is

(2.73)  Δν = lim(\frac{ρ^{2}}{4} σT a_{δ}) + lim(\frac{ρ^{2}d̄_{δ}}{2} ),

giving (2.65) with λ = 0.

Next we relate the scalar curvatures s̃_{ε} and s′_{ε}. A short calculation using (1.39) shows that

(2.74)  \frac{s̃_{ε}}{δ} = −4υ^{−3}Δν_{δ} + 2u^{−4}|dν_{δ}||dυ| + υ^{−2} \frac{s′_{ε}}{δ} ,

where again the right side of (2.74) is w.r.t the blow-up metric g′_{ε}. As noted above dν_{δ} → dν, and so is bounded, while dυ → 0 as ε → 0. Hence the second term in (2.74) tends to 0 as δ → 0. Further,

(2.75)  υ^{−2} \frac{s′_{ε}}{δ} = −υ^{−2} \frac{ρ^{2}}{δ} T σu,

so that (2.74) becomes

(2.76)  \frac{s̃_{ε}}{δ} = −4u^{−3}Δν − υ^{−2} \frac{ρ^{2}}{δ} T σu + o(1).

Referring to (2.72) and (2.73), ν · δ, a_{δ} · δ and d_{δ} · δ all converge to 0 as δ → 0. Hence the second term in (2.76) dominates the first, so that

(2.77)  \frac{s̃_{ε}}{δ} = −\frac{ρ^{2}}{δ} T σ(1 + o(1)).

In particular, one sees that s̃_{ε} is constant, to 1^{st} order in δ, which of course it must be by (2.68) and the Bianchi identity. The estimates (2.68) and (2.77) thus imply that g̃_{ε} is isometric, to first order in δ, to the space form g′_{−κ}, as in (2.67)-(2.68). The remainder of the proof follows as in Proposition 2.3.

We point out that if (2.71) holds, i.e. λ > 0 in (2.65), then the proof shows that ν_{δ} converges to ν, i.e. there is no constant or affine indeterminacy; in particular, ν ≤ 0 in this situation.

Observe that Theorem 2.11 implies that any allowable sequence {x_{ε}} gives rise to a non-trivial and uniformly controlled solution of the corresponding linearized equations, modulo addition of affine functions. This fact plays a crucial role throughout the rest of the paper.

In particular, at any allowable base point sequence {x_{ε}}, there is a constant κ_{o} > 0, independent of ε and x_{ε}, such that, for most y_{ε} ∈ B_{x_{ε}}(ρ(x_{ε})),

(2.78)  |dν_{δ(x_{ε})}|(y_{ε}) ≥ κ_{o}/ρ(x_{ε}).

Note that the estimate (2.78) is scale-invariant. There is a minor technical issue here in that (2.78) need not hold for all y_{ε} ∈ B_{x_{ε}}(ρ(x_{ε})); for instance, if y_{ε} is a critical point of u, then |dν_{δ(x_{ε})}|(y_{ε}) = 0. However, (2.78) does hold for generic y_{ε} ∈ B_{x_{ε}}(ρ(x_{ε})), since if not, the limit potential ν on the maximal limit (F, g′_{o}, x) would be constant, (modulo affine functions), contradicting Theorem 2.11. Thus, (2.78) holds for a large percentage of y_{ε} ∈ B_{x_{ε}}(ρ(x_{ε})), where κ_{o} depends only on the volume percentage. In particular, one can always slightly adjust the choice of x_{ε}, within ρ(x_{ε}), so that (2.78) holds at y_{ε} = x_{ε}. Such a choice of x_{ε}, (of course with u(x_{ε}) → 1), will be called generic, and it will always be assumed, (usually implicitly), in the work to follow that base points are generic.

The uniformity in Theorem 2.11 also implies that for generic base points there is a constant a_{o} > 0, depending only on the collapse ratio μ_{o} = ω(x_{ε})/ρ(x_{ε}) such that

(2.79)  a_{o} · ρ^{2}(x_{ε}) ≤ area(B_{x_{ε}}(ρ(x_{ε}) ∩ L)) ≤ a_{o}^{−1} · ρ^{2}(x_{ε}),

where L is the level set of ν_{δ} through x_{ε}.

Finally, it is worth pointing out that Theorem 2.11 is the reason for choosing δ = δ_{z} and not δ = δ_{r}; linearizations w.r.t. δ_{r} in the situation δ << ρ^{2} always lead to trivial solutions of the linearized equations, at least when σ(M) < 0.

3. Level Set Measures and Masses.

In this section, we describe natural measures and corresponding mass functions associated to the potential functions ν_{δ} and their limits ν from §2. These measures are supported on the level sets of u, and it is the distribution of these measures in relation to the geometry of the metrics g_{ε}, or g′_{ε}, which plays the central role in locating the 2-spheres in M.

First, recall that the choice of δ in (2.10) led to the constant and affine indeterminacy in the construction of the limit potential ν, at any given base point sequence x_{ε}, (as always satisfying (2.1)). At a later point in the paper in §5, it will be important to remove the affine indeterminacy of the potential ν_{δ}; however, until then this indeterminacy does not play a major role.

In analogy to the definition (2.10) of δ, define then

(3.1)  δ_{a} = δ_{a}(x_{ε}) = ∮_{B_{xε}( \frac{1}{2} )∩L_{ε}} |∇u|dA_{g′_{ε}} = ρ(x_{ε}) · ∮_{B_{xε}( \frac{1}{2} ρ(x_{ε}))∩L_{ε}} |∇u|dA_{g_{ε}},

where the subscript stands for affine, L_{ε} is the level set of u through the base point x_{ε} and ∮ is the average value, as before. The first integral is w.r.t g′_{ε} while the second is w.r.t. g_{ε}, so that δ_{a} is scale-invariant, as δ is. Theorem 2.11 implies that the potentials ν_{δ(x_{ε})}, at any allowable base point sequence x_{ε}, (sub)-converge, modulo affine functions, to non-trivial limit functions. As mentioned in connection with (2.78), it is also assumed henceforth that x_{ε} is generic. This implies immediately that

(3.2)  δ_{a} ≥ c_{1} · δ,

for a fixed numerical constant c_{1} > 0, independent of ε and the generic base point x_{ε}. The affine indeterminacy of ν_{δ} corresponds to the possibility that

(3.3)  δ_{a} >> δ.

The reason for preferring δ_{a} to δ, (at this stage), is that the potential function

(3.4)  ν_{δ_{a}(x_{ε})} = (u − 1)/δ_{a}(x_{ε}) = ν_{δ(x_{ε})} · (\frac{δ(x_{ε})}{δ_{a}(x_{ε})})

converges, modulo addition of constants, to a limit function ν = ν_{a}, so that the affine indeterminacy is reduced to a constant indeterminacy. When (3.3) occurs at x_{ε}, the potential ν_{δ_{a}} converges to a non-constant affine function ν_{a}, with |∇ν_{a}|(x) = 1, on the limit (F, g′_{o}, x_{ε}), while if

(3.5)  δ_{a} ≤ k_{o} · δ,

for some fixed constant k_{o} < ∞, (on base points x_{ε}), then the functions ν_{δ_{a}} and ν_{δ} have uniformly bounded ratios, and thus the limit functions ν and ν_{a} differ merely by a multiplicative factor. In particular, Theorem 2.11 holds also w.r.t. δ_{a} in place of δ, with the understanding that the limit function ν_{a} is affine in case (3.3) holds.

Note the following elementary result. Lemma 3.1. Suppose the base points x_{ε} ∈ (Ω_{ε}, g_{ε}) are allowable. Then there is a constant C_{o} < ∞, (independent of ε), such that

(3.6)  |du|(x_{ε}) ≤ C_{o} · \frac{δ_{a}(x_{ε})}{ρ(x_{ε})}.

Proof: To see this, observe that (3.6) is equivalent to the scale invariant estimate

ρ(x_{ε})|dν_{δ_{a}}|(x_{ε}) ≤ C_{o}.

Theorem 2.11 implies that the functions ν_{δ_{a}} (sub)-converge smoothly, modulo constants, to a limit function ν_{a} on the maximal limit (F, g′_{o}, x), x = lim x_{ε}, (unwrapping in the case of collapse as described in §1). By construction and elliptic regularity, the limit satisfies |dν_{a}|(x) ≤ C_{o}, with C_{o} independent of the base points x_{ε}. This gives (3.6), since the convergence to the limit is smooth.

For the same reasons, if y_{ε} are points with y_{ε} → y ∈ (F, g′_{o}), then there is a constant C_{1} < ∞, depending only on dist_{g′_{o}}(y, ∂F), and dist_{g′_{o}}(x, y) such that

(3.7)  |du|(y_{ε}) ≤ C_{1} · \frac{δ_{a}(x_{ε})}{ρ(x_{ε})}.

We now introduce Riesz-type measures and mass functions for the potentials ν_{δ_{a}} and limit potentials ν_{a} constructed above. Let L = L_{ε} be the level set of u through the base point x_{ε} and define

(3.8)  dμ = dμ_{δ_{a}}(L) = |∇ν_{δ_{a}}|dA_{g_{ε}},

where δ_{a} = δ_{a}(x_{ε}) and dA_{g_{ε}} is Lebesgue measure w.r.t. g_{ε} on the level L. This measure depends on the base point, both with regard to δ_{a} and the level L. Observe that dμ is unchanged when adding constants to ν_{δ_{a}}. For any compact set E ⊂ F, one has the associated mass

(3.9)  m(E) = ∫_{E∩L} |∇ν_{δ_{a}}|dA_{g_{ε}}.

In particular, for any q ∈ L, define the mass function m_{q} of dμ by

(3.10)  m_{q}(s) = ∫_{L∩B_{q}(s)} |∇ν_{δ_{a}}|dA_{g_{ε}}.

Similar measures and mass functions are defined on the other level sets L̄ of u as in (3.8)-(3.10), with dĀ the Lebesgue measure on L̄, and with the same potential ν_{δ_{a}}. These measures on distinct level sets can be related to each other by means of the divergence theorem applied to the trace equation (2.3).

The mass functions in (3.9)-(3.10) scale as distance. Thus, if g′_{ε} = ρ(x_{ε})^{−2} · g_{ε}, then m′(E) = ρ(x_{ε})^{−1}m(E) and m′_{q}(s) = ρ(x_{ε})^{−1}m_{q}(ρ(x_{ε}) · s).

When the blow-ups g′_{ε} (sub)-converge, to the limit (F, g′_{o}, x), the rescaled measure dμ′ and mass m′ converge to that of the limit potential ν_{a} satisfying (2.65). The same holds on each level set L̄_{ε} → L̄ ⊂ F. Note that in case Δν_{a} = 0, the limit measure dμ′ is the Riesz measure of the subharmonic function νL̄ = sup(ν_{a}, ν_{a}(L̄)) on F.

However, when the blow-ups g′_{ε} collapse at x_{ε}, the mass m′ as in (3.10) converges to 0. Thus, if the volume radius of x_{ε} is small compared with the curvature radius, i.e. ω(x_{ε}) ≤ μ_{o} · ρ(x_{ε}), for some μ_{o} > 0 small but fixed, unwrap the collapse by passing to large finite covers as described in §1, so that ω(x_{ε}) ∼ ρ(x_{ε}), and define the measure dμ̃ as in (3.8) in the cover. This is essentially equivalent to renormalizing the mass (3.9) as follows:

(3.11)  m̃L̄(E) = \frac{ρ^{2}(x_{ε})}{area_{g_{ε}}(L ∩ B_{x_{ε}}(\frac{1}{2}ρ(x_{ε})))} ∫_{E∩L̄} |∇ν_{δ_{a}}|dA_{g_{ε}},

where L is the level through x_{ε}. When the sequence based at x_{ε} is not μ_{o}-collapsed, then m̃(E) ∼ m(E), for sets E whose diameter is on the order of ρ(x_{ε}), c.f. (2.79). Observe again that the mass (3.11) scales as a distance. In particular, in the blow-up scale g′_{ε} = ρ(x_{ε})^{−2}g_{ε} attached to x_{ε}, one has, on L̄,

(3.12)  m′(E) = \frac{1}{area_{g′_{ε}}(L ∩ B′_{x_{ε}}(\frac{1}{2}))} ∫_{E∩L̄} |∇ν_{δ_{a}}|dA_{g′_{ε}}.

Note from the definitions (3.1), (3.10) and (3.12) that m′_{x_{ε}}(1/2) = 1, and correspondingly,

(3.13)  m̃_{x_{ε}}(\frac{1}{2}ρ(x_{ε})) = ρ(x_{ε}).

Each of these notions may be defined w.r.t. the ’curvature’ δ of (2.10) in place of the ’affine’ δ of (3.1). In situations where (3.5) holds, the corresponding measures and mass functions are uniformly bounded, in terms of k_{o}, with respect to each other. However, when (3.3) holds, the measures and mass functions w.r.t. δ become unbounded; this is why δ_{a} is used in place of δ, for the time being.

Next, we derive some estimates for the behavior of the limit potential ν_{a} on F. We begin with the following upper bound on ν_{a}. Lemma 3.2. Let x_{ε} be an allowable base point sequence satisfying (2.46). Then the potential function ν_{a} on the maximal limit (F, g′_{o}, x), normalized so that ν_{a}(x) = 0, is bounded above on compact subsets of F̄, i.e. for any compact set K ⊂ F̄,

(3.14)  ν_{a}(y) ≤ C,

for all y ∈ K, where C = C(K). The same estimate holds for the potentials ν_{δ_{a}(x_{ε})}, renormalized by additive constants, converging to ν_{a}. Proof: If δ_{a}(x_{ε}) >> δ(x_{ε}), i.e. (3.3) holds, then the limit function ν_{a} is affine and so (3.14) is immediate. Thus, one may assume that (3.5) holds and so replace δ_{a} by δ. The functions ν_{δ} thus sub-converge, modulo constants, to the limit function ν.

Recall from Theorem 2.11 that the potential ν satisfies the trace equation (2.65). If λ > 0, then as noted following Theorem 2.11, ν ≤ 0 and so again (3.14) follows. Hence, one may assume that (2.65) is given by

(3.15)  Δν = d̄_{o} ≤ 0,

where d̄_{o} = lim(ρ^{2}(x_{ε})/δ(x_{ε}))d̄_{ε} > −∞. Note that d̄_{ε} → 0, while the ratio ρ^{2}(x_{ε})/δ(x_{ε}) may remain bounded, or go to ∞; the product remains bounded as ε → 0, since x_{ε} is allowable.

Since ν is smooth on F, ν can diverge to +∞ on a compact set K in F̄ only on approach to ∂F. Let p be any fixed point in F ∩ K and let p_{ε} ∈ U^{υ_{o}} be a sequence converging to p. Let ν̂_{δ(p_{ε})} = ν_{δ(p_{ε})} − ν_{δ(p_{ε})}(p_{ε}) be the normalized potentials converging to the limit (normalized) potential ν_{p}. Then it suffices to prove that ν̂_{δ(p_{ε})} is uniformly bounded above on the component D′ = D′_{p_{ε}} of B′_{p_{ε}}(\frac{10}{9}) ∩ U^{υ_{o}} containing p_{ε}; here \frac{10}{9} may be replaced by 1 + μ, for any fixed μ > 0, and the ball B′ is the ball in the metric g′_{ε} = ρ(p_{ε})^{−2} · g_{ε} based at p_{ε}.

Suppose this is not the case, so that there exist q_{ε} ∈ D′_{p_{ε}}, for some sequence p_{ε} → p, such that

(3.16)  ν̂_{δ(p_{ε})}(q_{ε}) → ∞.

Then we claim that

(3.17)  \frac{ρ^{2}(q_{ε})}{δ(q_{ε})} << \frac{ρ^{2}(p_{ε})}{δ(p_{ε})}.

To see this, recall that δ(p_{ε}) is the L^{2} average of z on the rescaled ball B′_{p_{ε}}(\frac{1}{2}) based at p_{ε}. The relation (3.16) implies that |D_{o}^{2}ν_{δ(p_{ε})}| >> 1, near q_{ε} and hence by the relation (2.44), z/δ(p_{ε}) >> 1 near q_{ε}, in the scale based at p_{ε}. Transferring this estimate to the scale based at q_{ε} gives (3.17).

It follows from (3.17) and the definition of d̄_{o} in (3.15) that on the limit (F, g′_{o}, q), based at q = lim q_{ε}, g′_{ε} = ρ(q_{ε})^{−2} · g_{ε}, that the limit potential ν_{q} = lim ν̂_{δ(q_{ε})} is harmonic, (since d̄_{ε} is independent of base point). In particular, ν_{q} has no local maxima. Hence the approximating potentials ν̂_{δ(q_{ε})}, restricted to the rescaled balls B′_{q_{ε}}(1), have a maximal value a definite amount larger than the value at q_{ε}. Note that this gives a contradiction if for instance the component of ∂F containing q is compact.

Now observe this argument holds with respect to any allowable base point sequence y_{ε}. Thus, given x_{ε}, let L_{ε} be the u-level through x_{ε} and let R_{ε} be the component of the region bounded by L^{υ_{o}} and L_{ε} containing x_{ε}. For each y_{ε} ∈ L_{ε} ∩ ∂R_{ε}, consider the function ν̂_{δ(y_{ε})} on D′_{y_{ε}} as above. The potential ν̂_{δ(y_{ε})} has a maximal value on the closure D̄′_{y_{ε}} of D′_{y_{ε}}, and so one thus obtains a function ϕ_{ε} on L_{ε}. The function ϕ_{ε} itself has a maximal value, achieved, (or arbitrarily close to being achieved, since Ω_{ε} is complete), at say y_{ε}. Let q_{ε} ∈ D̄′_{y_{ε}} be a point realizing the maximum of ν̂_{δ(y_{ε})}. It follows that (3.16) and (3.17) hold, with y_{ε} in place of p_{ε}. As before, this gives a contradiction if q_{ε} is in the interior D′_{y_{ε}}. Suppose instead that q_{ε} ∈ ∂D′_{y_{ε}}. Observe that (3.17) gives

(3.18)  δ(q_{ε}) >> ρ′(q_{ε})^{2}δ(y_{ε}).

Hence, if y_{ε} is moved slightly to y′_{ε} so that q′_{ε} ∈ D′_{y_{ε}}, then (3.18) is still preserved. It follows that the limit harmonic potential ν_{q} = lim ν̂_{δ(q_{ε})} as above has a local maximum at q_{ε}, giving again a contradiction. This completes the proof of Lemma 3.2.

By construction, on the maximal limit (F, g′_{o}, x), one has B′_{x}(1) ⊂ F, c.f. also (2.19). Assume for the moment that F is flat, so that ∂F ∩ ∂B′_{x}(1) ≠ ∅. If B′_{x}(1) is simply connected, then the developing map D of the flat structure on F, c.f. [17], gives an isometric embedding of B′_{x}(1) onto the unit ball B_{0}(1) ⊂ ℝ^{3}, where D is normalized so that D(x) = 0. For any fixed R_{o} > 1, let

(3.19)  D(R_{o}) ⊂ B′_{x}(R_{o}) ⊂ F̄

be a maximal domain in B′_{x}(R_{o}), containing B′_{x}(1), such that D is an embedding of D(R_{o}) onto its image in ℝ^{3}. Such a maximal domain is not unique, (as for instance cuts in a Riemann surface are not unique), but the results below are independent of the particular choice of maximal domain.

A similar definition holds if B′_{x}(1) is not simply connected. Thus, if Γ = π_{1}(B′_{x}(1)), then Γ injects in π_{1}(F) since the unit ball B̃′ ; x̃(1) is convex in the universal cover F̃ of F. The group Γ is either ℤ or ℤ ⊕ ℤ, acting by translation or twist on F̃ or ℝ^{3}. The developing map descends to a map D : F̃/Γ → ℝ^{3}/Γ, and D induces an embedding of B′_{x}(1) onto the unit ball in ℝ^{3}/Γ. Thus, as in (3.19), define D(R_{o}) to be a maximal domain on which D is an embedding onto its image in ℝ^{3}/Γ.

The reason for constructing the domains D(R_{o}) is that they are natural compact analogues of the geodesic balls B′_{x}(R_{o}); as noted preceding Remark 2.2, the balls B′_{x}(R_{o}) may not be compact, (or have compact closure), in general. The same definition holds if the limit F is hyperbolic, using the developing map D of the hyperbolic structure. Domains D_{y}(R_{o}), centered at a point y ∈ F, may be defined in the same way.

The next two results give (local) upper bounds on the mass of ν_{a}; these will be important in §4, (in Lemma 4.6). Proposition 3.3. For (F, g′_{o}, x, ν_{a}) as in Lemma 3.2, let L̄ be any level set of the potential ν_{a}. Then there is a constant M_{o}, depending only on R_{o}, such that

(3.20)  m(D(R_{o}) ∩ L̄) ≤ M_{o},

where m is the mass of the limit potential ν_{a} on the level set L̄.

Proof: Assume the limit F is flat; the proof in the hyperbolic case is the same. Thus, the domain D(R_{o}) may be viewed as a domain in ℝ^{3}, (or ℝ^{3}/Γ).

As noted following Theorem 2.11, the function ν_{a} is uniformly controlled in B′_{x}(\frac{1}{2}). The function ν_{a} satisfies the elliptic equation (2.65) and is bounded above near ∂F by Lemma 3.2. Standard properties of positive (or negative) solutions of elliptic equations in domains in ℝ^{3} imply that ν_{a} is locally uniformly controlled, in that it satisfies a uniform local Harnack inequality and gradient estimates off ∂F, c.f. [9, Ch.8]. This implies that (3.20) holds for regions of L̄ ∩ D(R_{o}) not too close to ∂F.

In general, let H̄ be a compact region within D(R_{o}) ∩ L̄ and assume w.l.o.g. that ν_{a}(L̄) < ν_{a}(L), where L is the level through x. The flow lines of ∇ν_{a} starting in H̄ then tend to a local maximum of ν_{a}. (The small set of flow lines ending in critical points of ν_{a} may be ignored in the following). For those flow lines γ for which dist(γ(s), ∂F) ≥ t_{o}, for some small t_{o} > 0, the arguments above give a uniform upper bound on |∇ν_{a}|(γ(s)). Suppose instead a flow line γ stays near ∂F. Then since ν_{a} is bounded above, |∇ν_{a}|(γ(s)) is small, sufficiently far out along the flow. Hence, the region R consisting of the union of these flow lines which terminate where |∇ν_{a}| ≤ C, for some fixed constant C, is a compact set in F. Then (3.20) follows by applying the divergence theorem to the equation (2.65) over the domain K.

Proposition 3.3 also holds on the sequence (Ω_{ε}, g′_{ε}, x_{ε}) converging to (F, g′_{o}, x). Thus, let D_{ε}(R_{o}) be a sequence of domains in B′_{x_{ε}}(R_{o})∩U^{υ_{o}}, for U^{υ_{o}} as in (1.18), converging in the Gromov-Hausdorff topology to the domain D(R_{o}) as in (3.19). In addition, D_{ε}(R_{o}) may be chosen to that if ∂_{s}D_{ε}(R_{o}) is any subset of ∂D_{ε}(R_{o}) converging to a proper subset of ∂F ∩ D̄(R_{o}), then ∂_{s}D_{ε}(R_{o}) ⊂ L^{υ_{o}}. These conditions do not uniquely define D_{ε}(R_{o}), (even topologically), but this plays no role. If the sequence (Ω_{ε}, g′_{ε}, x_{ε}) is collapsing, or is μ_{o}-collapsed for ε sufficiently small, define D_{ε} so that in addition π_{1}(B′_{x_{ε}}(1)) ⊂ π_{1}(D_{ε}(R_{o})), so that coverings unwrapping the collapse of B′_{x_{ε}}(1) also unwrap D_{ε}(R_{o}).

Let L̄_{ε} be any level set of u with u(L̄_{ε}) → 1 and let m′_{δ_{a}} be the mass in the scale g′_{ε} of ν_{δ_{a}(x_{ε})}, lifted to covers if g′_{ε} is sufficiently collapsed at x_{ε}. Corollary 3.4. There is a constant M_{o} < ∞, such that on the levels L̄_{ε} as above,

(3.21)  m′_{δ_{a}}(D_{ε}(R_{o}) ∩ L̄_{ε}) ≤ M_{o}.

Proof: The proof is identical to the proof of Proposition 3.3; alternately, (3.21) follows from (3.20) and the continuity of the mass to the limit.

The results above give local upper bounds for the mass of the potential, within small g_{ε}- neighborhoods of base points x_{ε}. The following elementary result gives a crucial upper bound on the total mass, on the base scale (Ω_{ε}, g_{ε}). Theorem 3.5. Let L be any level set of u, δ > 0 and set ν_{δ} = (u − 1)/δ. Then on (Ω_{ε}, g_{ε}),

(3.22)  m_{δ}(L) ≡ ∫_{L} |∇ν_{δ}|dA_{g_{ε}} ≤ \frac{ε̄Z^{2}}{2δ} .

Proof: Let U^{+} = {x ∈ Ω_{ε} : u(x) ≥ u(L)} and apply the divergence theorem to the trace equation (2.3), renormalized by δ, over U^{+}. A little algebra then gives

(3.23)  ∫_{L} |∇ν_{δ}| = ∫_{U^{+}}(\frac{εZ^{2}}{4δT} − \frac{1}{8} \frac{σ}{T} (\frac{T^{2}u^{2} − 1}{δ}) − \frac{ε|z|^{2}}{4δT} ) ≤ ∫_{U^{+}}(\frac{εZ^{2}}{4δT} − \frac{1}{8} \frac{σ}{T} (\frac{T^{2}u^{2} − 1}{δ})).

The left side of (3.23) is the total mass of the potential ν_{δ} at this base scale g_{ε}. Recall from §1 that T u = w and w = −s/σ, where σ is the L^{2} norm of s; thus, the L^{2} norm of w equals 1. Since vol_{g_{ε}}Ω_{ε} = 1, this gives

∫_{Ω_{ε}}(\frac{T^{2}u^{2} − 1}{δ}) = 0.

Also, T^{2}u^{2} − 1 ≤ 0 precisely on the set W_{1} = {w ≤ 1}. Hence

− ∫_{U^{+}}(\frac{T^{2}u^{2} − 1}{δ}) ≤ − ∫_{W_{1}}(\frac{T^{2}u^{2} − 1}{δ}) = ∫_{W^{1}}(\frac{T^{2}u^{2} − 1}{δ}),

where W^{1} = {w ≥ 1}. Since volW^{1} ≤ 1 and u ≤ 1, it follows that

(3.24)  −\frac{1}{8} \frac{σ}{T} ∫_{U^{+}}(\frac{T^{2}u^{2} − 1}{δ}) ≤ \frac{1}{8} \frac{σ}{T} (\frac{T^{2} − 1}{δ}) ≤ \frac{1}{4} \frac{εZ^{2}}{δT} ,

where the last estimate follows from (1.8). This gives the bound (3.22).

Remark 3.6. Here a remark, not actually relevant to this paper, but relevant to the Sphere conjecture and the asymptotic geometry of Z_{c}^{2} solutions discussed in [1,2]. All of the analysis in §2-§3 carries over to the linearization at infinity of Z_{c}^{2} solutions (N, g′), provided sup_{N} u < ∞. In fact, this situation is considerably easier, since su ≡ 0 and there is no constant term in the Euler-Lagrange equations (1.14)-(1.15). In particular, one can define mass functions similar to (3.9), c.f. [2, (7.54)ff]. However, in this case, there is no upper bound estimate available for the mass - Theorem 3.5 is lacking. It is precisely the constant term c = c_{ε} of (1.6) in the Euler-Lagrange equations which leads to this bound. This information, which will be seen to be crucial, is thus lost when passing to Z_{c}^{2} limits.

4. Uniformity Estimates.

All of the results in §2 and §3, with the exception of Theorem 3.5, are local; they concern the geometry of blow-ups g′_{ε} and potentials ν_{δ} in small neighborhoods of base points x_{ε}. All of these results depend strongly on the size of δ, and hence on the choice of base point x_{ε}.

In order to relate these local behaviors at different base points, as well as relate them to the global estimate (3.22), it is important to obtain uniform control on δ on a given level set of u.

The purpose of this section is to prove the existence of base points y_{ε} where δ, (or more precisely δ_{a}), is not too small, both at y_{ε} and at points on the same level set as y_{ε} and within, say, unit g_{ε}-distance to y_{ε}. The main result in this section is Theorem 4.4. Some applications to the structure of ∂F are given in Proposition 4.8-Lemma 4.9.

To begin, the analysis is divided into two cases, according to the following: Definition 4.1. The sequence (Ω_{ε}, g_{ε}), ε = ε_{i}, satisfies the degeneration hypothesis if there is a sequence of base points x_{ε} ∈ Ω_{ε} such that, as ε → 0,

(4.1)  |u − 1|(x_{ε}) ≤ \frac{1}{(ln ε̄)^{2}} , and ρ(x_{ε}) → 0.

The sequence (Ω_{ε}, g_{ε}) satisfies the non-degeneration hypothesis if there exists a constant ρ_{o} > 0 such that whenever |u − 1|(x_{ε}) ≤ 1/(ln ε̄)^{2}, then

(4.2)  ρ(x_{ε}) ≥ ρ_{o}.

Thus, the given sequence (Ω_{ε}, g_{ε}), ε = ε_{i}, satisfies either the degeneration or the non-degeneration hypothesis; if it satisfies the degeneration (or non-degeneration) hypothesis, then so do all subsequences. Of course if (Ω_{ε}, g_{ε}) satisfies the degeneration hypothesis, then there still may well exist other base points z_{ε} ∈ Ω_{ε}, distinct from x_{ε}, such that |u − 1|(z_{ε}) ≤ 1/(ln ε̄)^{2} but ρ(z_{ε}) ≥ ρ_{o}, for some ρ_{o} > 0. The specific decay requirement (4.1) on |u−1| is used only in the proof of Proposition 4.8 below and for most purposes can be replaced by the weaker condition that u(x_{ε}) → 1 at any rate as ε → 0.

The analysis in these two cases is quite different, and in particular is much simpler when (Ω_{ε}, g_{ε}) satisfies the non-degeneration hypothesis. The non-degeneration case (4.2) will be analysed in §7, and throughout §4-§5, it is assumed that (Ω_{ε}, g_{ε}) satisfies the degeneration hypothesis. (It will be used however only in Theorem 4.4).

Further, in this section we use the assumption that M is σ-tame. Thus, it will always be assumed that the sequence (Ω_{ε}, g_{ε}), ε = ε_{i}, is chosen to satisfy (1.36), i.e.

(4.3)  ε̄Z^{2} ≡ ε̄Z^{2}(g_{ε}) ≤ ε̄^{4μ},

for some μ < \frac{1}{4} and ε sufficiently small; (4.3) follows from (1.36) since T ≥ 1 and hence T^{μ} ≤ T. Of course, (4.3) then remains valid for any subsequences.

Now for any given ε > 0 fixed, consider certain special base points as follows: Definition 4.2. A sequence of base points x_{ε} is called admissible if it satisfies ρ(x_{ε}) → 0 together with the following conditions. Let L = L_{ε} be the level set of u containing x_{ε}. Then

(4.4)  ε̄^{2μ} ≤ |u(L) − 1| ≤ \frac{2}{(ln ε̄)^{2}} ,

and, for the affine δ_{a} defined as in (3.1) and any point p_{ε} ∈ B_{x_{ε}}(1) ∩ L,

(4.5)  δ_{a}(p_{ε}) ≥ δ_{o} ≡ ε̄^{2μ}.

Observe that by Definition 2.7, admissible base points are allowable. Of course, δ_{o} → 0 as ε → 0 and in fact, from (4.3),

(4.6)  \frac{ε̄Z^{2}}{δ_{o}} ≤ ε̄^{2μ} → 0, as ε → 0.

Hence ε̄Z^{2}/δ(p_{ε}) → 0, for all p_{ε} ∈ L ∩ B_{x_{ε}}(1); compare with (3.22). The estimates (4.5) and (4.6) play the central role in the arguments of §5. Although (4.5) is not required to hold globally on L, it is important to recognize that (4.5) holds on a region in L of determined size about the base point x_{ε}.

The definition of δ_{o} in (4.5), natural in view of (4.6), is just a convenient choice, and other choices, such as δ_{o} = ε̄^{μ}, are equally possible. Nevertheless, it is necessary to make one choice, and so we choose (4.5).

An important related class of base points are the following: Definition 4.3. A sequence of base points x_{ε} is called well-separated (w.r.t. U^{υ_{o}}), if it satisfies (4.4) together with the following conditions: there exist scales l_{ε} > 0 and constants r_{o}, d_{o} > 0, (independent of ε), such that l_{ε} ≥ r_{o}ρ(x_{ε}), (4.5) holds for all p_{ε} ∈ B_{x_{ε}}^{υ_{o}}(l_{ε}) ∩ L, and

(4.7)  A_{x_{ε}}^{υ_{o}}(l_{ε}, (1 + d_{o})l_{ε}) ∩ L = ∅.

Here A_{x}^{υ_{o}}(r, s) is the geodesic annulus about x of inner and outer radii r, s, in the metric space (U^{υ_{o}}, g_{ε}), where U^{υ_{o}} = {x_{ε} ∈ Ω : u(x_{ε}) ≥ 1 − υ_{o}}, as in Proposition 1.1; similarly for the ball B_{x}^{υ_{o}}. While it is possible to work with the (more natural) full annulus A_{x_{ε}} in (Ω_{ε}, g_{ε}), it is somewhat simpler at this stage to work with the restricted annulus A_{x_{ε}}^{υ_{o}}; it will be seen later (in Lemma 5.7 below) that the difference between these two choices becomes irrelevant.

The region C = C(ε) = B_{x_{ε}}^{υ_{o}}(l_{ε}) ∩ L gives rise to a compact and isolated region of the level set L of ν_{a} in the limit (F, g′_{o}, x), where g′_{ε} = l_{ε}^{−2}g_{ε}; compare with Remark 2.2. A well-separated base point satisfies all the properties of admissible base points, except that (4.5) may not hold at all points in the 1-ball about x_{ε}, (and ρ(x_{ε}) is not required to go to 0).

Finally, a sequence of base points x_{ε} is called preferred if it is well-separated with l_{ε} ∼ ρ(x_{ε}), i.e. l_{ε}/ρ(x_{ε}) is bounded away from 0 and infinity as ε → 0.

The purpose of this section is to prove the existence of at least one of these two kinds of base points, either admissible or preferred, near any base points x_{ε} satisfying (4.1). In §5, it will then be proved that if y_{ε} is an admissible base point, then there exist well-separated base points z_{ε} near y_{ε}. As in Remark 2.2, it is the well-separated base points which are naturally associated with 2-spheres. Theorem 4.4. Suppose (Ω_{ε}, g_{ε}) satisfies the degeneration hypothesis (4.1), with ε = ε_{i} satisfying (4.3). Then for any base points x_{ε} satisfying (4.1), there are base points y_{ε} ∈ B_{x_{ε}}(1) which are either admissible or preferred.

The proof will proceed in severals steps, via two Lemmas. Let L = L(ε) be the the level set of u through x_{ε}. By Proposition 1.1, t_{υ_{o}}(x_{ε}) → 0, where t_{υ_{o}} = dist(·, L^{υ_{o}}). Let L_{1} = L_{1}(ε) be the level set of u defined by

(4.8)  u(L_{1}) = u(L) − ε̄^{2μ},

so that u(L_{1}) → 1, as ε → 0 also. By the converse of Proposition 1.1, c.f. (2.18), it follows that there are base points y_{ε} ∈ B_{x_{ε}}(\frac{1}{10}) ∩ L_{1} such that ρ(y_{ε}) → 0 as well. Thus, y_{ε} also satisfies (4.4).

It will be proved that such base points y_{ε} are either admissible, or there exist preferred base points q_{ε} ∈ B_{y_{ε}}(1), i.e. either all z_{ε} ∈ B_{y_{ε}}(1) ∩ L_{1} satisfy (4.5) or there exist preferred base points q_{ε} ∈ B_{y_{ε}}(1). Note that the Lipschitz property of ρ implies that ρ(z_{ε}) ≤ 1, for all z_{ε} ∈ B_{y_{ε}}(1).

Let L_{2} be the level set of u defined by

(4.9)  u(L_{2}) = u(L_{1}) − ε̄^{μ}.

Consider arbitrary (piecewise) C^{1} paths γ in U^{υ_{o}}, which start at some point in L^{υ_{o}}, end at a point z_{ε} in B_{y_{ε}}(1) ∩ L_{1}, and which satisfy the following cone condition:

(4.10)  s ≤ C · dist(γ(s), L^{υ_{o}}) = C · t_{υ_{o}}(γ(s)),

where s is the arclength parameter of γ and C is any fixed constant. Observe that (4.10) is scaleinvariant. It is clear that such paths exist; for instance one may choose γ to be a shortest path in U^{υ_{o}} realizing the distance of z_{ε} to L^{υ_{o}}. The condition (4.10) roughly means that γ does not zigzag too much across the levels of u. For any such γ, let s_{2} be the largest value of s such that γ(s_{2}) ∈ L_{2} and let a_{ε} = γ(s_{2}). Finally set γ^{2} = γ|_{[s_{2},b]}, where γ(b) = z_{ε}.

A simple but important idea in the proof is the following: Lemma 4.5. For any curve γ as above, there exist points p_{ε} ∈ γ^{2} such that

(4.11)  δ_{a}(p_{ε}) ≥ δ_{o}.

Proof: If not, then there is a curve γ satisfying (4.10) such that, ∀s ≥ s_{2}, δ_{a}(γ^{2}(s)) ≤ δ_{o}, and hence by Lemma 3.1,

(4.12)  |du|(γ^{2}(s)) ≤ C_{o} \frac{δ_{o}}{ρ(γ^{2}(s))}.

Observe again that the estimate (4.12) is scale-invariant. For the argument to follow, we work in the (base) scale g_{ε}. For t_{υ_{o}} as above, Proposition 1.1 implies that

\frac{1}{ρ(γ^{2}(s))} ≤ \frac{c_{1}}{t_{υ_{o}}} ≤ \frac{c_{2}}{s} ,

where the 2^{nd} inequality follows from (4.10). From (4.12), it then follows that, along γ^{2}(s),

|\frac{du}{ds}| ≤ c_{3} \frac{δ_{o}}{s} .

Integrating this along γ^{2} gives

(4.13)  |u(L^{2}) − u(L^{1})| ≤ c_{4}δ_{o} ln(\frac{1}{s_{2}} ).

Now s_{2} ≥ dist_{g_{ε}}(a_{ε}, L^{υ_{o}}) ≡ λ_{o} and again by (2.18), λ_{o} ≥ R_{o}^{−1}·ρ(a_{ε}). By Proposition 1.4, ρ^{2} ≥ κ·ε̄ at any base point and so in particular λ_{o} ≥ ε̄. Thus, using the definition of δ_{o}, one has

(4.14)  |u(L^{2}) − u(L^{1})| ≤ c_{5}ε̄^{2μ} ln(ε̄^{−1}) << ε̄^{μ},

for ε sufficiently small. This contradicts (4.9), and so proves the result.

Observe that the proof shows there are points q_{ε} on γ^{2} such that δ_{a}(q_{ε}) >> δ_{o}. More importantly, the proof also holds if γ is a curve satisfying (4.10), with, for instance,

(4.15)  C_{o} = C_{o}(ε) ≤ ε^{−μ/2},

in place of the ε-independent C_{o} of (4.12).

If all the points p_{ε} from Lemma 4.5 may be chosen to be the end points z_{ε} ∈ L_{1} ∩ B_{y_{ε}}(1) of curves γ, then the result is proved. Thus, suppose not, so that there exists z_{ε} ∈ L_{1} ∩ B_{y_{ε}}(1) such that

(4.16)  δ_{a}(z_{ε}) < δ_{o}.

We then construct a path γ, starting at z_{ε}, (i.e. in the reverse direction from previously), satisfying (4.10) and an analogue of (4.12). The point is to construct γ with a controlled increase of δ_{a}, i.e. the local mass, along γ, analogous to (4.15). Given the initial estimate (4.16), it is then shown that this path either terminates on L_{2}, giving a contradiction as in Lemma 4.5, or terminates on a preferred base point q_{ε}. This will then complete the proof.

The construction of γ and the growth estimate for δ_{a} is by an inductive procedure. Thus, let z_{ε} = z_{ε}^{1}, and let g_{ε}^{1} = ρ(z_{ε}^{1})^{−2} · g_{ε} be the rescaled metric based at z_{ε}^{1}. Let δ_{1} = δ_{a}(z_{ε}^{1}) and let ν_{δ_{1}} be the corresponding potential function. By adding a suitable constant, one may assume that ν_{δ_{1}}(L_{1}) = −1; recall that adding constants to ν_{δ_{a}} does not affect the measure or mass of the potential. Thus

(4.17)  ν_{δ_{1}}(L_{2}) ∼ −ε̄^{μ}/δ_{1} << −1,

for ε small. In particular, it follows that ρ^{1} → 0 on L_{2}, where ρ^{1} is the g_{ε}^{1} curvature radius, and so L_{2} converges, in the Hausdorff topology, to ∂F, where (F, g′_{o}, z^{1}) is the maximal limit based at z_{ε}^{1}.

Define intermediate level sets L^{j} by

(4.18)  ν_{δ_{1}}(L^{j+1}) = e^{k_{o}j}ν_{δ_{1}}(L^{j}),

with L^{1} = L_{1}. Here k_{o} is a fixed number which will be specified below, (see (4.23) and (4.26)). It follows that for ε > 0 fixed, the level L_{2} is reached after at most N iterations, i.e. L_{2} ∼ L^{N}, where

(4.19)  N = \frac{1}{k_{o}} ln \frac{ε̄^{μ}}{δ_{1}} .

The induction will be on j, i.e. down the level sets L^{j}, beginning with the levels L^{1} and L^{2}.

To start, recall the compact domains D_{ε}(R_{o}) defined following Proposition 3.3, with base point z_{ε}^{1}. If B_{z_{ε}^{1}}^{1}(1) is μ_{o}-collapsed, i.e. ω(z_{ε}^{1}) ≤ μ_{o}ρ(z_{ε}^{1}), then the collapse is unwrapped in covering spaces, as described following Proposition 3.3. Here μ_{o} is a fixed parameter, e.g. μ_{o} = 1/100. For convenience, set R_{o} = 2. Then Corollary 3.4 gives

(4.20)  m(D_{ε}(2) ∩ L̄) ≤ M_{o},

for any level set L̄ of u, e.g. L̄ = L^{1} or L^{2}, where m = m_{δ_{1}} is the mass w.r.t. δ_{1} in the scale g_{ε}^{1}.

The basic idea in the inductive step, i.e. in the construction of the next base point z_{ε}^{2} on the level L^{2} is to examine the distribution of the measure dμ_{δ_{1}} within L^{2} ∩ D_{ε}(2). By way of illustration, suppose dμ is a measure on I = [0, 1] of mass at most M_{o}. If the support of dμ satisfies supp dμ = I, then it is clear that for all 0 < t < 1, there exists x ∈ suppdμ such that

(4.21)  m_{dμ}([x, x + t]) ≤ M_{o} · t,

(i.e. the analogue of (4.20) holds on the scale t). If on the other hand the mass of dμ is concentrated, (for example the Dirac measure at a point in the most extreme case), then (4.21) does not hold. The only situation in which (4.21) can fail is when there are gaps in supp dμ, i.e. intervals J ⊂ I on which J∩suppdμ = ∅. This corresponds to the existence of well-separated base points.

Define a level set L̄, for instance L̄ = L^{2}, to be d_{o}-separated (w.r.t. U^{υ_{o}}), if there is a point p_{ε} ∈ L̄ and radius s_{ε} such that

(4.22)  A_{p_{ε}}^{υ_{o}}(s_{ε}, (1 + d_{o})s_{ε}) ∩ L̄ = ∅,

compare with (4.7). The condition (4.22) is scale-invariant, although we continue to work in the scale g_{ε}^{1}. The relation (4.22) implies that s_{ε} ≥ diam_{g_{ε}^{1}}C_{o}, where C_{o} is the component of L̄ containing p_{ε}, and diam is the extrinsic diameter, i.e. the diameter within (U^{υ_{o}}, g_{ε}^{1}). Let t(x) = t^{1}(x) = dist_{g_{ε}^{1}}(x, L^{υ_{o}}). Henceforth, assume that k_{o} is chosen sufficiently large so that

(4.23)  t^{1}(x) ≤ \frac{1}{4}μ_{o}, ∀x ∈ L^{2} ∩ D_{ε}(2),

where μ_{o} is the collapse parameter, defined preceding (4.20). Lemma 4.6. Suppose the level L^{2} has no d_{o}-separated components intersecting D_{ε}(2), for d_{o} ≤ 1. Then there exist base points z_{ε}^{2} ∈ L^{2} ∩ D_{ε}(2) such that

(4.24)  m(L^{2} ∩ D_{z_{ε}^{2}}(2t(z_{ε}^{2}))) ≤ μ_{1}M_{o}(1 + d_{o}) · t(z_{ε}^{2}),

where μ_{1} = μ_{1}(μ_{o}). Proof: Suppose first that L̂^{2} = L^{2} ∩ D_{ε}(2) is path connected. Since L^{2} has no d_{o}-separated components intersecting D_{ε}(2), the diameter, (within U^{υ_{o}}), satisfies diam_{g_{ε}^{1}} L̂^{2} ≥ μ_{o}/(1+d_{o}) ≥ μ_{o}/2. A curve τ realizing the intrinsic diameter of L̂^{2} thus has length l(τ) ≥ μ_{o}/2.

Now cover the path τ by a collection of smaller domains with disjoint interiors as follows. Let s be the arclength parameter for τ, with p_{0} = τ(0). Let D_{1} = D_{p_{1}}(2t(p_{1})), where p_{1} is the point on τ whose arclength distance to τ(0) equals 2t(p_{1}); here D_{1} is the domain as in (4.20), centered on p_{1}. Let s_{2} be the largest value of s such that τ(s) ∈ D_{1} and set p_{2} = τ(s_{2}); clearly s_{2} > 2t(p_{1}). Next let D_{2} = D_{p_{3}}(2t(p_{3})), where p_{3} is the point on τ whose arclength distance to p_{2} equals 2t(p_{3}). Observe that D_{1} ∩ D_{2} = p_{2}. One then continues inductively in this way until the maximal p_{k} is defined, for k odd. It follows from (4.23) that the domain V = ∪D_{j} covers at least \frac{1}{2} the length l(τ) of τ.

By construction, m(V) = ∑ m(D_{j}). If for each j, m(D_{j}) > μ_{1}M_{o}t(p_{j}), then

m(V) > μ_{1}M_{o} ∑ t(p_{j}).

But by constuction, ∑ t(p_{j}) is least \frac{1}{2}l(τ) ≥ \frac{1}{4}μ_{o}. Thus, if μ_{1} > 4μ_{o}^{−1}, this contradicts (4.20), and hence there is some j such that m(D_{j}) ≤ μ_{1}M_{o}t(p_{j}). This gives (4.24) in this situation, with z_{ε}^{2} = p_{j}.

If L̂^{2} is not path connected, let τ_{k} be a path realizing the intrinsic diameter of each path component C_{k} of L̂^{2}, and let τ be the union of the paths τ_{k}. Then perform the construction above on each component τ_{k}. Since there are no d_{o}-separated subsets, for any k, there must be a C_{j} and point q_{j} ∈ C_{j} such that dist_{D_{ε}(2)}(q_{j}, C_{k}) ≤ \frac{1+d_{o}}{2} (diam_{D_{ε}(2)}C_{k}). It follows that l(τ) ≥ (1 + d_{o})^{−1}μ_{o}, and the same argument as above then gives (4.24).

The domain D_{ε}^{1} ≡ D_{ε}(2) converges, (in a subsequence), to a limit domain D^{1} ≡ D_{o}(2) contained in the maximal limit (F, g′_{o}, z^{1}), z^{1} = lim z_{ε}^{1}; the limit F is either flat or hyperbolic. Since the limit level L^{2} ∩ D_{o}(2) ⊂ F has a definite diameter, bounded away from 0 and ∞, it follows that L^{2} ∩ D_{ε}(2) is d_{o}-separated only when (4.22) holds with

(4.25)  s_{ε} ∼ 1

in the scale g_{ε}^{1}, i.e. s_{ε} is bounded away from 0 and ∞ in this scale.

Lemma 4.6 provides the inductive step. Thus, fix any choice of d_{o} ∈ (0, 1] and set

(4.26)  k_{o} = ln[A_{o}μ_{1}(1 + d_{o})M_{o}],

where A_{o} = a_{o}^{−1} is the area constant from (2.79). (Note that k_{o} may be chosen to be large, by choosing M_{o} large). Suppose L^{2} has no d_{o}-separated components in D_{ε}(2). Then Lemma 4.6 gives the existence of the next base point z_{ε}^{2} ∈ L^{2} and domain D_{ε}^{2} ≡ D_{z_{ε}^{2}}(2t(z_{ε}^{2})) satisfying (4.24). Observe that, by construction, dist_{g_{ε}^{1}}(z_{ε}^{2}, z_{ε}^{1}) ≤ 2 and dist_{g_{ε}^{1}}(z_{ε}^{2}, L^{1}) ≥ \frac{1}{2}, where the 2^{nd} inequality follows from (4.23). Hence, one may choose a curve γ_{1}(s) ⊂ U^{υ_{o}} from z_{ε}^{2} to z_{ε}^{1}, with arclength parameter s, such that

(4.27)  s ≤ c · dist_{g_{ε}^{1}}(γ_{1}(s), L^{2}),

i.e. the analogue of (4.10) holds. Further, by (3.7) one has, along γ_{1},

(4.28)  |du|(γ_{1}(s)) ≤ C \frac{δ_{1}}{ρ(γ_{1}(s))},

where C is a fixed constant, possibly depending on k_{o} but not on ε.

Now at the endpoint z_{ε}^{2} of γ_{1}, we claim that for M_{1} = A_{o}μ_{1}M_{o}(1 + d_{o}),

(4.29)  δ_{2} ≡ δ_{a}(z_{ε}^{2}) ≤ M_{1}δ_{1}.

To see this, recall first that the limit based at z_{ε}^{1} is flat or hyperbolic. As noted in (2.19ff), without loss of generality, one may assume that

(4.30)  ρ^{1}(z_{ε}^{2}) = (1 + o(1))t(z_{ε}^{2}),

where ρ^{1} is the curvature radius in the g_{ε}^{1} scale. In the scale g_{ε}^{2} = ρ(z_{ε}^{2})^{−2} · g_{ε} where ρ^{2}(z_{ε}^{2}) = 1, by definition δ_{a}(z_{ε}^{2}) = ∮_{H( \frac{1}{2} )} |∇u|, where H(\frac{1}{2}) = B_{z_{ε}^{2}}(\frac{1}{2}) ∩ L^{2}. Further, in this scale D_{ε}^{2} = D_{z_{ε}^{2}}(2) and hence (4.24) and (4.30) give

m_{δ_{1}}(D_{ε}^{2}) = ∫_{D_{ε}^{2}∩L^{2}} |∇u|/δ_{1} ≤ μ_{1}M_{o}(1 + d_{o}).

Since H(\frac{1}{2}) ⊂ D_{ε}^{2}, (again in the g_{ε}^{2} scale), it follows that

δ_{a}(z_{ε}^{2}) ≤ (μ_{1}M_{o}(1 + d_{o})/area_{g_{ε}^{2}}H(\frac{1}{2}))/δ_{1}.

Via (2.79), this gives (4.29).

We now repeat this construction at the level 2 in place of level 1. Thus, as above, at the base point z_{ε}^{2} ∈ L^{2}, rescale the metric based at z_{ε}^{2}, so that g_{ε}^{2} = ρ(z_{ε}^{2})^{−2} · g_{ε} and work with the potential ν_{δ_{2}} in place of ν_{δ_{1}}. From (4.24), (4.29) and (4.30), one has the bound, in the g_{ε}^{2} scale,

(4.31)  m_{δ_{2}}(D^{2}) ≤ M_{1}^{2}.

Assuming that there are no d_{o}-separated subsets within L̂^{3} = L^{3} ∩ D_{ε}^{2}, one may then construct again by Lemma 4.6 the next level base point z_{ε}^{3} and domain D_{ε}^{3}, as well as the next part of γ, i.e. γ_{2}, from z_{ε}^{2} to z_{ε}^{3} satisfying the analogue of (4.27). The estimate (4.28) translates to the estimate

|du|(γ_{2}(s)) ≤ CM_{1}^{2} \frac{δ_{1}}{ρ(γ_{2}(s))}.

Thus, this process may be continued inductively until the level L_{2} is reached, provided at each stage there are no d_{o}-separated subsets.

Suppose then that this process continues to the level L_{2}, producing a curve γ starting at L_{2} and ending at z_{ε}^{1}. Since each part γ_{1}, γ_{2}, etc. of γ satisfies the analogue of (4.27), γ satisfies the cone condition (4.10). Further, along γ one has

(4.32)  δ_{a}(γ(s)) ≤ C · M_{1}^{N}δ_{1},

where N is given by (4.19). However, together with (4.26) this gives

δ_{a}(γ(s)) ≤ C · ε̄^{μ/2}δ_{1}^{1/2} ≤ C · ε̄^{3μ/2},

everywhere along γ(s), where the 2^{nd} inequality uses δ_{1} ≤ ε̄^{2μ} by (4.16); compare also with (4.15). The proof of Lemma 4.5 carries through as before, and again gives a contradiction.

Hence if (4.16) holds, (for some z_{ε}), there must be a (first) level L^{j−1}, with base points z_{ε}^{j−1} ⊂ L^{j−1}, and domains D_{ε}^{j−1} such that the next level L̂^{j} = L^{j} ∩ D_{ε}^{j−1} is d_{o}-separated, i.e. there exist z_{ε}^{j} ∈ L̂^{j} such that (4.22) holds, with s_{ε} ∼ 1, in the scale g_{ε}^{j−1} = ρ(z_{ε}^{j−1})^{−2} · g_{ε}. Suppose that, for all p_{ε}^{j} ∈ L̂^{j},

(4.33)  δ_{a}(p_{ε}^{j}) ≥ min(δ_{o}, δ_{a}(z_{ε}^{j−1})).

Then (2.78) implies that the potential function ν_{δ_{a}^{j−1}} ≡ ν_{δ_{a}(z_{ε}^{j−1})} satisfies |∇ν_{δ_{a}^{j−1}}|(p_{ε}^{j}) ≥ κ_{o}/t(p_{ε}^{j}) in

the g_{ε}^{j−1} scale. Since ν_{δ_{a}^{j−1}}(L^{j})/ν_{δ_{a}^{j−1}}(L^{j−1}) = e^{k_{o}} and k_{o} is fixed, it follows that ρ^{j−1}(z_{ε}^{j}) ≥ ρ_{o} = ρ_{o}(κ_{o}) > 0. This means that ρ^{j}(z_{ε}^{j}) ∼ 1, i.e. s_{ε} ∼ ρ(z_{ε}^{j}) in the g_{ε} scale, as ε → 0. It follows that z_{ε}^{j} is preferred, which completes the proof in this case.

Suppose instead that for some p_{ε}^{j} ∈ L̂^{j},

δ_{a}(p_{ε}^{j}) < max(δ_{o}, δ_{a}(z_{ε}^{j−1})).

In this case, the analogue of (4.16) holds, with p_{ε}^{j} in place of z_{ε}^{1} and one just continues the construction of the curve γ joining z_{ε}^{j−1} to z_{ε}^{j} ≡ p_{ε}^{j}. The point is that one must eventually come to points where (4.33) holds and the only way to do this is to come to a preferred base point, (unless x_{ε} is admissible).

Theorem 4.4 implies that there may be many admissible or preferred base points x_{ε}. In fact they are 1-dense in the region S_{ε} ⊂ Ω_{ε} defined by (4.4), i.e. ε̄^{2μ} ≤ |u − 1| ≤ 2/(ln ε̄)^{2}, and ρ ≤ χ_{ε}, where χ_{ε} is any sequence with χ_{ε} → 0 as ε = ε_{i} → 0. It is obvious from the proof that such points are η-dense in S_{ε}, for any given η > 0. This lack of uniqueness plays no role however. Any level set L of u satisfying (4.4) and containing some admissible or preferred base point will be called an admissible level set and denoted by

(4.34)  L = L_{o}.

Let ν_{δ_{o}} = (u−1)/δ_{o}, for δ_{o} as in (4.5), and let m_{o}, m̃_{o} be the mass of the potential ν_{δ_{o}}, as in (3.9) and (3.11) respectively. The following uniform lower bound on the mass is essentially immediate. Proposition 4.7. Let x_{ε} be any admissible or preferred base point. Then there is a constant κ_{o} > 0, independent of ε ≤ 1, such that for any p_{ε} ∈ L_{o} ∩ B_{x_{ε}}(1), or p_{ε} = x_{ε} respectively,

(4.35)  m̃_{o}(ρ(p_{ε})) ≥ κ_{o} · ρ(p_{ε}).

In particular, if (Ω_{ε}, g_{ε}) is not μ_{o} collapsed at x_{ε}, i.e. ω(x_{ε}) ≥ μ_{o} · ρ(x_{ε}), then

(4.36)  m_{o}(ρ(x_{ε})) ≥ κ_{o} · ρ(x_{ε}),

where κ_{o} depends only on μ_{o}.

Proof: By construction, i.e. (4.5), δ_{a}(p_{ε}) ≥ δ_{o}, and hence

|dν_{δ_{a}(p_{ε})}| ≤ |dν_{δ_{o}}|,

for all p_{ε} as above. Hence, the mass defined w.r.t. δ_{o} dominates the mass defined w.r.t. δ_{a}(p_{ε}). The estimates (4.35) and (4.36) are then an immediate consequence of (3.13).

We close this section with some applications of the proof of Theorem 4.4 to the structure of ∂F; these will be useful in §5.3 and §6. Let (F, g′_{o}, x) be a maximal limit, (flat or hyperbolic), of (Ω_{ε}, g′_{ε}, x_{ε}), with x_{ε} allowable, and let ν be the limit potential, ν = lim ν_{δ_{a}}, δ_{a} = δ_{a}(x_{ε}). If (Ω_{ε}, g_{ε}) is μ_{o}-collapsed at x_{ε}, i.e. ω(x_{ε}) ≤ μ_{o} · ρ(x_{ε}), it is assumed that the collapse is unwrapped in covering spaces, so that ω(x_{ε}) ∼ ρ(x_{ε}). Define

(4.37)  I_{∞} = \overline{∩_{N}\{U^{−N}\}},

where U^{−N} is the −N superlevel set of ν, N ∈ ℤ^{+}, and the closure is taken in F̄. Let dμ_{−N} be the measure of ν on the level L^{−N}, as in (3.8), but restricted to D_{x}(∞), c.f. (3.19), i.e. the maximal domain on which the developing map is an embedding. Thus dμ_{−N} is supported on the level set L_{−N} ∩ D_{x}(∞).

Let dμ_{∞} be the weak limit of the measures dμ_{−N} in F̄ as N → ∞. The limit exists since the measures dμ_{−N} are related via the divergence theorem applied to the trace equation (2.65). The measure dμ_{∞} is supported on ∂F. Proposition 4.8. Suppose the base points x_{ε} are allowable and satisfy |u − 1|(x_{ε}) ≤ 2/(ln ε̄)^{2}, and let (F, g′_{o}, x, ν) be the maximal limit as above. Then

(4.38)  ∂F = I_{∞}.

There is a constant m_{o} such that if B(r) is any r-ball in F̄ with center in ∂F, then ∀r ≤ 1,

(4.39)  m_{o} · r ≤ m_{∞}(B(r) ∩ ∂F),

where m_{∞} is the mass of dμ_{∞}. Further for any unit ball B(1) in F̄ centered at a point in ∂F,

(4.40)  m_{∞}(B(1) ∩ ∂F) ≤ M_{o},

for some constant M_{o} < ∞. The constants m_{o} and M_{o} are independent of the base point on ∂F. In particular, dim_{H}∂F ≤ 1 and H_{1}(∂F ∩ B(1)) ≤ M_{o}, where H_{1} is the 1-dimensional Hausdorff measure of the metric space (F̄, g′_{o}). Proof: It is clear that I_{∞} ⊂ ∂F and so only the reverse inclusion in (4.38) needs to be proved. Via Proposition 2.1, if y_{ε} ∈ U^{υ_{o}} converges to y ∈ F̄ (in the Hausdorff topology) then ν_{δ_{a}}(y_{ε}) → −∞ if and only if y ∈ ∂F. This does not immediately imply (4.38), (since ν_{δ_{a}} does not apriori converge to ν at ∂F). However, if for any y ∈ ∂F there exist y_{ε} → y as above such that, for some m_{1} > 0,

(4.41)  m̃_{δ_{a}}(\frac{1}{2}ρ(y_{ε})) ≥ m_{1} · ρ(y_{ε}),

then (4.38) does follow. Here m̃_{δ_{a}} is the mass of ν_{δ_{a}}, δ_{a} = δ_{a}(x_{ε}), on the level through y_{ε}, as in (3.11). Moreover (4.41) implies (4.39) provided the constant m_{1} is independent of y ∈ ∂F. Now if

(4.42)  δ_{a}(y_{ε}) ≥ c_{o} · δ_{a}(x_{ε}),

for some c_{o} > 0, then m̃_{δ_{a}(y_{ε})} ≤ c_{o}^{−1}m̃_{δ_{a}(x_{ε})}. But by (3.13), m̃_{δ_{a}(y_{ε})}(\frac{1}{2}ρ(y_{ε})) ≥ ρ(y_{ε}). Hence, (4.41) follows from (4.42).

The proof of (4.42) essentially follows from the proof of Lemma 4.5. Thus, consider paths γ = γ_{y,ε} in (Ω_{ε}, g_{ε}), parametrized by arclength s, starting at a point q_{ε} ∈ L^{υ_{o}} with q_{ε} → y, ending at x_{ε}, and satisfying the cone condition (4.10). If (4.42) were false along any such curve γ, then there is a constant C_{o} < ∞ such that, for all s sufficiently small and hence for all s ≤ S_{o}, S_{o} < ∞,

δ_{a}(γ(s)) ≤ C_{o} · δ_{a}(x_{ε}).

Let L_{ε} be the u-level set through x_{ε}. As in (4.12)-(4.14), it then follows that

(4.43)  |u(L^{υ_{o}}) − u(L_{ε})| ≤ C_{1}δ_{a}(x_{ε}) · | ln ε̄|,

where C_{1} depends only on C_{o} and S_{o}. However, in general one has the estimate

(4.44)  δ_{a}(x_{ε}) ≤ c_{2} · |u − 1|(x_{ε}),

for generic x_{ε} in the sense of (2.78), for some uniform c_{2} < ∞. To see this, (4.44) is equivalent to |ν_{δ_{a}(x_{ε})}|(x_{ε}) ≥ c_{2}^{−1}. If this estimate failed, i.e. |ν_{δ_{a}(x_{ε})}|(x_{ε}) << 1, then the limit function ν_{a} would be 0 at generic base points, contradicting the fact that ν_{a} is non-trivial, c.f. Theorem 2.11.

Since, by hypothesis, |u − 1|(x_{ε}) ≤ 2/| ln ε̄|^{2}, (4.43) and (4.44) give |u(L^{υ_{o}}) − u(L)| → 0 as ε → 0, a contradiction. This establishes (4.42) and hence (4.38)-(4.39), provided the point y ∈ ∂F is accessible, in the sense that y = lim y_{ε}, with y_{ε} joined to x_{ε} by a curve satisfying the cone condition (4.10). Let ∂_{a}F ⊂ ∂F denote the subset of accessible points.

By the definition of the measure dμ_{∞}, the estimate (4.40) follows immediately from Proposition 3.3 for balls B_{y}(1) with center y ∈ ∂F and dist_{g′_{o}}(y, x) ≤ 2, (for example). Together with the results above, this implies that ∂_{a}F has Hausdorff dimension at most 1, with locally finite H_{1}, within B̄_{x}(2), c.f. [13, Ch.4] for discussion of Hausdorff measures. In turn, this now implies that all points of ∂F within B̄_{x}(2) are accessible. Further, the arguments above may be applied at any other base point x′ ∈ F, with x′_{ε} → x′ and with δ_{a}(x_{ε}) replaced by δ_{a}(x′_{ε}). The limit of δ_{a}(x′_{ε})/δ_{a}(x_{ε}) as ε → 0 varies continuously as x′ varies over F, c.f. (the end of) Appendix B, and hence it follows that ∂F everywhere has Hausdorff dimension at most 1, with locally finite H_{1}.

To see that (4.39) holds globally on F, i.e. m_{o} is independent of center point of B(r), the local estimates above prove that (4.39)-(4.40) hold in any ball B_{x′}(2), when δ′_{a} is used in place of δ_{a} to define the local mass m_{∞}, and with m_{o}, M_{o} then independent of x′. This means that ∂F is not too dense in F, and it follows that there exists r_{o} = r_{o}(m_{o}, M_{o}) > 0 such that any point x′ ∈ F with dist(x′, ∂F) ≥ r_{o} may be joined to x via a path γ with dist(γ(s), ∂F) ≥ r_{o}. Hence, since any y ∈ F satisfies y ∈ B_{x′}(2) for some x′ as above, there also exist paths γ_{ε} from y_{ε} to x_{ε} satisfying the cone condition (4.10); the constant C in (4.10) of course depends on y, but c_{o} in (4.42) is independent of y, in the limit ε → 0.

Finally, to prove that (4.40) holds globally, i.e M_{o} is independent of the base point, suppose x′ is a base point in F such that m_{∞}(∂F ∩ B_{x′}(1)) >> 1, where of course m_{∞} is the mass w.r.t. the potential ν_{δ_{a}} based at x. This implies that δ_{a}(x′_{ε})/δ_{a}(x_{ε}) >> 1 in the limit on F. Interchanging the roles of x and x′, it follows that m′_{∞}(∂F ∩ B_{x}(1)) << 1, where m′_{∞} is the mass w.r.t. the potential ν_{δ′_{a}} based at x′. This contradicts the fact that (4.39) holds globally, (at the base point x′ in place of x), which gives (4.40).

We point out that the assumption |u − 1|(x_{ε}) ≤ 2/(ln ε̄)^{2} was used only in (4.43)-(4.44). Thus, Proposition 4.8 remains valid under the assumption δ_{a}(x_{ε}) ≤ 2/(ln ε̄)^{2}.

Recall that a subset of ℝ^{3} is polar if it is contained in a countable collection of compact sets, each of which has capacity 0. These are precisely the sets where non-positive subharmonic functions in domains in ℝ^{3} can become infinite, (i.e. −∞), c.f. [12]. A polar set has Hausdorff dimension at most 1. For the discussion below, suppose that the maximal limit F is flat; the discussion holds equally well for hyperbolic limits, with obvious changes. (The following remarks are not actually needed elsewhere in the paper).

The boundary ∂F may be divided into two parts, a regular part ∂_{r}F and a singular part, where p ∈ ∂_{r}F if and only if p ∈ ∂F and there is a neighborhood U of p in F̄ such that U embeds in ℝ^{3}. Note that such an extension of F past ∂_{r}F is unique. By Proposition 4.8, locally the regular boundary is contained in a closed set of finite H_{1} measure in ℝ^{3} and is of course a polar set. In particular, if ∂_{s}F = ∅ and F is flat, then F̄ is a complete, smooth flat manifold and hence

(4.45)  F̄ = ℝ^{3}/Γ,

where Γ ⊂ Isom(ℝ^{3}) is a discrete group, maybe trivial, acting freely on ℝ^{3}.

The local structure of ∂_{s}F is determined by the holonomy of the developing map D : F̃ → ℝ^{3}, where F̃ is the universal cover of F. For simplicity, assume here that F is maximally extended as a smooth flat manifold so that ∂F = ∂_{s}F. For q ∈ ∂F and r sufficiently small, let T_{r} denote the component of the r-tubular neighborhood of ∂F containing q, and let ∂T_{r} ⊂ F denote its boundary. Also, let S_{q}(r′) denote the geodesic sphere about q, r′ ≥ r. For generic r, r′, S_{q}(r′)∩∂T_{r} is a collection of embedded curves. Any such curve γ is either a loop, (linking ∂F), or a complete embedding of ℝ, i.e. γ has no endpoints. The former case occurs when B_{q}(r′) is compact, while the latter occurs when B_{q}(r′) is non-compact. Let γ̃ be a lift of γ to F̃, so that γ̃ is an embedding of ℝ in F̃. Then D(γ̃) is a closed loop in ℝ^{3}, and D|γ̃ acts as a covering projection ℝ → S^{1}. The loop D(γ̃) determines an isometric ℤ-action on ℝ^{3}.

Any isometric ℤ-action on ℝ^{3} is either rotational, i.e. rotation about an axis A, or a translation or a twist, i.e. a rotation about and translation along an axis A. In the latter two cases, the corresponding quotient spaces are the product S^{1} × ℝ^{2} and twisted product S^{1} ×_{α} ℝ^{2} respectively, where α is the rotation angle. These quotients are everywhere smooth and hence do not contribute ∂_{s}F. The only source of singularity to ∂F is that given by rotational holonomies. It follows that ∂_{s}F is given locally by a collection of lines, corresponding to rotation axes. A neighborhood of each line in the closure F̄ is a neighborhood of (0,0) in ℝ × C, where C is a flat cone of cone angle α; the angle α may assume any value in (0, ∞], the value α = ∞ corresponding to the universal cover of (ℝ^{2} \ {0}).

We close this section with the following useful result. Lemma 4.9. For F as in Proposition 4.8, let t(x) = dist_{g′_{o}}(x, ∂F). Then t is unbounded on F. Proof: The proof is by contradiction, so suppose that T_{o} = sup_{F} t < ∞. By Lemma 3.2, the limit potential ν_{a} is then bounded above on F, ν_{a} ≤ C, for some C < ∞. The potential ν_{a} satisfies the elliptic equation (2.65). Suppose first that Δν_{a} = 0, and consider a maximizing sequence {y_{j}} for ν_{a}, i.e. ν_{a}(y_{j}) → sup_{F} ν_{a}. The maximum principle implies that ν_{a} approches a constant function near y_{j}, for j sufficiently large, which in turn implies that the mass of ν_{a} in B_{y_{j}}(1) approaches 0. However, this contradicts the uniform lower bound (4.39) for the mass, again since t(y_{j}) ≤ T_{o} < ∞.

If ν_{a} is not harmonic, then Theorem 2.11 implies that ν_{a} is superharmonic. Let h be the harmonic function on F determined by the Riesz measure dμ_{∞} on ∂F. Since ν_{a} is superharmonic, h ≤ ν_{a} ≤ C. Then exactly the same argument as above with h in place of ν_{a}, gives the same contradiction.

5. Mass Gaps and Essential Spheres.

This is the most important part of the paper, and culminates in Theorems 5.9 and 5.12, which prove the existence of essential 2-spheres in M under two mild hypotheses. The first of these is the degeneration hypothesis (4.1) and the second, (which has two parts), is a non-collapse hypothesis.

In §5.1, we describe the non-collapse hypothesis (first part), and derive some elementary consequences for the structure of the level set L_{o} of admissible base points. Next, the work in §5.2 proves that if x_{ε} is any admissible base point, then there exists a well-separated base point y_{ε} on the same level L_{o}, (naturally constructed from the initial x_{ε}), c.f. Theorem 5.4. This crucial result is obtained by examining the mass distribution on the level L_{o} near x_{ε}, using the global mass estimate from Theorem 3.5 in combination with the local mass bound from Proposition 4.7.

In §5.3, it is then shown that well-separated base points are naturally associated to 2-spheres in M, under the 2^{nd} non-collapse hypothesis, c.f. Theorem 5.9. This leads to Theorem 5.12, proving that such 2-spheres are essential in M.

It is assumed throughout §5 that M is σ-tame. Thus, the sequence (Ω_{ε}, g_{ε}), ε = ε_{i}, is chosen to satisfy (4.3).

5.1. Theorem 4.4 gives the existence of special base points on (Ω_{ε}, g_{ε}), namely admissible or preferred. In this subsection, we derive some elementary consequences for the geometry near such base points, in the presence of the following assumption.

Non-Collapse Assumption I. There exists an admissible or preferred base point x_{ε} and a constant μ_{o} > 0, independent of ε, such that for all p_{ε} ∈ L_{o} ∩ B_{x_{ε}}(1), or p_{ε} = x_{ε} respectively,

(5.1)  \frac{volB_{p_{ε}}(ρ(p_{ε}))}{ρ(p_{ε})^{3}} ≥ μ_{o}.

where L_{o} is the level set through x_{ε}, as in (4.34).

The bound (5.1) means that (Ω_{ε}, g_{ε}) is not volume collapsing within ρ-balls centered at p_{ε}, i.e. the rescaled metrics (Ω_{ε}, g′_{ε}, p_{ε}), g′_{ε} = ρ(p_{ε})^{−2} ·g_{ε}, have a uniform lower bound on the volume radius at p_{ε}, and so one has convergence to the limit (F, g′_{o}, p), (in a subsequence). It is clear that (5.1) implies that there exists a_{o} = a_{o}(μ_{o}) > 0 such that, for H_{p_{ε}}(r) = B_{p_{ε}}(r) ∩ L_{o},

(5.2)  \frac{areaH_{p_{ε}}(ρ(p_{ε}))}{ρ(p_{ε})^{2}} ≥ a_{o}.

As always, (5.2) is understood to hold for generic points p_{ε}, as in (2.79), (although it follows from (5.5) below that it holds in fact for all p_{ε}). The ball B_{x_{ε}}(1) may be replaced by B_{x_{ε}}(η), for any fixed η > 0, throughout §5. In §6, we will deal separately with the possibility of collapse, where (5.1) does not hold, (for all admissible or preferred base points). The bound (5.1) is part of the assumptions of all of the results to follow in §5, but will not be mentioned explicitly each time.

Referring to Proposition 4.7, the following Corollary is now obvious. Corollary 5.1. Let x_{ε} be an admissible or preferred base point satisfying (5.1). There is a constant κ_{1} > 0, depending only on κ_{o} and μ_{o}, and in particular independent of ε, such that for any q_{ε} ∈ L_{o} ∩ B_{x_{ε}}(1), respectively q_{ε} = x_{ε},

(5.3)  m_{o}(ρ(q_{ε})) ≥ κ_{1} · ρ(q_{ε}).

Proof: This is an immediate consequence of (4.35) and (5.1), c.f. (4.36).

Remark 5.2. Observe that under the assumption (5.1), one has

(5.4)  ρ(q_{ε}) ≤ \frac{1}{2}κ_{1}^{−1}δ_{o} = \frac{1}{2}κ_{1}^{−1}ε̄^{2μ} → 0, as ε → 0,

for all q_{ε} ∈ L_{o}∩B_{x_{ε}}(1) or q_{ε} = x_{ε} respectively. To see this, by (5.3), the mass of ν_{δ_{o}} in B_{q_{ε}}(ρ(q_{ε}))∩L_{o} is at least κ_{1}ρ(q_{ε}). On the other hand, Theorem 3.5 implies that the total mass m_{o} of L_{o} is bounded by \frac{1}{2}δ_{o}, which gives (5.4). In particular, by (4.5), it follows that

(5.5)  ρ^{2}(q_{ε}) << δ_{a}(q_{ε}),

corresponding to Case (i) of (2.12). In the opposite direction, Proposition 1.4 gives a uniform lower bound on ρ, i.e. ρ^{2} ≥ ε̄ everywhere and so in particular on L_{o}.

Theorem 3.5 and (5.3) also combine to give the following result. Corollary 5.3. Let x_{ε} be an admissible or preferred base point and let C_{o} be any component of L_{o} ∩ B_{x_{ε}}(1), respectively L_{o} ∩ B_{x_{ε}}(ρ(x_{ε})). Then, ∀p_{ε} ∈ C_{o}, there exists r_{o} > 0 s.t.

(5.6)  diam_{g_{ε}}C_{o} ≥ r_{o} · ρ(p_{ε}),

where diam is the extrinsic diameter in (Ω_{ε}, g_{ε}), and r_{o} is independent of C_{o} and ε. Hence

(5.7)  diam_{g_{ε}}C_{o} ≥ r_{1}ε̄^{1/2}.

Further,

(5.8)  diam_{g_{ε}}C_{o} ≤ 5κ_{1}^{−1}ε̄^{2μ} → 0, as ε → 0.

Proof: The proof is the same in the admissible and well-separated cases, so assume x_{ε} is admissible. The estimate (5.6) follows exactly as in (2.79) from the uniform non-triviality of the limit potential ν_{a}. Note that (5.5) implies that the limit potential is harmonic, and so has no local maxima, (or minima). Thus (2.79) and (5.6) hold for all p_{ε} ∈ C_{o}, and not just generic p_{ε} in the sense of (2.78). The estimate (5.7) is an immediate consequence of (5.6) and Proposition 1.4.

To prove (5.8), consider the cover U of C_{o} by ρ-balls B_{z}(ρ(z)), z ∈ C_{o}. By a standard covering argument, c.f. [10, B33.3] or [13, §2.1], one may select a finite subcollection of disjoint balls B_{z_{j}}(ρ(z_{j})) such that the balls B_{z_{j}}(5ρ(z_{j})) cover C_{o}. Then

(5.9)  diam_{g_{ε}}(C_{o}) ≤ 5 ∑ ρ(z_{j}) ≤ 5κ_{1}^{−1} ∑ m_{o}(B_{z_{j}}(ρ(z_{j}))) ≤ 5κ_{1}^{−1}ε̄^{2μ},

where the second estimate uses (5.3) and the last uses (3.22) and (4.6).

Of course, combining (5.7) and (5.8) gives an upper bound, depending on ε, on the number of components of L_{o} in the ball B_{x_{ε}}(1) ⊂ (Ω_{ε}, g_{ε}).

It is worthwhile to keep in mind that Corollary 5.3 does not assert that diam_{g_{ε}}C_{o} ∼ ρ(q_{ε}), for some q_{ε} ∈ C_{o}. In fact, it may well be the case that for all q_{ε} ∈ C_{o}, diam_{g_{ε}}C_{o} >> ρ(q_{ε}). Nevertheless, Corollary 5.3 does show that all the components of L_{o} in B_{x_{ε}}(1) are being crushed (metrically) to points as ε → 0 and not to more complicated spaces. However, the distribution of these points might be very complicated; for instance they could apriori become arbitrarily dense in B_{x_{ε}}(1) as ε → 0, compare with Remark 1.3.

5.2. We now come to perhaps the central technical result of the paper, proving the existence of gaps of a definite relative size (independent of ε) in the level set L_{o} near an admissible base point. This result implies that if x_{ε} is admissible, then there is a well-separated base point y_{ε} ∈ L_{o} near to x_{ε}, c.f (4.7); in fact x_{ε} is well-separated w.r.t. all of Ω_{ε}, and not only w.r.t. U^{υ_{o}}. This allows one to pass from admissible base points to well-separated base points. Theorem 5.4. For any admissible base point x_{ε} ∈ L_{o} = L_{o}(ε) there exist points y_{ε} ∈ B_{x_{ε}}(1) ∩ L_{o} and scales λ_{ε} = λ_{ε}(y_{ε}) ∈(0, 1), λ_{ε} ≥ r_{o} · ρ(y_{ε}), for a fixed r_{o} > 0, such that

(5.10)  L_{o} ∩ A_{y_{ε}}(λ_{ε}, (1 + d_{o})λ_{ε}) = ∅,

where d_{o} = d_{o}(μ) > 0 is independent of ε and y_{ε}. Proof: Given an admissible base point x_{ε} ∈ L_{o}, consider the unit ball B = B_{x_{ε}}(1) w.r.t. g_{ε}. To each component C_{j} of L_{o} contained in B associate a subinterval I_{j} = [a_{j}, b_{j}] of I = [0, 1] by setting

a_{j} = min{r : C_{j} ∩ B_{x_{ε}}(r) ≠ ∅}, b_{j} = max{r : C_{j} ∩ B_{x_{ε}}(r) ≠ ∅}.

By (5.6) and (5.7), the length l(I_{j}) satisfies

(5.11)  l(I_{j}) = b_{j} − a_{j} ≥ r_{o}ρ(q_{j}) ≥ r_{1}ε̄^{1/2}, for some q_{j} ∈ C_{j}.

Further, to each I_{j} associate a mass m by setting m(I_{j}) ≡ m_{o}(C_{j}). Then the same estimates as in (5.9) give

(5.12)  m(I_{j}) ≥ κ_{2}l(I_{j}),

where κ_{2} = κ_{2}(κ_{1}). The intervals I_{j} may of course intersect, but this will only add further to the mass. Since the total mass of L_{o}(1) ≡ L_{o} ∩ B_{x_{ε}}(1) is small, i.e.

(5.13)  m(L_{o}(1)) ≤ ε̄^{2μ},

by (3.22), the total Lebesque measure of I_{o} ≡ ∪I_{j} is very small in I. Hence there exist gaps between the intervals I_{j}. However, there may be very many, very small gaps. To estimate the gap size, we essentially consider I_{o} as obtained from I by the usual Cantor set construction of deleting open intervals from I.

Thus, consider I_{o} as obtained from I by inductively dividing intervals, starting at I, into two parts obtained by removing an open interval of relative length ≤ d_{o}; the parameter d_{o} will be estimated below. Initially then, remove an open interval of length d_{1} from I, so there are left two closed intervals, of total length (1 − d_{1}). Remove from either or both of these again an interval of total relative length d_{2}, so that total length of removed intervals is then d_{1}+d_{2}(1−d_{1}) = 1−(1−d_{1})(1−d_{2}). Repeat this process inductively N times, until the collection of intervals I_{o} is reached.

A straightforward computation then shows that the total length of the removed intervals equals 1 − ∏_{1}^{N}(1 − d_{i}). It follows that the total length of the remaining closed intervals, and hence the total mass of I_{o} by (5.12), satisfies

(5.14)  m(I_{o}) ≥ κ_{2} ∏_{1}^{N}(1 − d_{i}).

Let d̄= sup_{i} d_{i}, so that (5.14) gives

m(I_{o}) ≥ κ_{2}(1 − d̄)^{N}.

Then from (5.13) one obtains the bound

(5.15)  (1 − d̄)^{N} ≤ κ_{2}^{−1}ε̄^{2μ}.

On the other hand, since each repetition of the process above produces a new closed interval of length < \frac{1}{2} that of its predecessor, some of the final closed intervals I_{j} in I_{o} must have length ≤ 2^{−N}. But by (5.11), the length of each interval I_{j} is at least ε̄, (in fact ε̄^{1/2}). Hence, one also has

(5.16)  ε̄ ≤ 2^{−N}.

An elementary calculation combining (5.15) and (5.16) then gives

d̄> 1 − κ_{2}^{−1/N}2^{−2μ}.

This implies that there are gaps in I_{o} at some scale, of relative size at least

(5.17)  d_{o} = 1 − κ_{2}^{−1}2^{−2μ}.

By construction, this means that there exist points y_{ε} ∈ L_{o} ∩ B_{x_{ε}}(1) and numbers λ_{ε} ∈ (0, 1] such that (5.10) holds. The fact that λ_{ε} ≥ r_{o} · ρ(y_{ε}) follows from (5.6).

Remark 5.5. It is clear from the proof of Theorem 5.4 that there could be many possible choices of base points y_{ε} and factors λ_{ε} satisfying (5.10). For the purposes of this paper, this lack of uniqueness, already occuring in §4, plays no fundamental role.

However, it is not asserted that there exist gaps forming as in (5.10) at some definite or determinable scale, (although see Lemma 5.6). For instance, it is not claimed that one can choose λ_{ε} to be bounded away from 0 as ε → 0, i.e. that there are gaps forming in L_{o} of a definite size w.r.t. g_{ε}, compare with Remark 1.3. At the other, small-scale extreme, it is not known that gaps form on the scale of ρ(y_{ε}), i.e. whether λ_{ε} ∼ ρ(y_{ε}), as is the case for preferred base points.

It is the existence of such gaps provided by Theorem 5.4 which is important, and not so much the scale at which they are visible. Without the σ-tame assumption in Theorem 5.4 the size of all relative gaps could approach 0 as ε → 0, as in the construction of Cantor sets of measure 0 in [0, 1] with Hausdorff dimension 1, c.f. [13, §4.10]. This is in fact the only place in the paper where the σ-tame assumption is used in an essential way; (the arguments in §4 and in §5 preceding Theorem 5.4 can be modified to hold without the σ-tame assumption).

The following (more technical) observation based on the proof of Theorem 5.4 will be used later in the final choice of base points.

Lemma 5.6. Let x_{ε} be any admissible base point. Then there exist well-separated base points y_{ε} ∈ B_{x_{ε}}(1) ∩ L_{o} satisfying (5.10), together with

(5.18)  λ_{ε} ≤ ε̄^{3μ/2},

and such that

(5.19)  m_{o}(B_{y_{ε}}(λ_{ε})) ≥ ε̄^{μ/2}λ_{ε}.

Proof: To prove (5.18), observe that the proof of Theorem 5.4 began with the ball B_{x_{ε}}(1). However, it can be applied equally well to balls B_{x_{ε}}(s_{ε}), with s_{ε} → 0 sufficiently slowly as ε → 0. For example, let s_{ε} = ε̄^{3μ/2} and rescale the balls B_{x_{ε}}(s_{ε}) to size 1, i.e. set ḡ_{ε} = s_{ε}^{−2} · g_{ε}. Since the mass scales as a distance, m̄_{o}((B_{x_{ε}}(1)) ≤ ε̄^{−3μ/2}ε̄^{2μ} ≤ ε̄^{μ/2}, where m̄_{o} is the mass m_{o} w.r.t the metric ḡ_{ε}. Now starting with (5.13), with ε̄^{μ/2} in place of ε̄^{2μ}, the same arguments as before in Theorem 5.4 apply to give (5.10), with λ_{ε} satisfying (5.18). (The estimate for 1 − d_{o} is smaller than that in (5.17) by a factor of 2).

The proof of (5.19) is similar. Thus, let A_{y_{ε}}(λ_{ε}) be an annulus satisfying (5.10) and (5.18) and consider the rescaled metrics ḡ_{ε} = λ_{ε}^{−2} · g_{ε}. If there is any fixed constant κ > 0, say κ = μ/2, s.t.

(5.20)  m̄_{o}(B̄_{y_{ε}}(1)) ≤ ε̄^{κ},

then exactly the same arguments as in the proof of (5.18) above may be repeated at this scale to produce a new smaller annulus A_{y_{ε}^{1}}(λ_{ε}^{1}) ⊂ B_{y_{ε}}(λ_{ε}), λ_{ε}^{1} << λ_{ε}, satisfying the gap condition (5.10), with λ_{ε}^{1} in place of λ_{ε}; as above, here d_{o} = d_{o}(κ) = d_{o}(μ). In this new scale g_{ε}^{1} = (λ_{ε}^{1})^{2}g_{ε}, the mass is of course much larger. Hence, by iterating this procedure a finite number of times, (depending on ε), one obtains base points y_{ε} and scale-factors λ_{ε} satisfying (5.10) and (5.18)-(5.19).

Theorem 5.4 and Lemma 5.6 prove the existence of well-separated base points, (of course under the assumptions (4.1), (5.1) and the σ-tame assumption), satisfying the properties (5.10) and (5.18)-(5.19). Preferred base points automatically satisfy (5.18)-(5.19); for such base points, since λ ∼ ρ, (5.18) follows from (5.4) while (5.19) follows from (5.3). From now on, it is assumed that the well-separated base point y_{ε} has been chosen to satisfy (5.18)-(5.19). One further observation is needed for the final determination of the base point.

The level L_{o} disconnects the manifold Ω_{ε} into two parts,

(5.21)  Ω_{ε} = U_{o} ∪ U^{o},

where U_{o} = {x ∈ Ω_{ε} : u(x) < u(L_{o})}, U^{o} = {x ∈ Ω_{ε} : u(x) > u(L_{o})}; of course neither of these sets is necessarily connected. Hence each component C_{y_{ε}}(λ_{ε}) of any annulus A_{y_{ε}}(λ_{ε}, (1 + d_{o})λ_{ε}) as in Theorem 5.4 is contained either in U_{o} or U^{o}.

The next result asserts the existence of such components contained in U^{o}. Lemma 5.7. Let A_{y_{ε}}(λ_{ε}, (1 + d_{o})λ_{ε})) be any annulus satisfying (5.10), with λ_{ε} satisfying (5.18). Then there is a component C_{ε} = C_{y_{ε}}(λ_{ε}) of A_{y_{ε}}(λ_{ε}, (1 + d_{o})λ_{ε}) such that

(5.22)  C_{ε} ⊂ U^{o}.

Proof: Let s_{ε} = (1 + \frac{d_{o}}{2} )λ_{ε}. By (5.10), the ball B(s_{ε}) = B_{y_{ε}}(s_{ε}) satisfies ∂B(s_{ε}) ∩ L_{o} = ∅, so that any component of ∂B(s_{ε}) is then contained either in U_{o} or U^{o}. It suffices then to prove that some component of ∂B(s_{ε}) is contained in U^{o}.

To see this, suppose instead the full boundary ∂B(s_{ε}) ⊂ U_{o}. Then some part of the level set L_{o} ∩ B(s_{ε}) bounds a connected domain D = D_{ε} ⊂ U^{o}, with D ⊂⊂ B(s_{ε}), ∂D ⊂ L_{o}. Recall that by (5.4), ρ ≤ ε̄^{2μ} on L_{o} ∩ B_{y_{ε}}(\frac{1}{2}), (ignoring constant factors here). Hence, by the Lipschitz property of ρ, ρ ≤ ε̄^{3μ/2} everywhere in B(s_{ε}), and so, at all base points in D,

(5.23)  ρ^{2}/δ_{o} → 0.

Choose a point z_{ε} ∈ D which realizes the maximal value of u on D, and work in the scale g′_{ε} where ρ′(z_{ε}) = 1. Since ρ ∼ dist(L^{υ_{o}}, ·) and L^{υ_{o}} lies outside D, it follows that dist_{g′_{ε}}(z_{ε}, L_{o}) ≤ 2 for ε sufficiently small. Consider the potential function ν_{δ_{o}} = (u − 1)/δ_{o}. If the mass m_{o} of ν_{δ_{o}} w.r.t. g′_{ε} is uniformly bounded over B′_{z_{ε}}(1/2), then ν_{δ_{o}} subconverges, modulo constants, to a limit ν_{o}. If the mass m_{o} is very large, then renormalize it by dividing by m_{o}(B′_{z_{ε}}(1/2)). (This has effect of increasing δ). In both cases, the estimate (5.23) shows that Case (i) of (2.12) holds and so Proposition 2.3 implies that the limit function ν_{o} satisfies Δν_{o} = 0. Since, ν_{o} has a maximal value at z = lim z_{ε}, it follows that ν_{o} is constant.

However, the estimate (4.35) on L_{o} passes to the limit, and shows that the limit potential ν_{o} has non-zero mass on L_{o}. This gives a contradiction.

Lemma 5.7 applies equally well to preferred base points; such base points are only well-separated w.r.t. U^{υ_{o}}, but Lemma 5.7 implies that there is a full component C_{ε} of the annulus A^{υ_{o}} contained in U^{o} and hence C_{ε} ∩ L^{υ_{o}} = ∅.

5.3. We are now ready to to fix the choice of the base point and prove, under a 2^{nd} natural non-collapse assumption, that such base points give rise to geometrically natural 2-spheres in M. Definition 5.8. A distinguished base point is a base point q_{ε} ∈ C_{ε} as in (5.22), with q_{ε} ∈ A_{y_{ε}}((1 + \frac{1}{4}d_{o})λ_{ε}, (1 + \frac{3}{4}d_{o})λ_{ε}), where y_{ε}, λ_{ε} satisfy the conclusions of Lemma 5.6.

For the remainder of §5, we work with a choice of distinguished base point. Again there are apriori many possible choices here, and the results to follow in §5.3 apply to any such choice. To any distinguished base point q_{ε} is associated the (center) base point y_{ε} and component C_{ε} ⊂ U^{o} of A_{y_{ε}}(λ_{ε}, (1 + d_{o})λ_{ε}), with q_{ε} ∈ C_{ε}.

Consider the rescaled metric

(5.24)  ḡ_{ε} = λ_{ε}^{−2} · g_{ε},

so that A_{ε}(λ_{ε}) = A_{y_{ε}}(λ_{ε}, (1 + d_{o})λ_{ε}) now becomes Ā_{ε}(1) = Ā_{y_{ε}}(1, 1 + d_{o}) and B_{ε}(λ_{ε}) = B_{y_{ε}}(λ_{ε}) becomes B̄_{ε}(1) = B̄_{y_{ε}}(1). For C_{ε} ⊂ Ā_{ε}(1) as above satisfying (5.22), Proposition 1.1 implies

(5.25)  ρ̄(p_{ε}) ≥ d_{o}/4,

for all p_{ε} ∈ Ā_{ε}(1 + \frac{1}{4}d_{o}, 1 + \frac{3}{4}d_{o}) ∩ C_{ε}. On the other hand, since λ ≥ r_{o} · ρ(y_{ε}),

ρ̄(y_{ε}) ≤ r_{o}^{−1},

and hence by the Lipschitz property of ρ,

ρ̄(q_{ε}) ≤ 2r_{o}^{−1}.

Thus, the rescaling (5.24) is uniformly homothetic to the natural blow-up metric based at q_{ε}, i.e.

(5.26)  g′_{ε} = ρ(q_{ε})^{−2} · g_{ε}.

We now make the following 2^{nd} non-collapse assumption.

Non-Collapse Assumption II. For some distinguished base point q_{ε}, there exists μ_{o} > 0 and some sequence K_{ε} → ∞, (slowly) as ε → 0, such that, for all 1 ≤ K ≤ K_{ε}

(5.27)  \frac{B_{q_{ε}}(Kρ(q_{ε}))}{(Kρ(q_{ε}))^{3}} ≥ μ_{o}.

The condition (5.27) means that the rescalings (5.26) based at q_{ε} do not collapse on large scales in the metric (5.26), i.e. ω′(q_{ε}) >> 1, as ε → 0. Although similar, the assumption NCA II does not quite follow from NCA I, since q_{ε} ∉ L_{o} and the collapse is on scales large compared with ρ(q_{ε}). However, the assumption NCA II is much weaker than NCA I, since it applies to the behavior at a single base point q_{ε}, while NCA I applies to all p_{ε} ∈ B_{x_{ε}}(1) ∩ L_{o}.

Under the NCA II, the pointed sequence (Ω_{ε}, g′_{ε}, q_{ε}) (sub)-converges to a maximal flat limit (F, g′_{o}, q). Similarly, the rescalings (Ω_{ε}, ḡ_{ε}, q_{ε}) (sub)-converge to a maximal flat limit (F, ḡ_{o}, q), and (F, ḡ_{o}) is homothetic to (F, g′_{o}). Recall that the convergence of the blow-ups (Ω_{ε}, g′_{ε}, q_{ε}) to the limit (F, g′_{o}, q) is smooth away from ∂F and that ∂F is formed by the limits of points z_{ε} where the curvature is concentrating at a higher rate, i.e. ρ′(z_{ε}) → 0, c.f. the discussion preceding (1.23). Of course ∂F ≠ ∅ and by Proposition 2.1, ∂F is the q_{ε}-based Hausdorff limit of the level L^{υ_{o}}; the same holds w.r.t. (F, ḡ_{o}). The center points y_{ε} converge to a point y ∈ F̄ with q ∈ B̄_{y}(1 + d_{o}), while the domains C_{ε} converge to the domain C_{o} ⊂ Ā_{y}(1).

The results above now combine easily to give the existence of natural 2-spheres. Theorem 5.9. Let q_{ε} be a distinguished base point of (Ω_{ε}, g_{ε}) satisfying the non-collapse assumption NCA II and let (F, g′_{o}, q) be a maximal flat blow-up limit constructed above. Then there is a nonempty compact subset Σ_{o} of ∂F, such that either ∂F \ Σ_{o} = ∅ or

(5.28)  dist_{g′_{o}}(Σ_{o}, ∂F \ Σ_{o}) ≥ d_{o} > 0.

Further, C_{o} is embedded in ℝ^{3},

(5.29)  C_{o} ⊂ ℝ^{3},

and C_{o} is isometric to a spherical annulus S^{2} × I about a point in ℝ^{3}. Proof: For convenience we work with the metric (F, ḡ_{o}) in place of (F, g′_{o}); as noted above, these metrics are homothetic. Returning to the sequence (Ω_{ε}, ḡ_{ε}) as in (5.24), suppose first that the annuli Ā_{ε}(1, 1 + d_{o}) are connected, (in a subsequence); more generally, one may suppose that Ā_{ε}(1, 1 + d_{o}) ⊂ U^{o}. Then we claim that L^{υ_{o}} ∩ B̄_{ε}(1) ≠ ∅, and so, (by Proposition 1.1 as usual),

(5.30)  ρ̄ → 0,

somewhere within B̄_{ε}(1). For if (5.30) does not hold, then L^{υ_{o}} lies outside B̄_{ε}(1), and ḡ_{ε} converges smoothly on B̄_{ε}(1) to the limit B̄_{y}(1). Since Ā_{ε}(1, 1 + d_{o}) ⊂ U^{o}, L_{o} ⊂ B̄_{ε}(1) bounds a domain D_{ε} in B̄_{ε}(1) and hence ν has a minimal value in D_{ε}. This gives the same contradiction as in Lemma 5.7 above.

Thus, a singularity forms at a point z_{ε} ∈ B̄_{ε}(1), i.e. there are points z_{ε} within B̄_{ε}(1) of much higher curvature concentration than that within C_{ε}. In particular, there is a non-empty component of ∂F within the limit B̄_{y}(1) of B̄_{ε}(1). As noted above, ρ̄ is bounded away from 0 in C_{ε}, so that

(5.31)  C_{o} ∩ ∂F = ∅.

Suppose instead L^{υ_{o}} ∩ B̄_{ε}(1) = ∅, so that the balls B̄_{ε}(1) converge smoothly to the limit B̄_{y}(1) and surrounding annulus Ā_{y}(1, 1 + d_{o}) in F. This limit annulus is not connected, since the annuli Ā_{ε}(1, 1 + d_{o}) are not connected. Hence, the limit B̄_{y}(1) is a geodesic ball in a smooth flat manifold, with disconnected boundary. This can only occur if B̄_{y}(1) is isometric to a ball in a flat product F_{prod} = ℝ × S^{1} × S^{1}. Now by Lemma 4.9, the distance function t to ∂F, (w.r.t. ḡ_{o}), is unbounded on F, while by (2.19), t = ρ on F. Hence, one may choose a sequence of points y_{i} with t(y_{i}) → ∞, approximate them by points y_{ε_{i}} ∈ (Ω_{ε_{i}}, g_{ε_{i}}), and consider the rescalings based at y_{ε_{i}}. This has the effect of blowing down F. However, the flat product structure of (F, ḡ_{o}) on B̄_{y}(1) extends to all of (F, ḡ_{o}) and thus it follows that the rescalings based at y_{ε_{i}} are collapsing. This contradicts the NCA II, (5.27). Actually, to apply Lemma 4.9, one needs to know that |u − 1|(q_{ε}) ≥ ε̄^{4μ}. By (4.44), it suffices to show that δ_{a}(q_{ε}) ≥ ε̄^{4μ}, which follows from (5.41) below; (the proof of (5.41) is independent of Theorem 5.9).

Thus Σ_{o} = ∂F ∩ B̄_{y}(1) is non-empty and the estimate (5.28), (w.r.t. ḡ_{o}), follows as in (5.25). The domain C_{o} from (5.31) is a connected geodesic annulus about the point y ∈ F̄, separating Σ_{o} from the rest of ∂F.

Suppose now that B̄_{y}(1 + d_{o}) is simply connected. Then the developing map D is an isometric immersion of B̄_{y}(1+d_{o}) into the ball B_{0}(1+d_{o}) about 0 in ℝ^{3}. To prove (5.29) and the fact that C_{o} is a spherical annulus, it suffices to prove that D is an embedding, or equivalently D has no holonomy on B̄_{y}(1 + d_{o}). As in the discussion following Proposition 4.8, each holonomy transformation is either a rotation, twist or translation. Now D can have no rotational holonomy, generated by rotation about an axis within B̄_{y}(1 + d_{o}). For such an axis, necessarily a line, must be contained in ∂F, which is impossible by the gap estimate (5.28). Hence the only holonomy transformations are those given by twists or translations. These both introduce non-trivial loops, and thus B̄_{y}(1 + d_{o}) could not be simply connected.

On the other hand, if B̄_{y}(1 + d_{o}) is not simply connected, then the arguments above imply that the developing map D maps F into a non-trivial quotient ℝ^{3}/Γ, for some Γ acting freely and isometrically on ℝ^{3}; hence Γ = ℤ or ℤ ⊕ ℤ acts by twists and/or translations. Via Lemma 4.9 as above, this contradicts the NCA II. This completes the proof.

Since the convergence of (Ω_{ε}, g′_{ε}, q_{ε}) to the limit (F, g′_{o}, q) is smooth away from ∂F, there are 2-spheres S^{2} = S_{ε}^{2} embedded in (Ω_{ε}, g′_{ε}) converging smoothly to the limit S^{2} ⊂ C_{o} ⊂ (F, g′_{o}). The metric g′_{ε} is just a rescaling of g_{ε}, and so the metrics g_{ε} are crushing the 2-spheres S_{ε}^{2} to points. Of course, since Ω_{ε} is weakly embedded in M, the spheres S_{ε}^{2} are also embedded in M, i.e.

(5.32)  S_{ε}^{2} ⊂ M.

It follows from Theorem 5.9 that the 2-sphere S_{ε}^{2} ⊂ Ω_{ε}, although it converges to a point as ε → 0, surrounds the formation of a singularity, in that ”within” S_{ε}^{2} there are regions of much higher curvature concentration than the curvature concentration on S_{ε}^{2} itself.

5.4. In this subsection, we prove the central result of the paper, Theorem 5.12, stating that the 2-spheres S_{ε}^{2} constructed above in (5.32) are essential 2-spheres in M.

To begin, one needs to understand the behavior of the potential functions ν_{δ} on the blow-ups g′_{ε} and the limit F in Theorem 5.9. To do this, recall that the spheres S_{ε}^{2} are contained in the connected region C_{ε} = A_{ε}(λ_{ε}) = A_{y_{ε}}(λ_{ε}, (1 + d_{o})λ_{ε}), where q_{ε} is a distinguished base point; the annulus is connected by the proof of Theorem 5.9. In particular, (5.18)-(5.19) hold. Let m′_{o} be the mass of the potential ν_{δ_{o}} = (u − 1)/δ_{o}, δ_{o} = ε̄^{2μ}, on the level set L_{o} containing y_{ε}, w.r.t. g′_{ε} in (5.26). Also let B′_{ε} = B′_{y_{ε}}(λ′_{ε}), where λ′_{ε} = λ_{ε} · ρ(q_{ε})^{−1}. Since the metrics ḡ_{ε} and g′_{ε} in (5.24) and (5.26) are uniformly homothetic, λ′_{ε} → λ′_{o} ∈ (0, ∞), as ε → 0.

Now there are three possibilities for the behavior of m′_{o}(B′_{ε}), as ε → 0, namely

(i) m′_{o}(B′_{ε}) → ∞,
(5.33)  (ii) n_{o} ≤ m′_{o}(B′_{ε}) ≤ n_{o}^{−1},
(iii) m′_{o}(B′_{ε}) → 0,

where n_{o} is some positive constant. Observe that in case (iii), (5.19) implies m′_{o}(B′_{ε}) ≥ ε̄^{μ/2}. On the other hand, one has the natural potential function w.r.t. the base point q_{ε}, i.e. as in (2.17),

(5.34)  ν_{δ(q_{ε})} = (u − 1)/δ(q_{ε}),

and its associated mass function m_{q_{ε}} on L_{o}, again in the g_{ε} scale. It is important to note that here we are using the ’curvature’ δ as in (2.10), and not δ_{a} from (3.1), used throughout §4 and so far in §5. Of course, for any fixed ε, the potentials ν_{δ_{o}} and ν_{δ(q_{ε})} are proportional, as are the mass functions.

We claim first that the two potentials and mass functions above have a uniformly bounded proportionality constant, when (5.33)(ii) holds. Lemma 5.10. If (5.33)(ii) holds, then there is a constant n_{1} = n_{1}(n_{o}) > 0, independent of ε, s.t.

(5.35)  n_{1}δ_{o} ≤ δ(q_{ε}) ≤ n_{1}^{−1}δ_{o}.

Proof: By assumption, the mass m′_{o} of ν_{δ_{o}} in B′_{ε} is on the order of 1. On the other hand, the mass m_{δ_{a}} of ν_{δ_{a}}, δ_{a} = δ_{a}(q_{ε}), is also on the order of 1 in B′_{ε}; this follows from (3.13), the NCA II in (5.27), together with Proposition 3.3. Hence, from the definitions of the masses, it follows that

(5.36)  δ_{a} ∼ δ_{o},

i.e. the ratio is uniformly bounded away from 0 and ∞ as ε → 0. Now recall from (3.2)-(3.3) that δ_{a} ≥ c_{1}·δ and δ_{a} >> δ if and only if the limit ν_{a} of ν_{δ_{a}} is an affine function; here of course δ = δ(q_{ε}). However, an affine function has no gap in the support of the (limit) measure dμ, or equivalently no mass gap. Hence by (5.28) one must have δ_{a} ≤ c_{1} · δ, so that

(5.37)  δ_{a} ∼ δ.

Combining the estimates (5.36)-(5.37) gives (5.35).

Observe that Lemma 5.10 implies that

(5.38)  m_{δ(q_{ε})}(B′_{ε}) ∼ 1, as ε → 0,

so that the part of the level L_{o} in B′_{ε} has a definite amount of mass for the potential ν_{δ(q_{ε})}. Since one has convergence to the limit F, and not collapse, observe that (5.5) holds here also, for the same reasons. Thus,

(5.39)  ρ^{2}(q_{ε}) << δ(q_{ε}),

so that the base points q_{ε} satisfy Case (i) of (2.12).

In the cases (5.33)(i) or (iii), one needs to renormalize the mass by setting

(5.40)  δ_{1} = δ_{o} · m′_{o}(B′_{ε}),

so that the mass of the potential ν_{δ_{1}} = (u − 1)/δ_{1} in B′_{ε} is on the order of 1. By the same reasoning as in Lemma 5.10, it then follows that δ_{1} ∼ δ_{a}(q_{ε}) ∼ δ(q_{ε}). We claim that in both cases (5.39) still holds. In the situation of (5.33)(i), this is obvious, since then δ(q_{ε}) ≥ δ_{o}. Suppose instead (5.33)(iii) holds. Then applying the estimate (5.19) to (5.40) gives

(5.41)  δ_{1} ≥ ε̄^{2μ+μ/2}.

On the other hand, on L_{o} ∩ B_{y_{ε}}(\frac{1}{2}), in the g_{ε} scale, one has ρ ≤ ε̄^{2μ} by (5.4). Since q_{ε} ∈ B_{y_{ε}}((1 + d_{o})λ_{ε}) and λ_{ε} ≤ ε̄^{3μ/2} by (5.18), the Lipschitz property of ρ, c.f. §1, thus gives

ρ(q_{ε}) ≤ ε̄^{3μ/2}.

Hence, for ρ = ρ(q_{ε}),

(5.42)  \frac{ρ^{2}}{δ_{1}} ≤ \begin{matrix} & ε̄^{3μ} \\  & ε̄^{2μ+μ/2} \end{matrix} → 0.

These arguments thus imply that the distinguished base points q_{ε} necessarily satisfy Case (i) of (2.12), i.e. satisfy the conclusions of Proposition 2.3. Further, (5.37) holds, so that the potentials ν_{δ} converge to the limit potential ν modulo constants. In sum, one has: Corollary 5.11. For q_{ε} a distinguished base point as in Theorem 5.9, the linearized equations of the blow-up limit (F, g′_{o}, ν, q) are the linearized static vacuum Einstein equations

(5.43)  r′ = D^{2}ν, Δν = 0.

For ε sufficiently small, and within B_{y}((1 + d_{o})λ′_{o}) ∩ F, λ′_{o} = lim λ′_{ε}ρ(q_{ε})^{−1}, the metric g′_{ε} has the form,

(5.44)  g′_{ε} = (1 − 2νδ)g′_{o} + o(δ),

modulo diffeomorphisms, where g′_{o} is the flat metric on F. The potential function ν is a nonconstant harmonic function and is the limit of ν_{δ(q_{ε})}, mod constants.

We are now in position to prove the main result of the paper; this result constitutes the most important part in the proof of the Main Theorem of §0.

Theorem 5.12. Let (Ω_{ε}, g_{ε}) be a sequence of minimizing pairs for M, with ε = ε_{i} → 0 satisfying (1.36) and suppose the degeneration hypothesis (4.1) and non-collapse assumptions NCA I, NCA II hold on some subsequence.

Then, for ε sufficiently small, the 2-sphere S_{ε}^{2} in M given by (5.32) is essential in M. In particular, M is a reducible 3-manifold. Proof: Suppose that S^{2} = S_{ε}^{2} is inessential in M, i.e. S^{2} bounds a 3-ball in M. By constructing a suitable metric glueing of a 3-ball onto S^{2} in M as a comparison metric, we will obtain a contradiction to the fact that g_{ε} is a minimizer of I_{ε}^{−}, for ε sufficiently small.

Given the preceding results of §5, most all of the work to prove Theorem 5.12 has already been done, explicitly for this purpose, in [1]. First, as in [1, Thm.3.1], the proof needs to be divided into two cases, according to whether or not S^{2} bounds a 3-ball B^{3} on the inside or outside. The 2-sphere S^{2} is said to bound B^{3} on the inside if the component B^{3} of M \ S^{2} contains the center base point y_{ε} associated to the distinguished base point q_{ε}; otherwise S^{2} bounds on the outside.

Since C_{ε} ⊂ U^{o} by (5.22), one has ν_{δ} > ν_{δ}(L_{o}) on C_{ε}, δ = δ(q_{ε}), while Theorem 5.9 implies that L_{o} ∩ B′_{y_{ε}}(λ′_{ε}) ≠ ∅ and L^{υ_{o}} ∩ B′_{y_{ε}}(λ′_{ε}) ≠ ∅. Since λ′_{ε} → λ′_{o} ∈ (0, ∞), the factor λ′_{ε} will be ignored, (i.e. set to 1), in the following; (this is the same as working with ḡ_{ε} in place of g′_{ε}). By Corollary 5.11, the potential functions ν_{δ} converge, modulo constants, to the limit potential ν on (F, g′_{o}) and by Proposition 4.8, ∂F = {ν = −∞}. It follows that there is a level set L′ = L′(ε) of ν_{δ} such that L′ ∩ B′_{y_{ε}}(1 + d_{o}) ⊂⊂ B′_{y_{ε}}(1 + d_{o}), i.e.

(5.45)  L′ ∩ ∂B′_{y_{ε}}(1 + d_{o}) = ∅,

with dist_{g′_{ε}}(L_{o}, L′) ≥ d_{1}, for some fixed d_{1} > 0.

Theorem 5.9 shows that the ball B_{y}(1 + d_{o}) is embedded in ℝ^{3}, and hence L′ ⊂ ℝ^{3} also. Thus L′ bounds a compact domain W ⊂ ℝ^{3}, ∂W = L′, containing the limit ∂F ∩ B′_{y}(1). In particular, ν(x) ≤ ν(L′), for all x ∈ W. Observe also that W is contained in the 3-ball B(1 + d_{o}) ⊂ ℝ^{3}.

Case I. S^{2} bounds on the inside. The proof of this case is given in [1, Thm.4.2], (which was developed just for this purpose). The idea here is to glue the flat metric g′_{o} onto the metric g′_{ε} along the compact level set L′. Namely the 2-sphere S̄^{2} = S^{2}(1 + \frac{1}{2}d_{o}) ⊂ C_{o} ⊂ F ⊂ ℝ^{3} bounds a flat 3-ball B̄^{3} ⊂ ℝ^{3} on the inside. Since S_{ε}^{2} bounds a 3-ball B^{3} ⊂ M on the inside, one may construct a comparison metric ḡ_{ε} to g′_{ε} essentially by replacing g′_{ε}|_{B^{3}} with the flat metric g′_{o}|B̄^{3}. For technical reasons, it is preferable to carry out this replacement argument on (W, L′) in place of (B^{3}, S^{2}). This construction is explained in complete detail in [1, (4.26)ff], and we refer there for details. (Note that the level L′ above corresponds to L_{o} in [1] and the level set results related to (5.45) are the same as the assumptions [1,(4.5)-(4.7)]). The upshot is that, for ε sufficiently small, the comparison metric ḡ_{ε} satisfies I_{ε}^{−}(ḡ_{ε}) < I_{ε}^{−}(g′_{ε}), contradicting the minimizing property of g′_{ε}.

Case II. S^{2} bounds on the outside. The proof in this situation consists of two (related) subcases, both based on the construction of a comparison metric on the outside. The 2-sphere S^{2} in M given by (5.32) bounds a 3-ball in M on the outside, and so in particular S^{2} separates M into two components, M = M′ ∪_{S^{2}} B^{3}, the outside one given by B^{3}. The comparison metrics are essentially the same as those already constructed, in essentially the same circumstances, in [1].

(i). Suppose first that, for some ε > 0 sufficiently small,

(5.46)  B^{3} ∩ ∂F ≠ ∅,

so that there are other components of ∂F outside S^{2} in F. Of course, ∂F ∩C_{o} = ∅ and ∂F ∩∂B^{3} = ∅.

We explain the idea first formally on the limit F, and then pass to the metrics (Ω_{ε}, g_{ε}). The round 2-sphere S^{2} = S^{2}(1 + \frac{d_{o}}{2} ) ⊂ ℝ^{3} isometrically embeds in a round 3-sphere S^{3}(R) of large radius R, R = 10(1 + \frac{d_{o}}{2} ) for instance. The surface S^{2} disconnects S^{3}(R) into a smaller, inside 3-ball B^{−} and a large, outside 3-ball B^{+}, so that, as above, M = M′ ∪_{S^{2}} B^{+}. Let ḡ_{o} be the round metric on B^{+} ⊂ S^{3}(R). This metric joins continuously to the flat metric g′_{o} along the seam S^{2}.

The resulting metric across S^{2} is then C^{o}, but not C^{1}. However, it is clear that this metric may be smoothed in a small neighborhood of the seam to give a smooth metric g̃_{o} with non-negative scalar curvature, s̃_{o} ≥ 0.

Observe that volg̃_{o}B^{+} < ∞, while vol_{g′_{o}}(F \ B(1 + \frac{d_{o}}{2} )) = ∞, (since F is unbounded). Thus, the volume of the comparison metric is much smaller than that of (F, g′_{o}). Similarly, z ≡ 0 on (B^{+}, ḡ_{o}) and the smoothing g̃_{o} has curvature z̃_{o} bounded near the seam, (depending only on the fixed smoothing). On the other hand, by (5.46), the metric g′_{o} formally has |z| = ∞ on ∂F, so that Z^{2} is much larger for (F, g_{o}) than for the comparison g̃_{o}. Finally, since the smoothing g̃_{o} satisfies s̃_{o} ≥ 0, one has (s̃_{o})^{−} ≡ 0, so that the comparison gives no added contribution to S_{−}^{2}. Thus, formally, the comparison metric has a smaller value of I_{ε}^{−} than g′_{o}.

It is now a simple matter to make this formal reasoning rigorous. Thus, choose ε sufficiently small so that g′_{ε} is very close to the limit (F, g_{o}) on a large compact domain in F. In particular g′_{ε} is almost flat away from ∂F. Since (S^{2}, g′_{ε}) has Gauss curvature strictly larger than that of S^{3}(R), the Weyl embedding theorem, c.f. [16], implies that (S^{2}, g′_{ε}) isometrically embeds in S^{3}(R) and hence bounds the domain B^{+} = (B^{+})_{ε} as above. It follows that the metric ḡ_{ε} = g′_{ε} ∪ ḡ_{o}, with g′_{ε} restricted to Ω_{ε} \ B^{3}, ∂B^{3} = S^{2}, is a C^{o} metric on Ω_{ε}. As above, this metric may be smoothed to a metric g̃_{ε}, satisfying (s̃_{ε})^{−} ≥ (s′_{ε})^{−} everywhere. By (5.46), the curvature Z^{2} of g′_{ε} is arbitrarily large, for ε sufficiently small. It then follows by the same arguments as above in the limit case that

(5.47)  I_{ε}^{−}(g̃_{ε}) < I_{ε}^{−}(g′_{ε}),

contradicting the minimizing property of (Ω_{ε}, g′_{ε}).

(ii). Thus, one may assume that

(5.48)  B^{3} ∩ ∂F = ∅.

Hence ∂F is compact and, (by Theorem 5.9),

(5.49)  F = ℝ^{3} \ ∂F,

with ∂F of Hausdorff dimension at most 1. It follows that the harmonic potential ν extends to a global subharmonic function on ℝ^{3} and so has the representation, c.f. [12],

(5.50)  ν(x) = − ∫_{∂F} \frac{1}{|x − y|}dμ_{y} + h,

where h is harmonic on ℝ^{3}, and dμ_{y} is the Riesz measure of ν on ∂F, as preceding Proposition 4.8. Note that all level sets of the Newtonian potential in (5.50) are compact, since ∂F is compact. Similarly, the function ν has a compact level set L′ in B_{y}(1 + d_{o}), by (5.45). It is then elementary to see that h also has a level set with a compact component. Since h is harmonic, this forces h to be constant. Hence, by adding a suitable constant to ν, one has

(5.51)  ν(x) = − ∫_{∂F} \frac{1}{|x − y|}dμ_{y}.

It follows that ν has an expansion of the form

(5.52)  ν = −m/r + O(r^{−2}),

where r(x) = |x| in ℝ^{3} and m > 0. In particular, the metric g′_{ε} has the form (5.44), with ν as in (5.52). The expression (5.51) also gives |D^{k}ν| = O(r^{−k−1}), for all k. (Note that by rescaling down the flat limit F by arbitrarily large factors, one may obtain a new flat blow-up limit F̂ with ∂F̂ = pt with ν̂ = −m/r; this new limit corresponds formally to blow-ups with an asymptotically flat end, c.f. Remark 1.6).

The argument in this case is again by a glueing to a large 3-ball on the outside, similar to (i) above. This situation is a little more complicated however, since there are no components to ∂F outside S^{2}, so that the comparison of the Z^{2} term is more delicate. However, the construction of a comparison metric g̃_{ε} satisfying (5.47) in this case is exactly that already given in Case II of [1, Thm.3.1], and so we refer there for further details.

In both cases one thus has a contradiction, completing the proof.

Theorem 5.12 implies that if M is σ-tame and irreducible then on some sequence of minimizing pairs (Ω_{ε_{i}}, g_{ε_{i}}) satisfying (4.3), either the degeneration assumption (4.1) fails, or one of the noncollapse assumptions NCA I or NCA II fails. To complete the proof of the Main Theorem, it remains to understand the situation where one of these assumptions fails. This is done in the final two sections.

6. The Collapse Situation.

In this section, we analyse in general the situation where there is collapse at base points x_{ε} ∈ (Ω_{ε}, g_{ε}), u(x_{ε}) → 1, on the scale of the curvature radius ρ(x_{ε}). This situation applies for instance when one of the two non-collapse assumptions NCA I or NCA II does not hold. The main point is that collapse at x_{ε} implies non-degeneration within bounded g_{ε}-distance to x_{ε}, c.f. Theorem 6.5. The remaining analysis of the collapse situation is then completed in §7 in connection with the non-degeneration hypothesis.

Let x_{ε} be any sequence of base points with u(x_{ε}) → 1. As discussed in §2, in this generality one may have ρ(x_{ε}) → 0, ρ(x_{ε}) → ρ_{o} > 0, or ρ(x_{ε}) → ∞, (in the case σ(M) = 0). Consider such sequences which collapse on the scale of ρ(x_{ε}), i.e. ω(x_{ε}) << ρ(x_{ε}) as ε → 0, where ω is the volume radius. As explained in §1, the collapse of (Ω_{ε}, g′_{ε}), g′_{ε} = ρ(x_{ε})^{−2} · g_{ε}, at x_{ε} may then be unwrapped by passing to sufficiently large, (depending on ε), finite covering spaces and on passing to a convergent subsequence, one obtains a maximal limit (F, g′_{o}, x). As previously, let ν_{δ_{a}} = (u − 1)/δ_{a}(x_{ε}), for δ_{a}(x_{ε}) as in (3.1) and recall from (3.2) that δ_{a} ≥ c · δ. For the remainder of the paper we work with δ_{a} as in (3.1), in place of δ in (2.10). By Theorem 2.11, the potential functions ν_{δ_{a}} sub-converge to a limit potential function ν_{a}, modulo addition of constant functions. The limit (F, g_{o}) has a free isometric S^{1} or T^{2} action, leaving the potential ν_{a} invariant.

The limit (F, g′_{o}, x) may be either flat or hyperbolic, i.e. of constant negative curvature, c.f. §2.1. The former case occurs whenever ρ → 0, or if ρ is bounded and σ(M) = 0. Conversely, if σ(M) < 0, then the limit is hyperbolic precisely when ρ ≥ ρ_{o}, for some ρ_{o} > 0. If ρ → ∞, forcing σ(M) = 0, limits could apriori be either flat or hyperbolic, c.f. §7.3 and Appendix A.

Isometric S^{1} actions on flat manifolds F may be classified as follows. First, by lifting to a cover if necessary, the S^{1} action may be converted to a proper ℝ-action. Since F is flat, an isometric ℝ-action is completely determined by its behavior in a neighborhood U of any given point x ∈ F. One may assume that U is embedded in ℝ^{3} and hence the action on U extends uniquely to an ℝ-action on ℝ^{3}. As discussed following Proposition 4.8, the ℝ-action on ℝ^{3} is generated either by a rotation about an axis A, a twist about an axis A, or a translation. Thus, the action is free everywhere except in the case of rotation, where the axis A consists of fixed points. The same classification holds for S^{1} actions on hyperbolic manifolds.

If the collapse at x_{ε} is a rank 2 collapse, (or a sequence of rank 1 collapses converging to a rank 2 collapse), then the the limit (F, g′_{o}, x) has a free isometric T^{2} action, obtained by unwrapping both collapsing circles. The results to follow hold for any S^{1} action with S^{1} ⊂ T^{2}.

We begin with the following result. Lemma 6.1. Any collapsing sequence of base points x_{ε} with u(x_{ε}) → 1 is allowable, w.r.t. δ_{a}, (in place of δ), c.f. Definition 2.7. Proof: Base points x_{ε} are not allowable only if, on a limit (F, g′_{o}, x), there exists a solution ν̄_{∞} to the equation

(6.1)  D^{2}ν̄_{∞} = g′_{o},

obtained by renormalizing ν_{δ_{a}}, c.f. (2.51)-(2.53). However, in the collapse case, ν̄_{∞} must, in addition, be invariant under the isometric S^{1}-action. The equation (6.1) only has S^{1}-invariant solutions in the case of a rotational S^{1}-action, and in this case, the only solution is of the form ν̄_{∞} = \frac{1}{2} ·r^{2}, where r is the distance from some point on the rotation axis, (modulo affine functions). However, then |∇ν̄_{∞}|(x) = 1, which implies, via the renormalization, that |∇ν_{a}| → ∞ at and near x_{ε}; this contradicts the definition of ν_{δ_{a}}, c.f. (3.13).

Throughout this section, let

(6.2)  L^{o} = {x_{ε} ∈ (Ω_{ε}, g_{ε}) : |u − 1|(x_{ε}) = 1/(ln ε̄)^{2}},

and U^{o} the corresponding u-superlevel set where |u − 1| ≤ 1/(ln ε̄)^{2}, i.e. u ≥ 1 − 1/(ln ε̄)^{2}, as in (4.1); these should not be confused with the levels L_{o} from (4.34) and their superlevels as in (5.21).

It is useful to separate the cases of flat and hyperbolic limits. We begin with the flat case. Proposition 6.2. Let x_{ε} be a collapsing sequence of base points in U^{o}, and let (F, g′_{o}, x, ν) be a maximal flat limit as above. Then ∂F consists of a finite number of orbits of the free S^{1} action, together possibly with the full axis A. In fact, there is a constant M < ∞, independent of x, such that

(6.3)  #(∂F) ≤ M,

where # denotes the number of components. In particular, away from A, ∂F is compact. Proof: Since x_{ε} is allowable, Proposition 4.8 and Lemma 4.9 hold on the limit (F, g′_{o}, x). The potential ν is invariant under the S^{1} action and by Proposition 4.8, ∂F = I_{∞}. Hence ∂F is a union of orbits of the action, so that the S^{1} action extends to F̄.

The measure dμ_{∞} of the potential ν at ∂F is also S^{1} invariant, c.f. (4.37ff). Hence, for orbits which are not fixed points of the action, i.e. all cases except the axis in the rotational case, this measure is a multiple of Lebesque measure on a circle. Proposition 4.8 then implies that the number of orbits not in A is locally finite, i.e. there is a uniform bound on the number of orbits in any compact set in F.

To obtain the global bound (6.3), recall from Lemma 4.9 that t = dist_{g′_{o}}(∂F, ·) is unbounded on F. Hence, one may choose a point y ∈ F with t(y) >> 1, as well as base points y_{ε} ∈ Ω_{ε} such that y_{ε} → y. The metrics g′_{ε} = ρ(y_{ε})^{−2} · g_{ε} then converge to a rescaling (F, g′_{1}, y) of (F, g′_{o}, x), (again unwrapping collapse). Applying the arguments above to (F, g′_{1}, y) gives a uniform bound on the number of orbits in balls of fixed size about y in (F, g′_{1}, y). Since y is arbitrary, this implies (6.3).

Finally, in the case of a rotational action, note that the full axis A is necessarily contained in ∂F, since the S^{1} action is free on F itself.

The next result shows that collapse at x_{ε} as in Proposition 6.2 propagates to collapse at larger scales in a natural way. Recall the definition of the levels L^{υ_{o}} and U^{υ_{o}} from (1.17). Lemma 6.3. For x_{ε} as above, suppose y_{ε} are base points in the component of U^{υ_{o}} containing x_{ε} such that u(y_{ε}) → 1 and

(6.4)  dist_{g_{ε}}(y_{ε}, x_{ε}) ≤ c · dist_{g_{ε}}(y_{ε}, L^{υ_{o}}),

for some constant c > 0. Then there are constants C_{ε}, with C_{ε} → ∞ as ε → 0 such that if

(6.5)  dist_{g_{ε}}(y_{ε}, x_{ε}) ≤ C_{ε},

then the metrics (Ω_{ε}, g_{ε}) also collapse at y_{ε}, i.e. ω(y_{ε}) << ρ(y_{ε}). Proof: Suppose first that ρ(x_{ε}) ≥ ρ_{o}, for some ρ_{o} > 0, so that by Proposition 1.1, dist_{g_{ε}}(x_{ε}, L^{υ_{o}}) ≥ ρ_{1}, for some ρ_{1} > 0. There is a curve γ joining x_{ε}, y_{ε}, satisfying (6.4) and hence ρ ≥ ρ_{2} > 0 everywhere along γ, so that the curvature of (Ω_{ε}, g_{ε}) is everywhere bounded in a neighorhood of γ. The result then follows from elementary comparison geometry, c.f. [10, Ch.8C] or [9, Ch.9.1].

If instead ρ(x_{ε}) → 0, we work with the metrics g̃_{ε} = u^{2}g_{ε}, as in the proof of Proposition 2.3. Note that g̃_{ε} is uniformly quasi-isometric to g_{ε} within U^{υ_{o}}, and hence the volume radii w.r.t. g_{ε} and g̃_{ε} are uniformly equivalent within U^{υ_{o}}. Similarly, Proposition 1.1 implies that the L^{2} curvature radii ρ, ρ̃, are uniformly equivalent and hence it suffices to establish collapse w.r.t. g̃_{ε}. The Ricci curvature of g̃ = g̃_{ε} is given by

r̃ = 2(d ln u)^{2} + \frac{1}{2}sg + \frac{2}{u}cg + ε∇Z^{2};

this follows from (1.38), via the Euler-Lagrange equations (1.14)-(1.15). Hence

(6.6)  r̃ ≥ \frac{1}{2}sg + ε∇Z^{2}.

From (1.18), one has |ε∇Z^{2}| ≤ cεt_{υ_{o}}^{−4}, where t_{υ_{o}} = dist_{g_{ε}}(L^{υ_{o}}, ·) so that (6.6) gives

(6.7)  r̃ ≥ \frac{1}{2}su^{−2}g̃ − cεt_{υ_{o}}^{−4} · g̃.

The standard volume comparison theorem, c.f. [10, Ch.5A] or [14, 9.1.3], implies that if the Ricci curvature of g̃_{ε} is uniformly bounded below, then collapse at x_{ε} implies collapse at y_{ε} satisfying (6.4)-(6.5). Note that the scalar curvature s of g_{ε} in (6.7) is uniformly bounded below, so only the second term in (6.7) needs to be considered. To understand this, if t_{υ_{o}} is sufficiently small, pass to the blow-up scale where ρ′(x_{ε}) = 1. Then (6.7) translates to

(r̃)′ ≥ \frac{1}{2}s′u^{−2}(g̃)′ − cα(t′_{υ_{o}})^{−4} · (g̃)′.

In this scale, t′_{υ_{o}}(x_{ε}) → 1. Since α → 0, the Ricci curvature (r̃)′ is almost non-negative; its negative part decays as (t′_{υ_{o}})^{−4}, for (t′_{υ_{o}}) large. Again, the volume comparison theorem implies that collapse at the base point x_{ε} implies collapse in cones as in (6.4) for t′_{υ_{o}} arbitrarily large. This procedure may then be iterated until the base scale g_{ε} is reached. The fact that one may allow C_{ε} → ∞ as ε → 0 then follows again from volume comparison at the base scale g_{ε}.

Lemma 6.3 implies that collapse at some initial x_{ε} ∈ U^{o} propagates to collapse at any y_{ε}, u(y_{ε}) → 1, in any connected cone region defined by (6.4)-(6.5). The limits (F, g′_{o}, y) - or (F, g_{o}, y) when no rescaling is involved - at such base points y_{ε} may either be flat or hyperbolic. Hence, one may iterate this process to increase t_{υ_{o}}, i.e. replace x_{ε} by y_{ε}, etc, provided the limits are flat, and provided the conclusions of Proposition 6.2 hold.

Suppose that all limits based at such y_{ε} satisfying (6.4)-(6.5) are flat; this is the case for instance if σ(M) = 0. Hence of course ρ(y_{ε})/t_{υ_{o}}(y_{ε}) → 1 as ε → 0. We then claim that

(6.8)  sup_{U^{υo}} t_{υ_{o}} → ∞, as ε → 0;

in fact (6.8) holds in cone regions about x_{ε} as in (6.4). This estimate may be viewed as an analogue of Proposition 6.2 at the base scale (Ω_{ε}, g_{ε}, x_{ε}).

To prove the claim, start with an initial collapsing sequence x_{ε} ∈ U^{o}. On the maximal flat limit (F, g′_{o}, x), Theorem 2.11 shows that either

(6.9)  Δν_{a} = 0, or Δν_{a} = −d_{o} < 0,

where d_{o} = lim ρ^{2}|d̄_{ε}|/δ_{a}; here ρ^{2}/δ_{a} is evaluated at x_{ε} and d̄_{ε} is from (2.5). The fact that x_{ε} is allowable by Lemma 6.1 implies that there is a constant D_{o} < ∞, independent of x_{ε}, such that d_{o} ≤ D_{o}.

Now if Δν_{a} = 0 on (F, g′_{o}, x), so that ν_{a} is an S^{1}-invariant harmonic function, then it is a standard consequence of the maximum principle that there are cones C ⊂ F as in (6.4), (with ∂F in place of L^{υ_{o}}), such that ν_{a}(y) > ν_{a}(x) for some y ∈ C with t′(y) = ρ′(y) arbitrarily large. Such points y may be approximated by points y_{ε} ∈ U^{o} ⊂ Ω_{ε} and hence the process above increasing t_{υ_{o}} may be iterated at the much larger scale based at y_{ε} in place of x_{ε}. This proves the estimate (6.8) in case the first alternative in (6.9) always holds in U^{o}. In fact, the same argument holds if

(6.10)  Δν_{a} = −μ,

where μ is sufficiently small, since all limits ν_{a} are non-trivial by Theorem 2.11.

Suppose instead, possibly at some stage of the iteration above, one has collapsing base points x_{ε} ∈ U^{o} such that Δν_{a} = −d_{o} < 0 on the limit (F, g′_{o}, x). Hence for ε small, (ρ^{2}/δ_{a})(x_{ε}) ≥ \frac{1}{2}d_{o}|d̄_{ε}^{−1}|. On the other hand, since d_{o} ≤ D_{o}, one has (ρ^{2}/δ_{a})(y_{ε}) ≤ 2D_{o}|d̄_{ε}^{−1}|, for ε small, for any collapsing base points y_{ε} with u(y_{ε}) → 1. Combining these estimates gives

(6.11)  δ_{a}(y_{ε}) ≥ \frac{d_{o}}{4D_{o}} (ρ′)^{2}(y_{ε})δ_{a}(x_{ε}),

where ρ′ is the L^{2} curvature radius in the scale g′_{ε} based at x_{ε}. Hence, δ_{a}(y_{ε}) is essentially increasing quadratically in the distance ρ′. However, Proposition 6.2 implies that ρ′ is unbounded on the limit (F, g′_{o}, x). If y_{ε} are chosen so that y_{ε} → y ∈ F with ρ′(y) >> 1, then (6.11) implies that the mass of the potential ν_{y} = lim ν_{δ_{a}(y_{ε})} based at y on (F, g′_{o}, y) becomes arbitrarily small. This contradicts Proposition 4.8. Hence (6.8) holds in all cases, as claimed.

As usual, Proposition 1.1 implies that whenever t_{υ_{o}} ≥ t_{o}, for any fixed t_{o} > 0, one has

(6.12)  ρ ≥ ρ_{o} > 0,

with ρ_{o} depending only on t_{o}. Thus, there is no degeneration in such regions.

Next, we turn to the case of hyperbolic limits of collapsing sequences {x_{ε}}. Under the assumption ρ(x_{ε}) ≤ K_{o}, for some K_{o} < ∞, this can occur only when σ(M) < 0 and ρ(x_{ε}) ≥ ρ_{o}, for some ρ_{o} > 0. Thus, essentially no rescalings of g_{ε} are involved and one may set g′_{ε} = g_{ε}. The following is the analogue of Proposition 6.2. Proposition 6.4. Let (F, g_{o}, x, ν) be a maximal hyperbolic limit arising from a collapse at x_{ε} ∈ U^{o}. Then ∂F consists of a uniformly locally finite number of orbits of the S^{1} action, together possibly with the full axis A. Thus, there exists M < ∞, independent of x, such that for any geodesic 1-ball B(1) in F̄,

(6.13)  #(∂F ∩ B(1)) ≤ M.

Proof: The proof is identical to the first part of Proposition 6.2.

Unlike the flat case, the existence of a global bound in (6.13) is not asserted in the hyperbolic case; the hyperbolic case is not scale-invariant, as is the flat case. However, Lemma 4.9 implies that (6.8) holds also in this case.

The results above may be summarized in the following result on the structure of the base metrics (Ω_{ε}, g_{ε}) near collapsing base points. Theorem 6.5. Let x_{ε} ∈ U^{o} be a collapsing sequence of base points. Then for ε small, the metrics (Ω_{ε}, g_{ε}) do not degenerate in the component of U^{υ_{o}} containing x_{ε} where

(6.14)  t_{υ_{o}} ≥ t, dist(x_{ε}, y_{ε}) ≤ C_{ε},

for any t > 0 and C_{ε} as in (6.5). For any t > 0, the region defined by (6.14) is non-empty and all limits of (Ω_{ε}, g_{ε}, y_{ε}) based in this region are constant curvature manifolds (F, g_{o}, y) with scalar curvature s_{o} ≡ σ(M), and with a free isometric S^{1} or T^{2} action.

If σ(M) = 0, all limits at base points in the region (6.14) with t_{υ_{o}} ∈ [t_{o}, T_{o}], for some T_{o} < ∞, are flat. If σ(M) < 0, then all limits based in the same region are hyperbolic, and such limits have boundary ∂F which is either empty, or consists of a uniformly locally bounded number of orbits of the action.

Theorem 6.5 allows one to pass to the situation where one has non-degeneration. The analysis in this situation is carried out next.

7. Non-Degeneration Situations.

In this final section, we complete the proof of the Main Theorem. Throughout this section, it is assumed that M is σ-tame and irreducible. These assumptions are not actually always necessary, but making these assumptions uniformly simplifies the overall logic of the arguments.

Recall from §4 that on (Ω_{ε}, g_{ε}), either the non-degeneration hypothesis (4.2) or the degeneration hypothesis (4.1) holds. In addition of course, either σ(M) < 0 or σ(M) = 0. The analysis is divided accordingly into three subsections.

In §7.1 we analyse the situation where non-degeneration (4.2) holds with σ(M) < 0, while §7.2 analyses the situation where degeneration (4.1) holds with σ(M) < 0. In §7.3 these two cases are handled together when σ(M) = 0. The Main Theorem is proved in Theorem 7.6, (the case σ(M) < 0), and in Theorem 7.7, (the case σ(M) = 0).

7.1.σ(M) < 0.

In this subsection, we examine the structure of (Ω_{ε}, g_{ε}) when σ(M) < 0 and the non-degeneration hypothesis holds. For convenience, recall the non-degeneration hypothesis (4.2): namely there exists a constant ρ_{o} > 0 such that if u(x_{ε}) ≥ 1 − 1/(ln ε̄)^{2}, i.e. x_{ε} ∈ U^{o} as following (6.2), then

(7.1)  ρ(x_{ε}) ≥ ρ_{o},

on (Ω_{ε}, g_{ε}); here ε = ε_{i} is some sequence satisfying (4.3), with ε_{i} → 0.

If ρ ≥ ρ_{o}, for some ρ_{o} > 0, on all of (Ω_{ε}, g_{ε}), then M is tame and the Main Theorem is proved in [1,Thm.0.2]. In fact in this situation, the sequence (Ω_{ε}, g_{ε}) is constant in ε and g_{ε} = g_{o} is hyperbolic, c.f. the discussion in §1. The work below is thus a generalization of that in [1].

Let L^{o} be the level set as in (6.2), with U^{o} the superlevel set as above. We claim that vol_{g_{ε}}U^{o} → 1 as ε → 0. This follows from the following more general result, which will also be needed later. Let L^{1} be the level of u given by

(7.2)  |u(L^{1}) − 1| = ε̄^{2μ}

with U^{1} = {u ≥ 1 − ε̄^{2μ}} the superlevel set of L_{1}. Hence U^{1} ⊂ U^{o}, (for ε small). Lemma 7.1. For U^{1} as above,

(7.3)  vol_{g_{ε}}U^{1} → 1, as ε → 0.

Proof: To see this, (1.8) and (4.3) imply that 1 ≤ T ≤ 1 + ε̄^{4μ}. Since w = uT and u ≤ 1, this gives w ≤ 1 + ε̄^{4μ}. Let v_{1} be the volume of the set U_{1} = {u ≤ 1 − ε̄^{2μ}} and v_{2} be the volume of U^{1} = {u ≥ 1 − ε̄^{2μ}}, so that v_{1} + v_{2} = 1. Then w ≤ 1 − ε̄^{2μ} + cε̄^{4μ} on U_{1} while w ≤ 1 + cε̄^{4μ} on U^{1}. Since, by definition, the L^{2} norm of w equals 1, one obtains

(7.4)  1 ≤ v_{1}(1 − ε̄^{2μ} + cε̄^{4μ})^{2} + v_{2}(1 + cε̄^{4μ})^{2},

which gives v_{1}ε̄^{2μ} ≤ v_{2}cε̄^{4μ}, for ε small. It follows that v_{2} ≥ 1 − cε̄^{2μ}, which gives (7.3).

Let V^{o} be the ρ_{o}/2 tubular neighborhood of U^{o}. We claim that the filling radius, (c.f. [10,3.35]), of L^{o} in U_{o} = Ω_{ε} \ U^{o} is at least ρ_{o}/2, i.e. no collection of components of L^{o} bound within V^{o} ∩ U_{o}. For if not, then there is a domain D_{ε} ⊂ V^{o} ∩ U_{o} with ∂D_{ε} ⊂ L^{o} and hence u achieves a minimal value at some point q_{ε} ∈ D_{ε}. Note that ρ ≥ ρ_{o}/4 everywhere on D_{ε}. Let (F, g_{o}, q) be a maximal limit based at q_{ε}. It follows that the limit potential function ν has a minimal value at q. This contradicts Theorem 2.11, via the trace equation (2.65).

Since U^{o} ⊂ V^{o}, vol_{g_{ε}}V^{o} → 1 also, and so vol_{g_{ε}}(V^{o} \ U^{o}) → 0. It follows in particular that

(7.5)  vol_{g_{ε}}T^{o} → 0,

where T^{o} is the ρ_{o}/4 tubular neighborhood of L^{o}.

Now let x_{ε} be any base point in U^{o}, so that ρ(x_{ε}) ≥ ρ_{o}. Any maximal limit (H_{x}, g_{o}, x) of (Ω_{ε}, g_{ε}, x_{ε}), ε = ε_{i}, is of constant curvature, with scalar curvature s_{o} ≡ σ(M) and with u ≡ 1, c.f. Proposition 1.2. In case the sequence collapses at x_{ε}, the limit has a free isometric S^{1} or T^{2} action, i.e. is a rank 1 or 2 hyperbolic cusp. If dist_{g_{ε}}(x_{ε}, L^{o}) → ∞, the limit (H_{x}, g_{o}) is complete, while if dist_{g_{ε}}(x_{ε}, L^{o}) remains bounded, the limit may either be complete or incomplete; however there is no boundary of H_{x} within the Hausdorff limit of the region V^{o} above. Observe also that (7.5) implies that all limits (H_{x}, g_{o}, x) for x_{ε} ∈ T^{o} are necessarily collapsing. Hence, if (Ω_{ε}, g_{ε}) is not collapsing at x_{ε}, then the standard volume comparison theorem, c.f. [10, Ch.5A], or [14, 9.1.3], implies that the limit H_{x} is complete.

The main result of this subsection is the following: Theorem 7.2. Suppose σ(M) < 0, M is irreducible and σ-tame, and the sequence (Ω_{ε}, g_{ε}), ε = ε_{i}, satisfies the non-degeneration hypothesis (7.1). Then (Ω_{ε}, g_{ε}) is constant in ε, i.e. (Ω_{ε}, g_{ε}) = (H, g_{o}), where H = ∪^{·} H_{k} is a finite collection of complete, connected, hyperbolic manifolds H_{k}, 1 ≤ k ≤ q, q = q(M) < ∞, and H is embedded in M. The metric g_{o} is hyperbolic, satisfying

(7.6)  s_{g_{o}} = σ(M), vol_{g_{o}}H = 1.

Proof: Suppose first that there is a (complete) component C_{ε} of Ω_{ε} such that C_{ε} ⊂ U^{o}, so that the sequence of metrics g_{ε}|_{C_{ε}} is tame. As mentioned at the beginning of §7, the work in [1,Prop.2.5,Rmk.2.10] then implies that g_{ε}|_{C_{ε}} is a constant sequence, so that (C_{ε}, g_{ε}) = (C_{o}, g_{o}) is a complete hyperbolic manifold of finite volume.

Thus, g_{ε} can only vary on the components of U^{o} ⊂ Ω_{ε} on which ∂U^{o} ≠ ∅. In this situation, we claim first that not all of U^{o} is collapsing, i.e. there are base points x_{ε} ∈ U^{o} such that (Ω_{ε}, g_{ε}) does not collapse at x_{ε}. For suppose instead all of U^{o} is collapsing as ε → 0. Then U^{o} is a graph manifold, and all based limits, when the collapse is unwrapped, are rank 1 or rank 2 hyperbolic cusps. (Of course, any domain of bounded diameter about any such x_{ε} has volume converging to 0, due to the collapse; however, there could be a large number of such regions, corresponding to a large number of limit cusps, whose total volume approaches 1).

Observe that the volume of geodesic balls in the expanding end of a hyperbolic cusp is proportional to the volume of the corresponding geodesic sphere. It then follows from (7.3) and (7.5) that most all (in terms of volume) of the components of U^{o} intersecting ∂U^{o} are cusps which are expanding into the interior of U^{o}. However, this implies that, for any ε > 0 sufficiently small, there must exist points realizing the maximal diameter of the orbits (circles or tori) of the F-structure, (i.e. graph manifold structure), within U^{o}. Choosing such points as base points, it follows that the limiting hyperbolic cusp also has a maximal value for the diameter of the associated S^{1} or T^{2} orbits. However, this is impossible, and thus proves the claim.

Let x_{ε} be any non-collapsing sequence of base points in U^{o}. It follows then from the observation preceding Theorem 7.2 that (Ω_{ε}, g_{ε}, x_{ε}) sub-converges to a complete hyperbolic manifold (H_{x}, g_{o}), of finite volume; the level set L^{o} satisfies dist_{g_{ε}}(L^{o}, x_{ε}) → ∞ as ε → 0 and so has no limit on H_{x}. The convergence is smooth and uniform on compact subsets. Any such manifold H_{x} is of finite topological type, with a finite number of cusp ends each of the form T^{2} × ℝ^{+}. Hence H_{x} embeds in Ω_{ε} and also in M. By the Margulis Lemma, c.f. [17], there is a fixed lower bound on the volume of any such H_{x}.

Now repeat this argument on Ω_{ε} \ H_{x}. It follows from the Margulis Lemma that after a finite number of iterations, one obtains a finite number of disjoint limit hyperbolic manifolds H_{k}, with H = ∪^{·} H_{k} satisfying vol_{g_{o}}H = 1, i.e. (7.6) holds. As above, H embeds in Ω_{ε} and so also in M. Thus, one may view the metrics g_{ε}|_{H} as a sequence of metrics on H.

The next result proves that the metrics g_{ε} are rigid. Proposition 7.3. Under the assumptions of Theorem 7.2, the metrics g_{ε}|_{H} are constant, up to isometry, i.e.

(7.7)  (Ω_{ε}, g_{ε}) = (H, g_{o}).

Proof: This is a simple application of Theorem 2.11 and the maximum principle. If (7.7) does not hold, then the metrics g_{ε} vary somewhere on H, say at base points x_{ε} → x ∈ H. Now the limit potential ν on H_{x}, satisfies (2.65), i.e.

(7.8)  Δν = λν − d,

where d ≥ 0 and λ = lim \frac{1}{4}ρ^{2}σT = \frac{1}{4}|σ(M)| lim ρ^{2}, for ρ = ρ(x_{ε}), c.f. (2.72). Since ρ(x_{ε}) is bounded away from 0 and ∞, observe that λ > 0. Further, as noted following Theorem 2.11, the potential ν_{δ(y_{ε})} = (u − 1)/δ(y_{ε}) converges to the limit ν, in that there is no constant or affine indeterminacy. Thus, ν ≤ 0. This implies that ν is also uniformly bounded below, i.e. there is a constant K < ∞ such that

(7.9)  ν ≥ −K > −∞,

c.f. the remark following (2.69). Hence, if p_{j} is a minimizing sequence for ν on H_{x}, i.e. ν(p_{j}) → inf ν, then one must have Δν(p_{j}) ≥ −μ_{j}, for some sequence μ_{j} → 0. It then follows from (7.8) that d = 0 and also λ = 0, giving a contradiction. Hence (7.7) must hold.

This completes the proof of Proposition 7.3 and hence also Theorem 7.2.

7.2.σ(M) < 0.

In this section, still assuming σ(M) < 0, we analyse the case where the degeneration hypothesis (4.2), opposite to (4.1) or (7.1) holds. The main result is that Theorem 7.2 also holds in this context. Theorem 7.4. Suppose σ(M) < 0, M is irreducible and σ-tame, and the sequence (Ω_{ε_{i}}, g_{ε_{i}}), satisfies the degeneration hypothesis (4.1). Then again the conclusions of Theorem 7.2 hold. Proof: In view of Theorem 5.12, one may assume that on the sequence (Ω_{ε}, g_{ε}), ε = ε_{i}, one of the non-collapse hypotheses NCA I or NCA II fails. The strategy is then to reduce to the proof to that of Theorem 7.2, using the results of §6.

First, recall that the degeneration hypothesis and Theorem 4.4 guarantee the existence of admissible level sets L_{o} as in (4.34), with corresponding superlevel sets U^{o}. Such levels are not unique, i.e. different admissible or preferred base points may lie on distinct levels L_{o}. (Similarly, recall that degeneration at some point x_{ε} ∈ L_{o} does not imply degeneration everywhere on L_{o}). However, it is always the case, by (4.4), that |u(L_{o}) − 1| ≥ ε̄^{2μ}. Thus for L^{1} and U^{1} as in (7.2), the domain U^{1} is contained in all superlevel sets U^{o} of admissible levels L_{o}.

Now let t^{1} = dist_{g_{ε}}(L^{1}, ·) and set

(7.10)  U^{2} = {x ∈ U^{1} : t^{1}(x) ≥ τ_{o}},

where τ_{o} is a small parameter. Then we have the following analogue of the non-degeneration hypothesis (7.1), Lemma 7.1 and (7.5) on U^{2}. Lemma 7.5. The domain U^{2} = U^{2}(τ_{o}) is non-empty and there is a constant ρ_{o} = ρ_{o}(τ_{o}) > 0 s.t.

(7.11)  ρ(x_{ε}) ≥ ρ_{o},

∀x_{ε} ∈ U^{2}. Further, the metrics g_{ε} collapse everywhere on the boundary L^{2} = ∂U^{2} as ε → 0, while

(7.12)  vol_{g_{ε}}U^{2} ≥ 1 − τ_{1},

where τ_{1} → 0 as τ_{o} → 0, on any sequence ε → 0. Proof: The estimate (7.11) is an immediate consequence of the definition (7.10) and Proposition 1.1. The main point is to prove that U^{2} is non-empty and that g_{ε} collapses everywhere on L^{2}. From this, (7.12) follows easily.

Thus, let x_{ε} be any sequence of base points on L^{1}. Suppose first that ρ(x_{ε}) ≥ ρ_{1}, for some ρ_{1} > 0 (in a subsequence). If (Ω_{ε}, g_{ε}, x_{ε}) does not collapse, then it (sub)-converges to a limit hyperbolic manifold (H, g_{o}, x), containing at least the full geodesic ball B_{x}(ρ_{1}). In particular then vol_{g_{ε}}B_{x_{ε}}(ρ_{1}) ≥ v_{1}, for some v_{1} > 0. However, a definite part of the volume of B_{x_{ε}}(ρ_{1}) is contained in U_{1} = Ω_{ε} \ U^{1}, which contradicts (7.3). Hence (Ω_{ε}, g_{ε}, x_{ε}) collapses as ε → 0. Theorem 6.5 then implies first that U^{2} = U^{2}(τ_{o}) is non-empty, for a fixed τ_{o} small, and further that (Ω_{ε}, g_{ε}) collapses at all points on L^{2} within bounded distance to x_{ε}. An examination of the proof of Lemma 7.1, c.f. (7.4ff), shows that the same conclusion holds if ρ(x_{ε}) → 0 sufficiently slowly as ε → 0, for instance ρ(x_{ε}) ≥ ε̄^{μ/10}.

Next, consider base points x_{ε} on L^{1} where ρ(x_{ε}) → 0, e.g. ρ(x_{ε}) ≤ ε̄^{μ/10}. Apriori all points on L^{1} may have this property. As mentioned following the proof of Theorem 4.4, the admissible and preferred base points are 1-dense, in fact η-dense, within the region

(7.13)  S_{ε} = {ε̄^{2μ} ≤ |u − 1| ≤ 2/(ln ε̄)^{2}} ∩ {ρ ≤ χ_{ε}}, where χ_{ε} → 0 as ε → 0.

To be definite, choose χ_{ε} = ε̄^{μ/10}. Clearly, x_{ε} ∈ S_{ε}. Choose η to be a fixed number with η < τ_{o}/3.

Now as mentioned above, one of the two non-collapse hypotheses fails. Suppose first that NCA I fails. This means that for all admissible or preferred base points y_{ε}, there is collapse at some p_{ε} ∈ B_{y_{ε}}(η) ∩ L_{o}, on the scale of ρ(p_{ε}), where L_{o} is the u-level through y_{ε}. The collection of balls B_{y_{ε}}(η), for y_{ε} admissible, covers S_{ε}. Now Lemma 6.3 and the analysis following it implies that the collapse at such p_{ε} propagates in cones to collapse on L^{1}. Together with the collapse result above when ρ(x_{ε}) ≥ ε̄^{μ/10}, it follows that L^{1} collapses on an η-dense set. Again, Lemma 6.3 or Theorem 6.5 then imply that L^{2} = L^{2}(τ_{o}) collapses everywhere and U^{2} is non-empty.

Suppose instead that NCA I holds, but NCA II fails. When NCA I holds, there are some admissible or preferred base points y_{ε} which satisfy (5.1) and possibly some that don’t. For those base points y_{ε} that don’t satisfy (5.1), the same conclusion as above holds, (i.e. U^{2} is non-empty and L^{2} collapses within bounded distance to y_{ε}). Thus, one only needs to consider those base points y_{ε} where (5.1) holds. For any such base point, all distinguished base points q_{ε} within g_{ε}-distance η to y_{ε} collapse, possibly on a scale large compared with ρ(q_{ε}). Since ρ(q_{ε}) → 0, this means there are points q′_{ε}, with dist_{g_{ε}}(q_{ε}, q′_{ε}) → 0, ρ(q′_{ε}) → 0, such that ω(q′_{ε}) << ρ(q′_{ε}). As above, Lemma 6.3 and Theorem 6.5 imply that L^{2} is non-empty and collapses within bounded distance to q′_{ε}. By the same η-density arguments as before, it follows that in all cases, L^{2} collapses everywhere w.r.t. g_{ε}.

Finally, to prove (7.12), all limits based at points in L^{2} are hyperbolic manifolds H with free isometric S^{1} or T^{2} action, with ∂H consisting of a locally uniformly bounded number of S^{1} orbits. In particular, ∂H has measure 0. Thus, by the continuity of the volume as ε → 0, (7.12) follows immediately from Lemma 7.1.

Lemma 7.5 proves that one has non-degeneration on U^{2}, with vol_{g_{ε}}U^{2} ≥ 1 − τ, where τ may be made small by choosing τ_{o} small, (for all ε sufficiently small). Further, all base points on L^{2} = ∂U^{2} collapse as ε → 0. The rest of the proof of Theorem 7.4 then proceeds exactly as in the proof of Theorem 7.2, by choosing τ smaller than the Margulis constant.

Theorems 7.2 and 7.4 now lead easily to the proof of the Main Theorem in case σ(M) < 0. Theorem 7.6. Suppose σ(M) < 0, and M is irreducible and σ-tame. Then M has a decomposition, unique up to isotopy, as

(7.14)  M = H ∪_{T} G,

where H is a finite collection of complete, connected, hyperbolic manifolds with scalar curvature σ(M) and total volume 1, and G is a finite collection of connected graph manifolds. The union is along a finite collection of tori T = ∂H = ∂G, each incompressible in M. The Sigma constant σ(M) is given by

(7.15)  σ(M) = −(6vol_{g_{−1}}H)^{2/3},

where vol_{g_{−1}}H is the total volume of H in the metric of constant curvature −1. Proof: Theorems 7.2 and 7.4 prove the existence of the hyperbolic manifold H embedded in M satisfying (7.7). In particular Ω_{ε} embeds in M. As mentioned in §1, the complement of Ω_{ε} in M is then a finite union G of connected graph manifolds, collapsed under a minimizing sequence for I_{ε}^{−} for any ε > 0. This gives the decomposition (7.14). The remaining parts of Theorem 7.6, namely that T = ∂H = ∂G is incompressible in M, that the decomposition (7.14) is unique up to isotopy, and that (7.15) holds, then follow from [1, Thm.0.2]; equivalently, the family (Ω_{ε}, g_{ε}) is tame, (since it is constant), and hence Theorem 7.6 follows from [1], as mentioned in §0.

7.3.σ(M) = 0.

In this section, the cases of non-degeneration (4.2) and degeneration on L_{o} are handled together. The following result proves the Main Theorem in case σ(M) = 0. Theorem 7.7. Suppose σ(M) = 0, and M is irreducible and σ-tame. Then M is a graph manifold and either Ω_{ε} = ∅, or Ω_{ε} = M and g_{ε} is a sequence of flat metrics on M.

Theorem 7.7 will be proved in a sequence of steps below, but logically is proved by contradiction. Thus, we assume that (Ω_{ε}, g_{ε}), ε = ε_{i}, is a non-empty sequence of minimizers for I_{ε}^{−} as in §1, with g_{ε} non-flat for all ε > 0. As noted in §1, this already implies that M is not a graph manifold.

Suppose first the non-degeneration hypothesis (7.1) holds on (Ω_{ε}, g_{ε}), so that there is nondegeneration everywhere on U^{o} = {x ∈ Ω_{ε} : u(x) ≥ 1/(ln ε̄)^{2}}. All based limits in this region are flat manifolds (F, g_{o}, x), x = lim x_{ε}, x_{ε} ∈ U^{o}. Such limits are necessarily non-compact; in fact by Lemma 4.9, dist_{g_{o}}(∂F, ·) is unbounded. Thus, (F, g_{o}) has infinite volume and so the sequence (Ω_{ε}, g_{ε}, x_{ε}) must collapse. (This is in strong contrast to the situation when σ(M) < 0). Hence all such limits have a free isometric S^{1} or T^{2} action. Further, by Proposition 6.2 or Theorem 6.5, the function t^{o} = dist_{g_{ε}}(L^{o}, ·) is unbounded, i.e. assumes arbitrarily large values as ε → 0.

Next, suppose instead that the degeneration hypothesis (4.1) holds. Then as in the proof of Theorem 7.4, one of the non-collapse assumptions must fail. In this case, let L^{1} and U^{1} be as in (7.2). The proof of all of Lemma 7.5 except the volume statement (7.1) carries over without any change and implies that one has non-degeneration on U^{2}, i.e. (7.11) holds, and there is collapse everywhere on L^{2} = ∂U^{2}, for U^{2} as in (7.10). Again, all limits (F, g_{o}, x) of (Ω_{ε}, g_{ε}, x_{ε}) for x_{ε} ∈ U^{2} are flat manifolds, collapsed and so with free isometric S^{1} or T^{2} action. As before, the distance function t^{1} from (7.10) is unbounded as ε → 0.

To unify these two situations, let L̄ = L^{o}, Ū = U^{o} as above in case (7.1) holds, while let L̄ = L^{2}, Ū = U^{2} as above in case (7.1) fails. Hence, all limits (F, g_{o}, x) of (Ω_{ε}, g_{ε}, x_{ε}), x_{ε} ∈ Ū are flat manifolds, with free isometric S^{1} or T^{2} action, with t̄= dist_{g_{ε}}(L̄, ·) unbounded as ε → 0.

As in the previous sections, it is then natural to consider the rescaled metrics

(7.16)  g′_{ε} = ρ(x_{ε})^{−2} · g_{ε}.

Of course ρ(x_{ε}) ≥ ρ_{o} > 0 within Ū. In the region where ρ(x_{ε}) → ∞ as ε → 0, (i.e. where t̄(x_{ε}) → ∞ as ε → 0), the metrics g_{ε} are then being “blown-down”. As before ρ′(x_{ε}) = 1, and any sequence (Ω_{ε}, g′_{ε}, x_{ε}) has a subsequence converging smoothly to a maximal limit (N_{x}, g′, x). Since the unscaled sequence (Ω_{ε}, g_{ε}, x_{ε}) is collapsing, the rescaled sequence (7.16) is collapsing even more, so that all limits (N_{x}, g′, x) necessarily have a free isometric S^{1} or T^{2} action.

The metrics g′_{ε} satisfy the Euler-Lagrange equations (2.24). Here ᾱ = ε/T ρ^{2}, ρ = ρ(x_{ε}) so that obviously ᾱ → 0. Thus, as before in §2, the terms in (2.24) containing ᾱ∇Z^{2} and ᾱ|z|^{2} converge to 0. On the other hand, the constant term c̄′_{ε} in (2.24) is given by c̄′_{ε} = ρ^{2} · c̄_{ε}, so that c̄′_{ε} >> c̄_{ε} as ε → 0. Recall from (1.6) that

(7.17)  c̄_{ε} = \frac{1}{12}σ̄ + \frac{1}{6}ε̄Z^{2} → 0, as ε → 0.

It follows that the limit metric g′ is a solution of the equations

(7.18)  L^{∗}u = (−\frac{1}{4}su + c̄_{∞})g,

(7.19)  2Δu + \frac{1}{4}su = −3c̄_{∞}.

Here s, (= s_{g′}), is the scalar curvature of the limit metric g′, and the prime has been dropped from the notation. The constant c̄_{∞} is given by c̄_{∞} = lim c̄′_{ε}. Apriori, this could be infinite, but the following simple Lemma shows it to be finite. Lemma 7.8. There is a constant K < ∞ such that, for any sequence of base points x_{ε} in Ū,

(7.20)  0 ≤ c̄_{∞} ≤ K.

Proof: To see this, return to (2.24) in the scale (7.16) and note that, by definition, 0 ≤ u ≤ 1. Let η be a bounded smooth cutoff function supported in B′_{x_{ε}}(1). Then multiply the trace equation (2.24) by η and integrate by parts over B′_{x_{ε}}(1). Since ᾱ → 0, while u and Δη are bounded, it follows that ∫ ηc̄′_{ε} is also bounded as ε → 0, independent of x_{ε}. Hence c̄′_{ε} is also uniformly bounded, which gives (7.20).

As noted in §2.1, Proposition A below shows that all maximal limits (N_{x}, g′, x) of (Ω_{ε}, g′_{ε}, x_{ε}), x_{ε} ∈ Ū, are of constant curvature, either flat or hyperbolic, i.e. of constant negative curvature. As previously, denote such limits as (F, g′_{o}, x). The limit F is flat when c̄_{∞} = 0, hyperbolic when c̄_{∞} > 0. Recall, c.f. Remark 2.5(ii), that Theorem 2.11 holds for such limits, as does Proposition 6.2.

Now, for any given ε > 0, c̄′_{ε} = ρ^{2}c̄_{ε} > 0, and hence Lemma 7.8 implies that ρ has a uniform upper bound on Ū, depending on ε, i.e. for ε sufficienty small,

(7.21)  ρ_{max} = sup_{Ū} ρ(x_{ε}) ≤ 2(K/c̄_{ε})^{1/2} < ∞.

On the other hand, ρ_{max} → ∞ as ε → 0, since t̄ is unbounded and σ(M) = 0. The next result characterizes the geometry of limits at points x_{ε} ∈ Ū (almost) realizing ρ_{max}. Lemma 7.9. If x_{ε} satisfies ρ(x_{ε})/ρ_{max} → 1, then the limit (F, g′_{o}, x) of (Ω_{ε}, g′_{ε}, x_{ε}) cannot be flat. Proof: If the limit (F, g′_{o}, x) is flat, then Proposition 6.2 holds and so the distance function t′ to ∂F assumes arbitrarily large values on F. Hence, so does ρ′, and so there exist y_{ε} ∈ Ω_{ε} such that ρ(y_{ε}) >> ρ(x_{ε}). If there exist such y_{ε} ∈ Ū, then one has a contradiction. On the other hand, this can only not be the case if Δν_{x} ≤ −μ < 0, as in (6.10), which leads to the same contradiction as following (6.11).

Thus, blow-downs based at points x_{ε} (almost) realizing ρ_{max} must be hyperbolic. Note that the scalar curvature s′_{ε} = ρ^{2}s_{ε} of g′_{ε}, ρ = ρ(x_{ε}), satisfies s′_{ε} → const < 0 on such limits. Of course, since s_{ε} = −σT u and u → 1, one has

(7.22)  ρ^{2}σT → const > 0.

There are now several ways to complete the proof of Theorem 7.7, but perhaps the simplest is the following. For x_{ε} as above, the limit (F, g′_{o}, x) is hyperbolic. Proposition 6.4 (or Theorem 6.5) describes the structure of the limit. By Lemma 4.9, the function t = dist_{g′_{o}}(∂F, ·) is unbounded on F. Hence, by choosing new base points y_{j} ∈ F if necessary, with t(y_{j}) → ∞, and approximating them by points y_{ε} = y_{ε_{j}} ∈ Ω_{ε}, one obtains a new hyperbolic limit (F, g′_{o}, y) with ∂F = ∅, i.e. (F, y) is complete, without boundary. (Note that ρ′ is constant on hyperbolic limits and does not increase with t, as is the case with flat limits).

This brings one exactly to the situation of Proposition 7.3. Namely, the limit potential ν on F satisfies (7.8)-(7.9), and the potentials ν_{δ(y_{ε})} = (u−1)/δ(y_{ε}) converge to the limit ν. The same proof as in Proposition 7.3 then gives a contradiction to (7.22). This completes the proof of Theorem 7.7.

Appendix A.

In this Appendix, we prove the analogue of Proposition 1.2 when

(A.1)  u(x_{ε}) → 1 and ρ(x_{ε}) → ∞.

Of course, this situation is only possible when σ(M) = 0 and the metrics (Ω_{ε}, g_{ε}) are collapsing at x_{ε}. For x_{ε} satisfying (A.1), let

(A.2)  g′_{ε} = ρ(x_{ε})^{−2} · g_{ε},

and let (F, g′, x) be the maximal limit of a convergent subsequence of (Ω_{ε}, g′_{ε}, x_{ε}), unwrapped in covering spaces to obtain convergence. From §7.3, g′ is a solution of the equations (7.18)-(7.19).

Proposition A. Let (F, g′_{o}, x) be a maximal limit constructed above, satisfying the equations (7.18)- (7.19). If c̄_{∞} = 0, then (F, g_{o}) is flat, with u ≡ 1. If c̄_{∞} > 0, then (F, g_{o}) is hyperbolic, i.e. of constant negative sectional curvature, again with u ≡ 1.

Proof: From (7.19) and (2.3), (2.6), at every point of (F, g_{o}) one has

(A.3)  −\frac{1}{4}su − 3c̄_{∞} ≤ 0.

Further by the definition of u, −su ≥ 0 on F. Hence, if c̄_{∞} = 0, then su ≡ 0 on F and the equations (7.18)-(7.19) are the static vacuum equations (1.25). The maximum principle then implies that the limit is flat, with u ≡ 1.

Thus suppose c̄_{∞} > 0. The proof in this case is a rather long computation. Suppose first that at the base point x where u(x) = 1 = sup u,

(A.4)  −\frac{1}{4}s(x) − 3c̄_{∞} = 0,

or equivalently, Δu(x) = 0, by (7.19) We then claim that essentially the same proof as Proposition 1.2 then shows that g′ is hyperbolic and u ≡ 1. To see this, recall that by definition on the sequence (Ω_{ε}, g_{ε}), u = −(s^{−})/σT = −(s^{−})ρ^{2}/σT ρ^{2}, ρ = ρ(x_{ε}). The term s^{−}ρ^{2} is the non-positive part of the scalar curvature s′_{ε} of g′_{ε} in (A.2). Passing to the limit, it follows that the trace equation (7.19) may be rewritten in the form

(A.5)  2Δs + \frac{1}{4}s^{2} − 3c′ = 0,

on the limit (F, g_{o}) where c′ = lim ρ^{4}σc_{ε}. Similarly, the equation (7.18) becomes

(A.6)  L^{∗}s + (\frac{1}{4}s^{2} + c′)g = 0.

Write (A.5) in the form

(A.7)  2Δs + \frac{1}{4}(s − (12c′)^{1/2})(s + (12c′)^{1/2}) = 0.

Let w = s + (12c′)^{1/2} and note that (A.3) implies that w ≥ 0 while (A.4) implies that w(x) = 0. Thus (A.7) is of the form

(A.8)  2Δw + fw = 0,

with f = \frac{1}{4}(s − (12c′)^{1/2}) ≤ 0. This equation is exactly analogous to (1.26) and the same argument as following (1.26) implies that w ≡ 0, so that (A.4) holds everywhere on F. Hence u ≡ 1 and (7.18) or (A.6) implies that g′ is hyperbolic.

Thus, it remains to prove that (A.4) holds at x. To do this, observe first that (A.6) implies that D_{o}^{2}s = sz, where D_{o}^{2}s = D^{2}s − \frac{Δs}{3} · g is the trace-free Hessian, and hence

(A.9)  δdD_{o}^{2}s = δd(sz).

Here d is the exterior derivative on vector valued 1-forms defined by the metric, and δ is the adjoint of d, c.f. [6,Ch.1]. All the computations to follow are at the base point x where u(x) = 1. Since sup u = 1, one has ds = 0 at x and an elementary computation then shows that δdsz = −(Δs)z + sδdz. Consequently, tr(δdsz) = 0 at x, so that

(A.10)  tr(δdD_{o}^{2}s) = 0.

at x. The rest of the proof is a computation of tr(δdD_{o}^{2}s). First, dD^{2}s(X, Y, Z) = R(X, Y, ds, Z), which vanishes at x. Hence (A.6) and the definition of L^{∗} in (1.5) imply that dr = δR = 0 at x, where the first equality is the Bianchi identity. A straightforward computation then gives

δdD^{2}s = −R ∘ (D^{2}s),

where R∘ is the action of the curvature on symmetric 2-tensors, c.f [6, Ch.1]. Consequently,

trδdD^{2}s = −tr(R ∘ (D^{2}s)) = − < r, D^{2}s >,

at x. Also, δd(Δs · g) = −(ΔΔs) · g and so it follows from (A.10) that

(A.11)  − < r, D^{2}s > +ΔΔs = 0,

at x. Expanding r as r = z + \frac{s}{3}g and using the equations (A.5)-(A.6), this implies that

−s|z|^{2} − \frac{1}{3}sΔs − \frac{1}{4}sΔs = 0,

so that s|z|^{2} = −\frac{7}{12}sΔs. However, s < 0, Δs ≥ 0 at x, and so Δs = 0 and z = 0, at x. This proves (A.4), as required.

Appendix B.

Let u(x_{ε}) → 1 and let (F, g′_{o}, x) be a maximal limit of (Ω_{ε}, g′_{ε}, x_{ε}), as in §2. In this Appendix, we prove that, for δ = δ(x_{ε}),

(B.1)  ᾱ\frac{∇Z^{2}}{δ} → 0,

uniformly in L^{2} on compact subsets of F, proving the claims in (2.25) and (2.43). We also indicate that the covariant derivatives ∇^{k}z/δ, k ≥ 0, remain bounded locally in L^{2} on compact subsets of F. Here and below, all metric quantities are w.r.t. g′_{ε} or g′_{o}.

We first prove (B.1) at the base point x_{ε} and within B′_{x_{ε}}(\frac{1}{2}). Given this, the proof that (B.1) holds near any y ∈ F, with δ = δ(x_{ε}), follows easily and is given at the end.

To begin, Proposition 1.4 implies that ᾱ → 0 while Proposition 1.1 implies that for any k ≥ 0, ∇^{k}z → 0 in L^{2} on compact subsets of F. Also, by definition, z/δ is uniformly bounded in L^{2}(B), B = B′_{x_{ε}}(\frac{1}{2}). Hence

(B.2)  ᾱ\frac{|z|^{2}}{δ} → 0 in L^{2}(B).

Suppose first that δ ≥ c · ρ^{2}, for some c > 0, i.e. Case (i) or (ii) of (2.12) holds. Then the same argument as in (2.36) implies that

(B.3)  Δν_{δ} → 0,

strongly in L^{2}(B). By [6, Ch.4H] or [2,(3.8)],

(B.4)  ∇Z^{2} = D^{∗}Dz + \frac{1}{3}D^{2}s − 2R ∘ z + \frac{1}{2}(|z|^{2} − \frac{1}{3}Δs)g,

so that the Euler-Lagrange equation (2.24) in the scale g′_{ε} gives

(B.5)  \frac{ᾱ}{δ}D^{∗}Dz + u\frac{z}{δ} = D^{2}ν_{δ} + o(1)

in L^{2}(B). Now integration by parts shows that

(B.6)  \frac{ᾱ}{δ}D^{∗}Dz → 0 in L^{−2,2},

i.e. it converges to 0 when paired with any sequence of symmetric 2-tensors which are uniformly bounded in L^{2,2}, (since ᾱ → 0). Thus, from (B.5),

(B.7)  D^{2}ν_{δ} = 0(1) in L^{−2,2}.

It then follows from (B.3) and elliptic regularity theory that D^{2}ν_{δ} is bounded in L^{2}(B) and converges strongly in L^{2}(B) to a limit D^{2}ν ≡ h. (Here and in similar arguments below, one must actually go to slightly smaller balls B′ ⊂ B; since this is of no consequence in these arguments, this will be left aside). Returning to (B.5), this means that ᾱD^{∗}Dz/δ + uz/δ converges strongly in L^{2} to h. In particular, since z/δ is bounded in L^{2}, ᾱD^{∗}Dz/δ is bounded in L^{2}, and thus converges weakly to 0 in L^{2}.

To prove that ᾱD^{∗}Dz/δ converges strongly to 0 in L^{2}, i.e. (B.1), since h is fixed, one has

< ᾱD^{∗}Dz/δ, η^{2}h >→ 0,

where <, > denotes the L^{2} inner product w.r.t. g′_{ε} and η is a fixed cutoff function supported in B. Letting || · || denote the L^{2} norm, it follows that < u(z/δ), η^{2}h >→ ||ηh||^{2} and so the Cauchy-Schwarz inequality gives

(B.8)  ||ηu\frac{z}{δ} ||^{2} → K ≥ ||ηh||^{2}.

On the other hand, ||(ηᾱD^{∗}Dz/δ) + (ηuz/δ)||^{2} → ||ηh||^{2}. Expanding out the norm on the left, the cross term has the form

(B.9)  \frac{ᾱ}{δ^{2}} ∫ < D^{∗}Dz, uη^{2}z > = \frac{ᾱ}{δ^{2}} ∫ < Dz, D(uη^{2}z) > = \frac{ᾱ}{δ^{2}} ∫ uη^{2}|Dz|^{2}

+\frac{1}{δ^{2}} ∫ < ᾱ^{1/2}ηDz, ᾱ^{1/2}ηdu ⊗ z) > +\frac{2}{δ^{2}} ∫ < ᾱ^{1/2}ηDz, ᾱ^{1/2}udη ⊗ z) > .

The last two terms are estimated as follows. Since z/δ is bounded in L^{2} and u → 1, du → 0 smoothly, the Hölder and Young inequalities (ab ≤ \frac{1}{2}(a^{2} + b^{2})) imply that the 4^{th} term in (B.9) is the sum of two terms, one small compared with the 3^{rd} term and the other of order ᾱ → 0. The same holds w.r.t. the last 5^{th} term, since dη is bounded.

Thus the cross term is positive, or goes to 0 as ε → 0. It follows that ||ηᾱD^{∗}Dz/δ||^{2}+||ηuz/δ||^{2} ≤ ||ηh||^{2} in the limit and hence (B.1) now follows from (B.8).

The argument in case δ << ρ^{2} is similar, although further work is required since (B.3) may not hold. Since z/δ is uniformly bounded in L^{2}(B), div(z/δ) is uniformly bounded in L^{−1,2}(B). By the Bianchi identity, div(z/δ) = \frac{1}{6}d(s/δ), where s = s′ is the scalar curvature of g′_{ε}. Hence there are constants m_{ε} such that (s − m_{ε})/δ remains uniformly bounded in L^{2}(B); the constant m_{ε}/δ is the mean value of s/δ on B, and may be unbounded as ε → 0. It follows that

(B.10)  (\frac{s − m_{ε}}{δ}) → ζ,

weakly in L^{2}(B) as ε → 0, for some ζ ∈ L^{2}(B). Returning to the trace equation (1.31), it follows from (B.2) and (B.10) that there are constants ĉ_{ε}, possibly unbounded as ε → 0, such that

(B.11)  Δν_{δ} = ζ_{δ} + ĉ_{ε},

where ζ_{δ} is bounded in L^{2}(B) and converges to ζ weakly in L^{2}(B). Suppose the limit (F, g′_{o}, x) is flat. Then as in (2.53), the function ϕ = \frac{1}{6} ĉ_{ε}r^{2} satisfies Δϕ = ĉ_{ε} on the limit F. Define ν̂_{δ} = ν_{δ} − \frac{1}{6} ĉ_{ε}r^{2}; it then follows that Δν̂_{δ} is uniformly bounded in L^{2}(B) and

(B.12)  Δν̂_{δ} → ζ,

weakly in L^{2}(B). The same argument holds for hyperbolic limits, using (2.57) in place of (2.53).

Now (B.6) holds as before and hence (B.7) holds, with ν̂_{δ} in place of ν_{δ}, or with D_{o}^{2} in place of D^{2}. Together with (B.12), it follows again from elliptic regularity that D^{2}ν̂_{δ} is uniformly bounded in L^{2}(B) which in turn implies that ᾱ\frac{∇Z^{2}}{δ} is uniformly bounded in L^{2}(B).

To prove (B.1), by the same arguments as above, it suffices to prove that the convergence in (B.12) is in the strong L^{2} topology. To prove this, return to the trace equation (2.69), i.e.

Δν_{δ} = \frac{1}{2}σT ρ^{2}ν_{δ} + c_{ε} + o(1) = ζ_{δ} + ĉ_{ε} + o(1),

where the constants may be unbounded as ε → 0 and o(1) → 0 in L^{2} as ε → 0. In particular, ζ_{δ} = \frac{1}{2}σT ρ^{2}ν_{δ}, mod additive constants. The coefficient σT ρ^{2} is uniformly bounded as ε → 0, since −σT ρ^{2}u is the scalar curvature of g′_{ε}. It follows that Δζ_{δ} = \frac{1}{2}σT ρ^{2}ζ_{δ} + \frac{1}{2}σT ρ^{2}ĉ_{ε}. If one pairs this equation with a cutoff function η and uses the self-adjointness of Δ, together with the fact that ζ_{δ} is weakly bounded in L^{2}, it follows that \frac{1}{2}σT ρ^{2}ĉ_{ε} is bounded as ε → 0. (This is the same proof as Lemma 7.8). This implies that Δζ_{δ} is bounded in L^{2} and hence, by elliptic regularity, ζ_{δ} converges strongly in L^{2} to its limit ζ.

This completes the proof of (B.1) at x_{ε} within B. To prove that (B.1) holds on all of (F, g′_{o}, x) choose any y ∈ F and let y_{ε} → y. Choosing y_{ε} to be new center points, the validity of (B.1) at y_{ε} implies that ᾱ∇Z^{2}/δ(y_{ε}) → 0 in L^{2}(B′_{y_{ε}}(\frac{1}{2})), where B′_{y_{ε}}(\frac{1}{2}) the the \frac{1}{2}-ball in the scale ρ′(y_{ε}) = 1. Suppose these balls B′_{x_{ε}}(\frac{1}{2}), B′_{y_{ε}}(\frac{1}{2}) have non-empty intersection. If x_{ε} is allowable, then Theorem 2.11, i.e. (2.64)-(2.65) holds for each on the intersection region. These equations and elliptic regularity imply that δ(y_{ε})/δ(x_{ε}) ∼ 1, i.e. the ratio is bounded away from 0 and ∞ as ε → 0. Hence, a covering argument shows that (B.1) holds on all of F, provided it holds on B′_{x_{ε}}(\frac{1}{2}). In fact, the arguments above following (B.10) give the same conclusions even if x_{ε} is not allowable.

Finally, suppose the base points x_{ε} are allowable, so that, after adding an affine term to ν_{δ}, ν_{δ} converges to a limit function ν. It then follows that one obtains C^{∞} convergence of z/δ to its limit. This will not be detailed here, (since its not actually needed), but the proof is exactly the same as that given in [2, Thm.4.2, Rmk.4.3].

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December, 2001

Department of Mathematics S.U.N.Y. at Stony Brook Stony Brook, N.Y. 11794-3651 anderson@math.sunysb.edu