EINSTEIN METRICS WITH PRESCRIBED CONFORMAL INFINITY ON 4-MANIFOLDS

MICHAEL T. ANDERSON

Abstract. This paper considers the existence of conformally compact Einstein metrics on 4manifolds. A reasonably complete understanding is obtained for the existence of such metrics with prescribed conformal infinity, when the conformal infinity is of positive scalar curvature. We find in particular that general solvability depends on the topology of the filling manifold. The obstruction to extending these results to arbitrary boundary values is also identified. While most of the paper concerns dimension 4, some general results on the structure of the space of such metrics hold in all dimensions.

1. Introduction.

This paper is concerned with the existence of conformally compact Einstein metrics on a given manifold M with boundary ∂M. The main results are restricted to dimension 4, although some of the results hold in all dimensions.

This existence problem was raised by Fefferman and Graham in [19] in connection with a study of conformal invariants of Riemannian manifolds. More recently, the study of such metrics has become of strong interest through the AdS/CFT correspondence, relating gravitational theories on M with conformal field theories on ∂M, cf. [18], [38] and references therein.

Let M be a compact, oriented manifold with non-empty boundary ∂M; M is assumed to be connected, but apriori ∂M may be connected or disconnected. A defining function ρ for ∂M in M is a non-negative C^{∞} function on the closure M̄ = M ∪ ∂M such that ρ^{−1}(0) = ∂M and dρ ≠ 0 on ∂M. A complete Riemannian metric g on M is conformally compact if there is a defining function ρ such that the conformally equivalent metric

(1.1)  g̃ = ρ^{2} · g

extends at least continuously to a Riemannian metric g̃ on M̄. The metric γ = g̃|_{∂M} induced on ∂M is the boundary metric induced by g and the compactification ρ. Any compact manifold with boundary carries many conformally compact metrics; for instance, one may let g̃ be any smooth metric on M̄ and define g by g = ρ^{−2} · g̃, for any choice of defining function ρ.

Defining functions for ∂M are unique only up to multiplication by positive smooth functions on M̄ and hence the compactification (1.1) is not uniquely determined by (M, g). On the other hand, the conformal class [γ] of a boundary metric is uniquely determined by g; the class [γ] is called the conformal infinity of (M, g).

A conformal compactification is called C^{m,α} or L^{k,p} if the metric g̃ extends to a C^{m,α} or L^{k,p} metric on the closure M̄; here C^{m,α} and L^{k,p} are the usual Hölder and Sobolev function spaces.

In this paper, we consider complete conformally compact Einstein metrics g on (n+1)-dimensional manifolds M, normalized so that

(1.2)  Ric_{g} = −ng.

Partially supported by NSF Grant DMS 0604735.

It is easy to see, cf. the Appendix, that any C^{2} conformally compact Einstein metric g satisfies |K_{g}+ 1| = O(ρ^{2}), where K_{g} denotes the sectional curvature of g. Hence, such metrics are asymptotically hyperbolic (AH), in that the local geometry tends to that of hyperbolic space at infinity.

Let E_{AH} = E_{AH}^{m,α} be the space of AH Einstein metrics on M which admit a C^{m,α} compactification g̃ as in (1.1). We require that m ≥ 3, α ∈ (0, 1) but otherwise allow any value of m, including m = ∞ or m = ω, (for real-analytic). The space E_{AH}^{m,α} is given the C^{m,α′} topology on metrics on M̄, for any fixed α′ < α, via a fixed compactification as in (1.1). Let E_{AH} = E_{AH}/D_{1}(M̄), where D_{1}(M̄) is the group of orientation preserving C^{m+1,α} diffeomorphisms of M̄ inducing the identity on ∂M, acting on E_{AH} in the usual way by pullback.

As boundary data, let Met(∂M) = Met^{m,α}(∂M) be the space of C^{m,α} metrics on ∂M and C = C^{m,α}(∂M) the corresponding space of pointwise conformal classes, as above endowed with the C^{m,α′} topology. There is a natural boundary map, (for any fixed (m, α)),

Π : E_{AH} → C (1.3)

Π[g] = [γ],

which takes an AH Einstein metric g on M to its conformal infinity on ∂M.

It is proved in [7] that when dimM = 4, and π_{1}(M, ∂M) = 0, the space E_{AH}^{m,α} is a C^{∞} infinite dimensional Banach manifold, and the boundary map Π is C^{∞} of Fredholm index 0, provided E_{AH}^{m,α} is non-empty. Moreover, the spaces E_{AH}^{m,α} are essentially independent of (m, α), in that these spaces are all diffeomorphic, with E_{AH}^{m′,α′} dense in E_{AH}^{m,α} whenever m′ + α′ > m + α. Essentially the same results also hold in in all dimensions n + 1 ≥ 4, see §2 for a more detailed discussion.

The issues of existence and uniqueness of AH Einstein metrics with a given conformal infinity are thus equivalent to the surjectivity and injectivity of the boundary map Π.

The main result of the paper which is used to approach the existence question is the following. Let C^{o} be the space of non-negative conformal classes [γ] on ∂M, in the sense that [γ] has a nonflat representative γ of non-negative scalar curvature. Let E_{AH}^{o} = Π^{−1}(C^{o}) be the space of AH Einstein metrics on M with conformal infinity in C^{o}. Thus, one has the restricted boundary map Π^{o} = Π|_{E_{AH}^{o}} : E_{AH}^{o} → C^{o}.

Theorem A. Let M be a 4-manifold satisfying π_{1}(M, ∂M) = 0 and for which the inclusion ι : ∂M → M̄ induces a surjection

(1.4)  H_{2}(∂M, 𝔽) → H_{2}(M̄, 𝔽) → 0,

for all fields 𝔽. Then for any (m, α), m ≥ 4, the boundary map

(1.5)  Π^{o} : E_{AH}^{o} → C^{o}

is proper.

Unfortunately (or fortunately), the boundary map Π is not proper in general. In fact, a sequence of distinct AH Einstein metrics {g_{i}} on M = ℝ^{2} × T^{2} was constructed in [6], whose conformal infinity is an arbitrary fixed flat metric on ∂M = T^{3}, but which has no convergent subsequence to an AH Einstein metric on M. These metrics are twisted versions of the AdS T^{2} black hole metrics, cf. Remark 2.6. The sequence {g_{i}} converges, (in a natural sense), to a complete hyperbolic cusp metric

(1.6)  g_{C} = dr^{2} + e^{2r}g_{T^{3}}

on the manifold N = ℝ×T^{3}. Since the boundary metric on T^{3} is flat and M satisfies (1.4), Theorem A is sharp, at least without further restrictions.

We will show in Theorem 5.4 that this behavior is the only way that Π is non-proper, in that divergent sequences {g_{i}} of AH Einstein metrics on a fixed 4-manifold M and with a fixed conformal infinity not in C^{o}, (or a convergent sequence of conformal infinities outside C^{o}), necessarily converge to AH Einstein manifolds (N, g) with cusps, although the cusps are not necessarily of the simple form (1.6). Thus, the completion Ē_{AH} of E_{AH} in a natural topology consists of AH Einstein metrics on M, together with AH Einstein metrics with cusps, cf. Theorem 5.4 and Corollary 5.5 for further details. Using this, we show in Corollary 5.6 that the boundary map Π is proper onto the subset Ĉ of C consisting of conformal classes which are not the boundary metrics of AH Einstein metrics with cusps.

The infinite sequence of AH Einstein metrics g_{i} on ℝ^{2} × T^{2} above all lie in distinct components of the space E_{AH}, so that this space has infinitely many components. On the other hand, the fact that Π^{o} is proper of course implies that E_{AH}^{o} has only finitely many components mapping into any compact region of C^{o}.

The topological condition (1.4) is used solely to rule out orbifold degenerations of AH Einstein sequences {g_{i}}. This condition may not be necessary, but it has been included in order to not overly complicate the paper; some situations where it is not needed are described in §7, cf. Remark 7.4. Similarly it may well be possible to weaken or drop the condition π_{1}(M, ∂M) = 0 on the fundamental group, but this will not be pursued further here.

Theorem A implies that at least on E_{AH}^{o}, the map Π^{o} is a proper Fredholm map of index 0. It follows from a result of Smale [34] that Π^{o} has a well-defined mod 2 degree, deg_{2}Π^{o} ∈ ℤ_{2}, on each component of E_{AH}^{o}. In fact, building on the work of Tromba [35] and White [36], [37] we show in Theorem 6.1 that Π^{o} has a ℤ-valued degree

(1.7)  degΠ^{o} ∈ ℤ,

again on components of E_{AH}^{o}. Tautologically, if degΠ^{o} ≠ 0, then Π^{o} is surjective.

We compute degΠ^{o} in certain cases via symmetry arguments. Namely, it is proved in [8], (cf. also Theorem 2.5 below) that any connected group of isometries of the conformal infinity [γ] on ∂M extends to a group of isometries of any AH Einstein filling metric (M, g), with Π[g] = [γ], in any dimension, again provided π_{1}(M, ∂M) = 0. Using this, and a straightforward classification of Einstein metrics with large symmetry groups leads to the following result.

Theorem B. Let M = B^{4} be the 4-ball, with ∂M = S^{3}, and, (abusing notation slightly), let C^{o} be the component of the non-negative C^{m,α} conformal classes containing the round metric on S^{3}. Also let E_{AH}^{o} be the component of Π^{−1}(C^{o}) containing the Poincaré metric on B^{4}. Then

(1.8)  deg_{B^{4}}Π^{o} = 1.

In particular, for any (m, α), m ≥ 4, any conformal class [γ] ∈ C^{o} on S^{3} is the conformal infinity of an AH Einstein metric on B^{4}.

Explicit examples of AH Einstein metrics on B^{4}, whose the conformal infinity is an arbitrary Berger sphere, were constructed by Pedersen [32]. More generally, Hitchin [26] has constructed such metrics with conformal infinity an arbitrary left-invariant metric on S^{3}.

It remains an open question whether the boundary map Π is surjective onto all of C when M = B^{4}. Based on reasoning from the AdS/CFT correspondence, Witten [38] has remarked that this may not be the case. It appears that physically, the natural class of boundary metrics are those of positive scalar curvature; in this regard, see also the work of Witten-Yau [39] proving that ∂M is necessarily connected when the conformal infinity has a component of positive scalar curvature. In any case, as remarked above in connection with Theorem 5.4, the only obstruction to surjectivity is the possible existence of AH Einstein cusp metrics associated to B^{4}, cf. also Remark 6.8.

Theorem B also holds when M is a disc bundle over S^{2}, so that ∂M = S^{3}/ℤ_{k}, provided k ≥ 10, with again C^{o} the component containing the standard round metric on S^{3}/ℤ_{k}, cf. Proposition 7.6.

We conjecture this result holds for any k ≥ 3. However, for M = S^{2} × ℝ^{2}, (i.e. k = 0), it is shown in Proposition 7.2, that

(1.9)  deg_{ℝ^{2}×S^{2}}Π^{o} = 0,

and further that Π^{o} is not surjective onto C^{o}. In fact, the conformal class of S^{2}(1) × S^{1}(L), for any L > 2π/\sqrt{3} is not the boundary metric of any AH Einstein metric on S^{2} × ℝ^{2}. This result is based on the work of Hawking-Page [23] on the AdS-Schwarzschild metric. Similarly, for M the degree 1 disc bundle over S^{2}, i.e. M = ℂℙ^{2} \ B^{4}, as well as the degree 2 disc bundle, Π^{o} is not surjective. These results for Π^{o} on disc bundles use properties of the AdS-Taub Bolt metrics analysed in [24],[31]. These results and others on deg Π^{o} are given in §7.

Theorem B and the related results above represent the first global results on an existence theory for complete Einstein metrics, outside the context of Kähler-Einstein metrics.

I would like to thank Piotr Chruściel, Robin Graham, Rafe Mazzeo, Michael Singer for their interest and comments on this work. Thanks also to the referees for a careful examination of the paper leading to improvements in the exposition.

2. Background Results

In this section, we discuss a number of background results from [6]-[8], [11] [19], [21] needed for the work to follow.

We begin with the structure of the moduli space E_{AH}^{m,α} of C^{m,α} conformally compact Einstein metrics on a given (n + 1)-dimensional manifold M. The results discussed below are from [7].

Let E_{AH}^{m,α} be the space of AH Einstein metrics on M which have a C^{m,α} conformal compactification with respect to a smooth (or C^{ω}) defining function ρ, as in (1.1). The space E_{AH}^{m,α} is given the C^{m,α′} topology on the space of C^{m,α} metrics Met^{m,α}(M̄) on M̄, for some α′ < α. Thus, a neighborhood of g ∈ E_{AH}^{m,α} is defined to be the set of metrics g′ whose compactification g′ by ρ lie in a neighborhood of g̃ in Met^{m,α}(M̄), where Met^{m,α}(M̄) is given the C^{m,α′} topology. Let Met^{m,α} be the corresponding space of metrics on ∂M, again with the C^{m,α′} topology. One thus has a natural boundary map

(2.1)  Π = Π^{m,α} : E_{AH}^{m,α} → Met^{m,α}(∂M),

sending g to its boundary metric γ.

An element [g] in the moduli space E_{AH}^{m,α} is an equivalence class of a C^{m,α} conformally compact Einstein metric g on M, where g ∼ g′ if g′ = ϕ^{∗}g, with ϕ a C^{m+1,α} diffeomorphism of M̄, equal to the identity on ∂M, i.e. ϕ ∈ D_{1}. Changing g by a diffeomorphism in D_{1}, or changing the defining function ρ in (1.1) changes the boundary metric conformally. Hence if C^{m,α} denotes the space of conformal classes of C^{m,α} metrics on ∂M, then the boundary map

(2.2)  Π = Π^{m,α} : E_{AH}^{m,α} → C^{m,α}

is well defined. When there is no danger of confusion, we will usually work with any given representative g ∈ [g] and γ ∈ [γ].

Suppose dimM = n + 1 = 4, and the 4-manifold M satisfies π_{1}(M, ∂M) = 0. Then if E_{AH}^{m,α} is non-empty, the spaces E_{AH}^{m,α} and E_{AH}^{m,α} are C^{∞} infinite dimensional separable Banach manifolds, and the boundary maps Π in (2.1) and (2.2) are C^{∞} smooth maps of Banach manifolds, of Fredholm index 0. This result holds for any m ≥ 3, and α ∈ (0, 1), and, replacing Banach by Fréchet, also for m = ∞ and m = ω, (i.e. the real-analytic case). Implicit in this statement is the following boundary regularity result: an AH Einstein metric with a C^{2} conformal compactification which has a C^{m,α} boundary metric, m ≥ 2, has a C^{m,α} conformal compactification.

In addition, the spaces E_{AH}^{m,α} are all diffeomorphic and for any (m′, α′) with m′ + α′ > m + α, the space E_{AH}^{m′,α′} is dense in E_{AH}^{m,α}. In particular,

(2.3)  \overline{E_{AH}^{ω}} = E_{AH}^{m,α},

where the completion is taken in the C^{m,α′} topology.

If dimM = n+1 > 4, then these results hold when 3 ≤ m ≤ n−1, using the boundary regularity results in [30]. In addition, the smooth manifold result holds when m = ∞, if, when n is even, E_{AH}^{∞} is understood to be the space of AH Einstein metrics which are C^{∞} polyhomogeneous; this uses the boundary regularity result of [16], cf. also the discussion following (2.10) below.

Let 𝕊^{m,α}(M̄) denote the space of C^{m,α} symmetric bilinear forms on M̄, and similarly for M and ∂M. A tangent vector h to the Banach manifold E_{AH}^{m,α} at a representative g ∈ [g], i.e. an infinitesimal Einstein deformation of g, is a form h ∈ 𝕊^{m,α}(M̄) satisfying the equation

(2.4)  L(h) = \frac{1}{2}D^{∗}Dh − R(h) = 0, β_{g}(h) = 0,

where R is the action of the curvature tensor of g on symmetric bilinear forms and β_{g} = δ_{g} + \frac{1}{2}dtr_{g} is the Bianchi operator with respect to g. Such h are transverse to the orbits of the diffeomorphism group D_{1} acting on M̄. Let T_{g}E_{AH}^{m,α} denote the space of such tangent vectors.

An Einstein metric (M, g) is non-degenerate if the operator L has no kernel in L^{2}(M, g),

(2.5)  K = L^{2} − KerL = 0.

For g ∈ E_{AH}^{m,α}, the L^{2} kernel K equals the kernel of the linear map DΠ : T_{g}E_{AH}^{m,α} → T_{Π(g)}C^{m,α}. Hence, g is non-degenerate if and only if g is a regular point of the boundary map Π and thus Π is a local diffeomorphism near g. Any element κ ∈ K is transverse-traceless, i.e.

(2.6)  δ_{g}κ = 0, tr_{g}κ = 0,

and so satisfies the Bianchi gauge condition (2.4).

In the next sections, we will frequently consider compactifications ḡ of g by a geodesic defining function t, for which

(2.7)  t(x) = distḡ(x, ∂M).

A compactification ḡ = t^{2}g satisfying (2.7) is called a geodesic compactification and t is a geodesic defining function. Such compactifications are natural from a number of viewpoints. In particular the curvature of ḡ has a particularly simple form; this and related issues are discussed in the the Appendix. It is not difficult to see that given a C^{2} conformally compact Einstein metric g with boundary metric γ in some compactification, there is a unique geodesic compactification ḡ of g with boundary metric γ. Further if g̃ is some C^{m,α} compactification, then then the geodesic compactification ḡ is at least C^{m−1,α} off the cutlocus C̄ of ∂M in (M, ḡ).

Since the integral curves of ¯∇t are geodesics, the metric ḡ splits as

(2.8)  ḡ = dt^{2} + g_{t},

within a collar neighborhood U (inside the cutlocus) of ∂M; here g_{t} is a curve of metrics on the boundary ∂M with g_{0} = γ. Setting r = − log t, a simple calculation shows that the integral curves of ∇r are also geodesics in (M, g) and so the metric g also splits as g = dr^{2} + g_{r}.

The Fefferman-Graham expansion [19] is the formal expansion of the curve g_{t} in a Taylor-like series: the specific form of the expansion depends on n. If dimM = n+1 is even, then the expansion reads

(2.9)  g_{t} ∼ g_{(0)} + t^{2}g_{(2)} + ⋯ + t^{n−1}g_{(n−1)} + t^{n}g_{(n)} + ⋯,

where the coefficients are symmetric bilinear forms on ∂M. The expansion is in even powers of t up to order n − 1 and the terms g_{(2k)} are intrinsically determined by the boundary metric γ = g_{(0)} and its tangential derivatives up to order 2k, for 2k ≤ n − 1. The term g_{(n)} is transverse-traceless; however, beyond this, g_{(n)} is not locally determined by γ; it depends on global properties of the Einstein metric (M, g). For example, when n = 3, one has

g_{(3)} = \frac{1}{6} ¯∇_{N} R̄ic,

where R̄ic is the Ricci curvature and N = ¯∇t is the unit normal of ∂M in (M̄, ḡ). This term is not computable in terms of γ; it depends on the global properties of (M, g). On the other hand, the higher terms g_{(k)}, k > n depend on two tangential derivatives of g_{(k−2)}.

If dimM = n + 1 is even, then the expansion is

(2.10)  g_{t} ∼ g_{(0)} + t^{2}g_{(2)} + ... + t^{n}g_{(n)} + t^{n}(log t)H + ⋯,

where again this is an even expansion up to order n, with coefficients up to order (n − 2) locally determined by γ. The term H, called the obstruction tensor in [19], is naturally identified as the stress-energy tensor of the integral L of the conformal anomaly, (see (2.16) below), via the AdS/CFT correspondence, cf. [18], [33]; it is transverse-traceless and also locally determined by γ. The trace and divergence of g_{(n)} are determined by γ; in fact there is a symmetric bilinear form r_{(n)} and scalar function a_{(n)}, both explicitly computable from γ and its derivatives, such that

(2.11)  δ(g_{(n)} + r_{(n)}) = 0, tr(g_{(n)} + r_{(n)}) = a_{(n)}.

The term a_{(n)} is the conformal anomaly, (so that ∫_{∂M} a_{(n)} = L). However, as above, g_{(n)} is otherwise only globally determined by (M, g). The higher order terms in (2.10) contain terms of the form t^{m}(log t)^{p}. The expansion is even in powers of t, and each coefficient g_{(k)}, k ≠ n, depends on two derivatives of g_{(k−2)}. Note that (2.11) also holds when n is odd, in which case r_{(n)} = a_{(n)} = 0. In both cases, all terms in the expansion are thus determined by the two terms g_{(0)} and g_{(n)} or τ_{(n)}.

Let

(2.12)  τ_{(n)} = g_{(n)} + r_{(n)},

so that τ_{(n)} is divergence-free. This term will play an important role thoughout much of the paper, in particular in §4. In the AdS/CFT correspondence τ_{(n)} corresponds, (up to a constant), to the stress-energy tensor of the corresponding CFT on ∂M, cf. (2.19) below.

The boundary regularity results of [6], [7], [16] discussed above give the existence of the formal series (2.9) and (2.10); in the C^{∞} case, these are well-defined asympotic series for the curve g_{t}. If the data (g_{(0)}, g_{(n)}) are real-analytic, a result of Kichenassamy [28] shows that the series (2.9) and (2.10) converge to g_{t}, so that such data determine a solution to the Einstein equations defined in a neighborhood of ∂M.

The next local unique continuation result from [8] shows that the terms g_{(0)} and g_{(n)} uniquely determine an AH Einstein metric up to isometry, near the boundary.

Proposition 2.1. Let g^{1} and g^{2} be two Einstein metrics defined on a half-ball U ≃ (ℝ^{n+1})^{+}, which have a C^{3,α} conformal compactification to the closed half-ball Ū. If

(2.13)  |g^{1} − g^{2}| = o(t^{n}),

where t is the distance to ∂U ≃ ℝ^{n}, (in either compactified metric), then g^{1} is isometric to g^{2} in U.

Of course (2.13) holds if g^{1} and g^{2} are C^{n} polyhomogeneous conformally compact with identical g_{(0)} and g_{(n)} terms. Since Einstein metrics are real-analytic in local harmonic coordinates, local unique continuation in the interior, away from the boundary, is well-known.

Observe that the terms g_{(k)} depend on the choice of the boundary metric γ ∈ [γ]. If γ̃ = λ^{2}γ, then the coefficients g̃_{(k)} are determined explicitly by λ and γ, cf. [18]. For n odd, the tranformation rule for the undetermined coefficient g_{(n)} has the simple form

(2.14)  g̃_{(n)} = λ^{−(n−2)}g_{(n)}.

In particular |g̃_{(n)}|g̃ = λ^{−n}|g_{(n)}|_{g}. For n even, the transformation rule is more complicated; it has the form (2.14) at leading order, but has lower order terms depending explicitly on the derivatives of γ and λ. Exact formulas in low dimensions are given in [18], [33].

Next we consider certain global issues associated with AH Einstein metrics which arise via the AdS/CFT correspondence. Let g be an AH Einstein metric on an (n + 1)-manifold M with boundary metric γ, and such that the corresponding geodesic compactification is at least C^{n} in the polyhomogeneous sense. The expansions (2.9)-(2.10) easily lead to an expansion for the volume of the region B(t) = {x ∈ M : t(x) ≤ t}. As before, the form of the expansion depends on the parity of n. For n odd,

(2.15)  volB(t) = v_{(n)}t^{−n} + v_{(n−2)}t^{−(n−2)} + ⋯ + V + o(1),

while for n even,

(2.16)  volB(t) = v_{(n)}t^{−n} + v_{(n−2)}t^{−(n−2)} + ⋯ + L log t + V + o(1).

The constant term V is called the renormalized volume. More importantly for our purposes, let I_{EH} be the Einstein-Hilbert action, (with Gibbons-Hawking-York boundary term), given by

(2.17)  I_{EH} = ∫_{M}(s − 2Λ)dvol + \frac{1}{2} ∫_{∂M} Hdv,

where s is the scalar curvature, Λ is the cosmological constant and H is the mean curvature of the boundary. In the normalization (1.2), Λ = −n(n − 1)/2. Einstein metrics, with scalar curvature −n(n + 1), are critical points of I_{EH}, among variations fixing the boundary metric. (This is of course not the case for the volume functional). The action I_{EH} is infinite on M, but since s is constant on Einstein metrics, (i.e. on shell), the cut-off action has an expansion of the same form as (2.15) or (2.16). Subtracting the divergent contributions as in (2.15)-(2.16) gives the renormalized action I_{EH}^{ren}. On the space E_{AH}, i.e. on shell, when n is odd one has

I_{EH}^{ren} = −2nV,

as functionals on E_{AH}; the boundary term in (2.17) renormalizes to 0. Similarly in this case, the renormalized action or volume is in fact independent of the defining function t, so these functionals descend to functionals on the moduli space E_{AH}. Neither of these facts hold however for n even.

When dimM = 4, it is proved in [5] that on E_{AH}, I_{EH}^{ren} and V are related to the square of the L^{2} norm of the Weyl curvature W of (M, g) in the following simple way:

(2.18)  \frac{1}{8π^{2}} ∫_{M} |W |^{2} = χ(M) + \frac{1}{8π^{2}} I_{EH}^{ren} = χ(M) − \frac{3}{4π^{2}} V.

The relation (2.18) will play an important role in understanding the structure of the space E_{AH}, in particular in §3 and §5.

It is proved in [5], [18], cf. also [33], that as a 1-form on E_{AH}^{m,α} the differential of the renormalized action is given by

(2.19)  d_{g}I_{EH}^{ren}(h) = c_{n} ∫_{∂M}⟨τ_{(n)}, h_{(0)}⟩dv,

where τ_{(n)} is as in (2.12), the inner product and volume form are with respect to γ and c_{n} is a constant depending only on n. The term h_{(0)} is the variation of the boundary metric induced by h, i.e. h_{(0)} = Π_{∗}(h).

For later purposes, we need a slight but important extension of (2.19). Thus, given g ∈ E_{AH}, define an enlarged “tangent” space T̃_{g}E_{AH} roughly as follows: h ∈ T̃_{g}E_{AH} if h ∈ 𝕊^{m,α}(M) and h is an infinitesimal Einstein deformation to order at least n at t = 0, in the sense that the expansion (2.9) or (2.10) holds to order n in t, to 1^{st} order in the variation h. More precisely, consider the curve g_{u} = g + uh and assume without loss of generality that the geodesic defining function associated to g_{u} is the fixed function t; (this can always be achieved by modifying g_{u} by a curve of diffeomorphisms in D_{1} if necessary). Then ḡ_{u} has the expansion

(2.20)  ḡ_{u} = dt^{2} + (g_{(0)} + uh_{(0)}) + t(g_{(1)} + uh_{(1)}) + t^{2}(g_{(2)} + uh_{(2)}) + ⋯ + t^{n} log t(H_{g+uh})
+t^{n}(g_{(n)} + uh_{(n)}) + O(t^{n+α}).

Then h ∈ T̃_{g}E_{AH} means that the coefficients h_{(i)} = \frac{d}{du}(g_{(i)} + uh_{(i)})|_{u=0} are the linearizations of the coefficients g_{(i)} and H in (2.9)-(2.10) in the direction h_{(0)}, for i ≤ n. Similarly, the linearization of the trace condition in (2.11) should hold in the direction h_{(0)}. Here we recall that the coefficients g_{(i)}, H and trg_{(n)} are explicitly determined from the Einstein equations by the boundary metric γ = g_{(0)}.

Lemma 2.2. For any boundary variation h_{(0)} ∈ 𝕊^{2}(∂M), there exists h ∈ T̃_{g}E_{AH} such that h induces h_{(0)} at ∂M, i.e. for Π as in (2.1), Π_{∗} : T̃_{g}E_{AH} → 𝕊^{2}(∂M) is surjective.

Moreover, at any g ∈ E_{AH}, the equation (2.19) holds for any h ∈ T̃_{g}E_{AH}.

Proof: The first statement follows immediately from the fact that the equations for the terms h_{(i)} in the expansion of h can easily be solved algebraically. For example, the g_{(2)} term is given explicitly by

g_{(2)} = \frac{1}{n − 2}(Ric_{γ} − \frac{s_{γ}}{2(n − 1)}γ).

The linearization of the right-hand side of this equation in the direction h_{(0)} then defines the term h_{(2)}. The same reasoning applies to all the other coefficients, including the log coefficient H, as well as the trace constraint (2.11). Of course h_{(i)} = 0 if i is odd, i < n.

The second statement follows directly from the proof of (2.19) in [5], to which we refer for some further details. First, it is easy to see that I_{EH}^{ren} is well-defined, and so finite, for any metric g_{u} = t^{−2}ḡ_{u} as in (2.20) which is Einstein to n^{th} order at ∂M; see also [5, Remark 1.2]. Further, as noted above, Einstein metrics are critical points of I_{EH}^{ren}, and so its variation in any direction depends only on the variation of the metric at the boundary. Thus, dI_{EH}^{ren} depends only on the boundary terms g_{(j)} in the Fefferman-Graham expansion (2.9)-(2.10). Since it is exactly the form of these terms which leads to the formula (2.19), and since this form is preserved for the metrics g_{u}, (to 1st order in u), it is essentially clear that (2.19) holds for h ∈ T̃_{g}E_{AH}. Alternately, this may verified directly by an examination of the proof of (2.19) in [5, Lemma 2.1, Theorem 2.2], with I_{EH} in place of volume.

The divergence constraint in (2.11) is not used in the computation of I_{EH}^{ren} or its first variation. (It does arise however in the second variation). Let D_{g}E_{AH} ⊂ 𝕊^{m,α}(M) be the subspace consisting of forms satisfying the linearized divergence constraint (2.11) at ∂M, so that

(2.21)  δ′(τ_{(n)}) + δτ′_{(n)} = 0,

where δ′ = \frac{d}{du}δ_{(g_{(0)}+uh_{(0)})} and τ′_{(n)} = g′_{(n)} + r′_{(n)} = h_{(n)} + s_{(n)}, s′_{(n)} = \frac{d}{du}(r_{(n)})_{(g_{(0)}+uh_{(0)})}. Of course s_{(n)} = 0 for n odd.

The space F_{g}E_{AH} = D_{g}E_{AH} ∩ T̃_{g}E_{AH} represents the space of formal solutions of the linearized Einstein equations near ∂M, in that any h ∈ F_{g}E_{AH} defines uniquely a formal series solution as in (2.9)-(2.10) of the linearized Einstein equations. If h_{(0)} and h_{(n)} are real-analytic on ∂M, the result of Kichenassamy [28] mentioned above implies that the series converges to an actual linearized Einstein deformation h defined near ∂M. Note that while the divergence and trace of h_{(n)} are determined by h_{(0)}, the transverse-traceless part of h_{(n)} may be freely chosen.

Proposition 2.3. For g ∈ E_{AH}, the map Π_{∗} : D_{g}E_{AH} → 𝕊^{2}(∂M), Π_{∗}(h) = h_{(0)}, is surjective.

Proof: Given Lemma 2.2, one needs to show that (2.21) is solvable, for any boundary variation h_{(0)}, i.e. δ′(τ_{(n)}) ∈ Imδ, for all variations h_{(0)} of γ. The space of 1-forms on ∂M has the splitting

Ω^{1}(∂M) = Imδ ⊕ Kerδ^{∗},

so that if Kerδ^{∗} = 0, i.e. (∂M, γ) has no Killing fields, then the result is clear. When (∂M, γ) does have Killing fields, this result is far from clear; using Theorem 2.5 below, this result is proved in [8], to which we refer for details.

We will see later in §4, (in and following (4.45)), that Π_{∗} is surjective on the full formal space F_{g}E_{AH}. It is worth noting that the global boundary map Π_{∗} : T_{g} → 𝕊^{2}(∂M) is not surjective in general, cf. §7. Thus, the formal solutions of the linearized Einstein equations in F_{g}E_{AH}, even if they give rise to actual linearized solutions, do not extend in general to smooth solutions on the compact manifold M.

On the other hand, Proposition 2.3 is false for AH Einstein metrics defined only in a neighborhood or thickening of ∂M. The proof of Proposition 2.3 in [8] is global; it requires that (M, g) is conformally compact, as does Theorem 2.5 below.

The following result will play an important role in §4, cf. the proof of Proposition 4.6.

Proposition 2.4. For g ∈ E_{AH} and h ∈ F_{g}E_{AH}, let σ_{(n)} = τ′_{(n)}, for τ′_{(n)} as in (2.21). Similarly, let a′_{(n)} be the variation of the conformal anomaly a_{(n)} in the direction h. Then for any boundary metric (∂M, γ), γ ∈ [γ] = Π[g], and smooth vector field X on ∂M, one has

(2.22)  ∫_{∂M}⟨L_{X}τ_{(n)} + [(1 − \frac{2}{n})divX]τ_{(n)}, h_{(0)}⟩dV = ∫_{∂M}⟨σ_{(n)} + \frac{1}{2}trh_{(0)}τ_{(n)}, L̂_{X}γ⟩dV +
\frac{1}{n} ∫_{∂M} divX(a_{(n)}trh_{(0)} + 2a′_{(n)})dV,

where L̂_{X}γ is the conformal Killing operator on ∂M, L̂_{X}γ = L_{X}γ − \frac{2divX}{n} γ. The formula (2.22) holds for any variation h_{(0)} of γ.

Proof: This is a simple consequence of a result proved in [8]. Namely, from [8, Prop.5.4], one has

(2.23)  ∫_{∂M}⟨L_{X}τ_{(n)}, h_{(0)}⟩ = −2 ∫_{∂M}⟨δ′(τ_{(n)}), X⟩ + ∫_{∂M} δX⟨τ_{(n)}, h_{(0)}⟩ + ⟨τ_{(n)}, δ^{∗}X⟩trh_{(0)}dV,

where δ′ = \frac{d}{du}δ_{γ+uh_{(0)}}. Using the definition of L̂_{X}γ, this gives

∫_{∂M}⟨L_{X}τ_{(n)} + divXτ_{(n)}, h_{(0)}⟩ = −2 ∫_{∂M}⟨δ′(τ_{(n)}), X⟩
+\frac{1}{2} ∫_{∂M}⟨τ_{(n)}, L̂_{X}γ⟩trh_{(0)}dV + \frac{1}{n} ∫_{∂M} a_{(n)}divXtrh_{(0)}dV.

Next, by (2.21) and Proposition 2.3, δ′τ_{(n)} = −δσ_{(n)}, for any variation h_{(0)} of the boundary metric γ, so that

−2 ∫_{∂M}⟨δ′τ_{(n)}, X⟩ = 2 ∫_{∂M}⟨δσ_{(n)}, X⟩ = 2 ∫_{∂M}⟨σ_{(n)}, δ^{∗}X⟩ = ∫_{∂M}⟨σ_{(n)}, L_{X}γ⟩.

Write L_{X}γ = L̂_{X}γ + \frac{2divX}{n} γ. Then ⟨σ_{(n)}, γ⟩ = trσ_{(n)} = −tr′(τ_{(n)}) + a′_{(n)} = ⟨τ_{(n)}, h_{(0)}⟩ + a′_{(n)}, where

the middle equality follows from the linearization of (2.11) with h ∈ T̃_{g}E_{AH}. Combining these computations gives (2.22).

Observe that the first term on the right-hand side of (2.22) vanishes when X is a conformal Killing field of (∂M, [γ]). Using (2.14), it is easy to see that both sides of (2.22) are conformally invariant when n is odd; on the other hand, both sides of (2.22) depend on the representative γ ∈ [γ] when n is even. Note also that (2.22) is invariant under the addition of transverse-traceless terms to σ_{(n)}.

The following result, also from [8] and used to prove Proposition 2.3, will be frequently used in §7.

Theorem 2.5. Let g be an AH Einstein metric on an (n + 1)-manifold M, with C^{∞} boundary metric γ, (in some compactification). Then any connected Lie group of isometries of (∂M, γ) extends to an action by isometries on (M, g).

Remark 2.6. Finally, for the work in §5, we mention briefly the class of toral AdS black hole metrics on M = ℝ^{2} × T^{n−1}. These are AH Einstein metrics given explicitly by

(2.24)  g_{m} = V^{−1}dr^{2} + V dθ^{2} + r^{2}g_{T^{n−1}},

where V (r) = r^{2} − \frac{2m}{r^{n−2}} , θ ∈ [0, 4π/nr_{+}], r_{+} = (2m)^{1/n} > 0 and g_{T^{n−1}} is any flat metric on the torus T^{n−1}, cf. [6, Prop.4.4] and references therein. The boundary metric of g_{m} is a flat (product) metric on the n-torus T^{n}.

Twisted versions of these metrics, obtained by taking suitable covering spaces and then passing to (different) discrete quotients, give rise to infinitely many isometrically distinct AH Einstein metrics g_{i} on ℝ^{2} × T^{n−1} with a fixed flat boundary metric T^{n−1}. These metrics are also the examples discussed following the statement of Theorem A. It is worth noting that the metrics g_{m} are all locally isometric.

3. Compactness I: Interior Behavior.

The purpose of the next two sections is to lay the groundwork for the proof of Theorem A; this section deals with the interior behavior, while §4 is concerned with the behavior near the boundary. The property that the boundary map Π is proper is a compactness issue. Thus, given a sequence of boundary metrics γ_{i} converging to a limit metric γ, one needs to prove that a sequence of AH Einstein metrics (M, g_{i}) with boundary metrics γ_{i} has a subsequence converging, modulo diffeomorphisms, to a limit AH Einstein metric g on M, with boundary metric γ.

In §3.1, we summarize background material on convergence and degeneration of sequences of metrics in general, as well as sequences of Einstein metrics. This section may be glanced over and then referred to as necessary. The following section §3.2 then applies these results to the interior behavior of AH Einstein metrics.

§3.1. In this section, we discuss L^{p} Cheeger-Gromov theory as well as the convergence and degeneration results of Einstein metrics on 4-manifolds from [1]-[3].

We begin with the L^{p} Cheeger-Gromov theory. The L^{∞} Cheeger-Gromov theory [15], [22], describes the (moduli) space of metrics on a manifold, (or sequence of manifolds), with uniformly bounded curvature in L^{∞}, i.e.

(3.1)  |R_{g}|(x) ≤ Λ < ∞,

in that it describes the convergence or possible degenerations of sequences of metrics satisfying the bound (3.1). The space L^{∞} is not a good space on which to carry out analysis, and so we replace (3.1) by a corresponding L^{p} bound, i.e.

(3.2)  ∫_{M} |R_{g}|^{p}dV ≤ Λ < ∞.

The curvature involves 2 derivatives of the metric, and so (3.2) is analogous to an L^{2,p} bound on the metric. The critical exponent p with respect to Sobolev embedding L^{2,p} ⊂ C^{o} is (n + 1)/2, where n + 1 = dimM and hence we will always assume that

(3.3)  p > (n + 1)/2.

In order to obtain local results, we need the following definitions of local invariants of Riemannian metrics, cf. [4].

Definition 3.1. If (M, g) is a Riemannian (n + 1)-manifold, the L^{p} curvature radius ρ(x) ≡ ρ^{p}(x) at x is the radius of the largest geodesic ball B_{x}(ρ(x)) such that, for all B_{y}(s) ⊂ B_{x}(ρ(x)), with s ≤ dist(y, ∂B_{x}(ρ(x))), one has

(3.4)  \frac{s^{2p}}{volB_{y}(s)} ∫_{B_{y}(s)} |R|^{p}dV ≤ c_{0},

where c_{0} is a fixed sufficiently small constant. Although c_{0} is an essentially free parameter, we will fix c_{0} = 10^{−2} throughout the paper. The left-side of (3.4) is a scale-invariant local average of the curvature in L^{p}.

The volume radius ν(x) of (M, g) at x is given by

(3.5)  ν(x) = sup\{r : \frac{volB_{y}(s)}{ω_{n+1}s^{n+1}} ≥ μ_{0}, ∀B_{y}(s) ⊂ B_{x}(r)\},

where ω_{n+1} is the volume of the Euclidean unit (n + 1)-ball and again μ_{0} > 0 is a free small parameter, which will be fixed in any given discussion, e.g. μ_{0} = 10^{−2}.

The L^{k,p} harmonic radius r_{h}^{k,p}(x) is the largest radius such that on the geodesic ball B_{x}(r), r = r_{h}^{k,p}(x), there is a harmonic coordinate chart in which the metric components satisfy

(3.6)  C^{−1}δ_{ij} ≤ g_{ij} ≤ Cδ_{ij}
r^{kp−(n+1)} ∫_{B_{x}(r)} |∂^{k}g_{ij}|^{p}dV_{g} ≤ C,

where C is a fixed constant; again C may be arbitrary, but will be fixed to C = 2, (for example).

Observe that ρ(x), ν(x) and r_{h}(x) scale as distances, i.e. if g′ = λ^{2} · g, for some constant λ, then ρ′(x) = λ · ρ(x), ν′(x) = λ · ν(x) and r′_{h}(x) = λ · r_{h}(x). By definition, ρ, ν and r_{h} are Lipschitz functions with Lipschitz constant 1; in fact for y ∈ B_{x}(ρ(x)), it is immediate from the definition that

(3.7)  ρ(y) ≥ dist(y, ∂B_{x}(ρ(x))),

and similarly for ν, r_{h}.

One may define L^{k,p}, (or C^{k,β}) curvature radii ρ^{k,p} in a manner completely analogous to (3.4). For the L^{k,p} radius, the curvature |R| term in (3.4) is replaced by ∑_{j≤k} |∇^{j}R| and the power of s is chosen to make the resulting expression scale invariant. Clearly

ρ^{k,p} ≥ ρ^{k′,p′},

whenever k + p ≥ k′ + p′. It is proved in [3] that the curvature radius and harmonic radius are essentially equivalent, given a lower bound on the volume radius. Thus

(3.8)  r_{h}^{2,p}(x) ≥ r_{0}ρ^{p}(x),

where r_{0} depends only on a lower bound ν_{0} for ν(x). The same statement holds for the radii r_{h}^{k+2,p} and ρ^{k,p}.

A sequence of Riemannian metrics (Ω_{i}, g_{i}) is said to converge in the L^{k,p} topology to a limit L^{k,p} metric g on Ω if there is an atlas A for Ω and diffeomorphisms F_{i} : Ω → Ω_{i} such that F_{i}^{∗}(g_{i}) converges to g in the L^{k,p} topology in local coordinates with respect to the atlas A. Thus, the local components (F_{i}^{∗}(g_{i}))_{αβ} → g_{αβ} in the usual L^{k,p} Sobolev topology on functions on ℝ^{n+1}. Similar definitions hold for C^{m,α} convergence, and convergence in the weak L^{k,p} topology. Any bounded sequence in L^{k,p} has a weakly convergent subsequence, and similarly any bounded sequence in C^{m,α} has a convergent subsequence in the C^{m,α′} topology, for any α′ < α. By Sobolev embedding,

L^{k,p} ⊂ C^{m,α},

for m + α < k − \frac{n+1}{p}. In particular, in dimension 4, for p ∈ (2, 4), L^{2,p} ⊂ C^{α}, α = 2 − \frac{4}{p}, and for p > 4, L^{2,p} ⊂ C^{1,α}, for α = 1 − \frac{4}{p}.

One then has the following result on the convergence and degeneration of metrics with bounds on ρ, cf. [4].

Theorem 3.2. Let (Ω_{i}, g_{i}, x_{i}) be a pointed sequence of connected Riemannian (n + 1)-manifolds and suppose there are constants ρ_{o} > 0, d_{o} > 0 and D < ∞ such that, for a fixed p > (n + 1)/2,

(3.9)  ρ^{p}(y) ≥ ρ_{o}, diamΩ_{i} ≤ D, dist(x_{i}, ∂Ω_{i}) ≥ d_{o}.

Then for any 0 < ϵ < d_{o}, there are domains U_{i} ⊂ Ω_{i}, with ϵ/2 ≤ dist(∂U_{i}, ∂Ω_{i}) ≤ ϵ, for which one of the following alternatives holds.

(I). (Convergence) If there is constant ν_{o} > 0 such that,

ν_{i}(x_{i}) ≥ ν_{o} > 0,

then a subsequence of {(U_{i}, g_{i}, x_{i})} converges, in the weak L^{2,p} topology, to a limit L^{2,p} Riemannian manifold (U, g, x), x = lim x_{i}. In particular, U_{i} is diffeomorphic to U, for i sufficiently large.

(II). (Collapse) If instead,

ν(x_{i}) → 0,

then U_{i} has an F-structure in the sense of Cheeger-Gromov, cf. [15]. The metrics g_{i} are collapsing everywhere in U_{i}, i.e. ν_{i}(y_{i}) → 0 for all y_{i} ∈ U_{i}, and so in particular the injectivity radius inj_{g_{i}}(y_{i}) → 0. Any limit of {g_{i}} in the Gromov-Hausdorff topology [22] is a lower dimensional length space.

Remark 3.3. (i). The local hypothesis on ρ^{p} in (3.9) can be replaced by the global hypothesis

(3.10)  ∫_{Ω_{i}} |R|^{p}dV ≤ Λ,

in case the volume radius satisfies ν_{i}(y_{i}) ≥ ν_{o}, for some ν_{o} > 0 and all y_{i} ∈ Ω_{i}; in fact under this condition (3.9) and (3.10) are then equivalent, with Λ = Λ(ρ_{o}, D, ν_{o}).

(ii). It is an easy consequence of the definitions that the L^{p} curvature radius ρ^{p} is continuous under convergence in the (strong) L^{2,p} topology, i.e. if g_{i} → g in the L^{2,p} topology, then

(3.11)  ρ_{i}(x_{i}) → ρ(x),

whenever x_{i} → x, cf. [4] and references therein. However, (3.11) does not hold if the convergence is only in the weak L^{2,p} topology.

If in addition to a bound on ρ^{p} one has L^{p} bounds the covariant derivatives of the Ricci curvature up to order k, then in Case (I) one obtains convergence to the limit in the L^{k+2,p} topology. Analogous statements hold for convergence in the C^{m,α′} topology.

Next we discuss the convergence and degeneration of Einstein metrics. If M is a closed 4-manifold and g an Einstein metric on M, then the Chern-Gauss-Bonnet theorem gives

(3.12)  \frac{1}{8π^{2}} ∫_{M} |R|^{2}dV = χ(M),

where χ(M) is the Euler characteristic of M. This gives apriori control on the L^{2} norm of the curvature of Einstein metrics on M. However the L^{2} norm is critical in dimension 4 with respect to Sobolev embedding, cf. (3.3) and so one may not expect Theorem 3.2 to hold for sequences of Einstein metrics on M. In fact, there is a further possible behavior of such sequences.

Definition 3.4. An Einstein orbifold (X, g) is a 4-dimensional orbifold with a finite number of cone singularities {q_{j}}, j = 1, ..., k. On X_{o} = X \ ∪{q_{j}}, g is a smooth Einstein metric while each singular point q ∈ {q_{j}} has a neighborhood U such that U \ q is diffeomorphic to C(S^{3}/Γ) \ {0}, where Γ is a finite subgroup of O(4) and C denotes the cone with vertex {0}. Further Γ ≠ {e} and when lifted to the universal cover B^{4} \ {0} of C(S^{3}/Γ) \ {0}, the metric g extends smoothly over {0} to a smooth Einstein metric on the 4-ball B^{4}.

There are numerous examples of sequences of smooth Einstein metrics g_{i} on a compact manifold M which converge, in the Gromov-Hausdorff topology, to an Einstein orbifold limit (X, g). Such orbifold metrics are not smooth metrics on the manifold M, but may be viewed as singular metrics on M, in that M is a resolution of X.

We then have the following result describing the convergence and degeneration of Einstein metrics on 4-manifolds, cf. [3].

Theorem 3.5. Let (Ω_{i}, g_{i}, x_{i}) be a pointed sequence of connected Einstein 4-manifolds satisfying

(3.13)  diamΩ_{i} ≤ D, dist(x_{i}, ∂Ω_{i}) ≥ d_{o},

and

(3.14)  ∫_{Ω_{i}} |R_{g_{i}}|^{2}dV_{g_{i}} ≤ Λ_{o},

for some constants d_{o} > 0, D, Λ_{o} < ∞. Then for any 0 < ϵ < d_{o}, there are domains U_{i} ⊂ Ω_{i}, with ϵ/2 ≤ dist(∂U_{i}, ∂Ω_{i}) ≤ ϵ, for which exactly one of the following alternatives holds.

(I). (Convergence). A subsequence of (U_{i}, g_{i}, x_{i}) converges in the C^{∞} topology to a limit smooth Einstein metric (U, g, x), x = lim x_{i}.

(II). (Orbifolds). A subsequence of (U_{i}, g_{i}, x_{i}) converges to an Einstein orbifold metric (X, g, x) in the Gromov-Hausdorff topology. Away from the singular variety, the convergence g_{i} → g is C^{∞}, while the curvature of g_{i} blows up in L^{∞} at the singular variety.

(III). (Collapse). A subsequence of (U_{i}, g_{i}, x_{i}) collapses, in that ν(y_{i}) → 0 for all y_{i} ∈ U_{i}. The collapse is with “generalized” bounded curvature, in that inj_{g_{i}}^{2}(x)|R|_{g_{i}}(x) → 0, metrically away from a finite number of singular points; however such singularities might be more complicated than orbifold cone singularities.

The cases (I) and (II) occur if and only if

(3.15)  ν_{i}(x_{i}) ≥ ν_{o},

for some ν_{o} > 0, while (III) occurs if (3.15) fails. One obtains C^{∞} convergence in (I), and (II) away from the singularities, since Einstein metrics satisfy an elliptic system of PDE, (in harmonic coordinates).

When Ω_{i} is a closed 4-manifold M, the bound (3.14) follows immediately from (3.12). When (Ω_{i}, g_{i}) are complete AHE metrics on a fixed 4-manifold M, then of course (3.14) does not hold with Ω_{i} = M. However, in the normalization (1.2), one has |W |^{2} = |R|^{2} − 6 where W is the Weyl tensor of (M, g) and so by (2.18)

(3.16)  \frac{1}{8π^{2}} ∫_{M}(|R_{g}|^{2} − 6)dV_{g} = χ(M) − \frac{3}{4π^{2}} V.

Thus, a lower bound on the renormalized volume V (or upper bound on the renormalized action) and an upper bound on χ(M) give a global bound on the L^{2} norm of W. In particular, under these bounds (3.14) holds for Ω_{i} a geodesic ball B_{x_{i}}(R) of any fixed radius R about the base point x_{i} ∈ (M, g_{i}).

We point out that Theorem 3.5 is special to dimension 4. A similar result holds in higher dimensions only if one has a uniform bound on the L^{n/2} norm of curvature in place of (3.14); however there is no analogue of (3.16) for (general) Einstein metrics in higher dimensions.

Remark 3.6. Theorems 3.2 and 3.5 are local results. However, we will often apply them globally, to sequences of complete manifolds, and with complete limits. This is done by a standard procedure as follows, in the situation of Theorem 3.2 for example. Suppose {g_{i}} is a sequence of complete metrics on a manifold M, or on a sequence of manifolds M_{i}, with base points x_{i}, and satisfying ρ(y_{i}) ≥ ρ_{o}, for all y_{i} ∈ (M, g_{i}). One may then apply Theorem 3.2 to the domains (B_{x_{i}}(R), g_{i}, x_{i}), where B_{x_{i}}(R) is the geodesic R-ball about x_{i} to obtain a limit manifold (U(R), g_{∞}, x_{∞}), in the non-collapse case. Now take a divergent sequence R_{j} → ∞ and carry out this process for each j. There is then a diagonal subsequence B_{x_{i}}(R_{i}) of B_{x_{i}}(R_{j}) which converges to a complete limit (N, g_{∞}, x_{∞}). Similar arguments apply in the case of collapse and the cases of Theorem 3.5.

The convergence in these situations is also convergence in the pointed Gromov-Hausdorff topology, [22].

§3.2. In this section, we study the behavior of sequences of AH Einstein metrics in the interior, i.e. away from ∂M. We first discuss the hypotheses, and then state and prove the main result, Theorem 3.7.

Essentially for the rest of the paper, we will assume the topological condition on M = M^{4} that

(3.17)  H_{2}(∂M, 𝔽) → H_{2}(M̄, 𝔽) → 0,

where the map is induced by inclusion ι : ∂M → M̄ and 𝔽 is any field. As will be seen, this serves to rule out any orbifold degenerations of AH Einstein sequences. While many of the results of this paper carry over to the orbifold setting, we prefer for simplicity to exclude this possibility here.

Next, let g_{i} ∈ E_{AH} be a sequence of AH Einstein metrics on M with C^{m,α} boundary metrics γ_{i}, m ≥ 3. As noted in §2, the geodesic compactification ḡ_{i} of g_{i} with boundary metric γ_{i} is at least a C^{m−1,α} compactification. We assume the boundary behavior of {g_{i}} is controlled, in that

(3.18)  γ_{i} → γ,

in the C^{m,α′} topology on ∂M, for some α′ ≤ α. Next assume that the inradius of (M, ḡ_{i}) has a uniform lower bound, i.e.

(3.19)  Inḡ_{i}(∂M) = distḡ_{i}(C̄_{i}, ∂M) ≥ τ̄,

for some constant 0 < τ̄ ≤ 1, where C̄_{i} is the cutlocus of ∂M in (M̄, ḡ_{i}), and also assume an upper diameter bound

(3.20)  diamḡ_{i}S(t_{1}) ≤ T,

where t_{1} = τ̄/2 and S(t_{1}) = {x ∈ (M, ḡ_{i}) : t_{i}(x) = t_{1}} is the t_{1}-level set of the geodesic defining function t_{i} for (g_{i}, γ_{i}).

Finally, we assume that the Weyl curvature of g_{i} is uniformly bounded in L^{2}, i.e.

(3.21)  ∫_{M} |W_{g_{i}}|^{2}dV_{g_{i}} ≤ Λ_{0} < ∞.

All of these assumptions will be removed in §4 and at the beginning of §5. The main result of this subsection is the following:

Theorem 3.7. Let {g_{i}} be a sequence of metrics in E_{AH} on M, satisfying (3.17)-(3.21) and let x_{i} be a base point of (M, g_{i}) satisfying

(3.22)  d ≤ distḡ_{i}(x_{i}, ∂M) ≤ D,

for some constants d > 0 and D < ∞, where ḡ_{i} is the geodesic compactification associated to γ_{i}.

Then a subsequence of (M, g_{i}, x_{i}), converges to a complete Einstein metric (N, g, x_{∞}), x_{∞} = lim x_{i}. The convergence is in the C^{∞} topology, uniformly on compact subsets, and the manifold N weakly embeds in M,

(3.23)  N ⊂⊂ M,

in the sense that smooth bounded domains in N embed as such in M.

Proof: By Theorem 3.5 and the discussion following (3.16), together with Remark 3.6, a subsequence of {(M, g_{i})} based at x_{i} either, (i): converges smoothly to a complete limit Einstein manifold (N, g, x_{∞}), (ii): converges to an Einstein orbifold, smoothly away from the singular variety, or (iii): collapses uniformly on domains of arbitrary bounded diameter about {x_{i}}.

The first (and most important) task is to rule out the possibility of collapse. To do this, we first prove a useful volume monotonicity formula in the Lemma below; this result will also be important in §4 and holds in all dimensions.

Let (M, g) be any AH Einstein manifold of dimension n + 1, with C^{2,α} geodesic compactification

(3.24)  ḡ = t^{2}g

and boundary metric γ. Let Ē be the inward normal exponential map of (M, ḡ) at ∂M, so that σ_{x}(t) = Ē(x, t) is a geodesic in t, for each fixed x ∈ ∂M. As discussed in §2, for r = − log t, the curve σ_{x}(r) is a geodesic in the Einstein manifold (M, g). Let J̄(x, t) be the Jacobian of Ē(x, t), so that J̄(x, t) = dV̄ (x, t)/dV̄ (x, 0), where dV̄ is the volume form of the ’geodesic sphere’ S(t), i.e. the t-level set of t as following (3.20). Also, let τ_{0} = τ_{0}(x) be the distance to the cutlocus of Ē at x ∈ ∂M.

Lemma 3.8. In the notation above, the function

(3.25)  \frac{J̄(x, t)}{τ_{0}^{n}(1 − (\frac{t}{τ_{0}} )^{2})^{n}}

is monotone non-decreasing in t, for any fixed x.

Proof: As above, let r = log(\frac{1}{t}) and, for any fixed x, set r_{0} = r_{0}(x) = log(\frac{1}{τ_{0}} ). Then the curve σ_{x}(r) is a geodesic in (M, g) and S(r) is the r-rlevel set of the distance function r. Let J(x, r) be the corresponding Jacobian for S(r) along σ_{x}(r), so that J(x, r) = t^{−n}J̄(x, t) by (3.24). Since Ric_{g} = −ng, the infinitesimal form of the Bishop-Gromov volume comparison theorem [22] implies the ratio

\frac{J(x, r)}{sinh^{n}(r − r_{0})} ↓

i.e. the ratio is monotone non-increasing in r, as r → ∞. Converting this back to (M, ḡ), it follows that, in t,

\frac{t^{−n}J̄(x, t)}{t^{n} sinh^{n}(log(\frac{τ_{0}}{t}))} ↑,

15

is increasing, since as r increases to ∞, t decreases to 0. But sinh^{n}(log(\frac{τ_{0}}{t})) = \frac{1}{2^{n}} (\frac{τ_{0}}{t})^{n}(1 − (\frac{t}{τ_{0}} )^{2})^{n}, which gives (3.25).

Let q be any point in ∂M and consider the metric s-ball B_{x}(s) = {y ∈ M̄ : distḡ(y, q) ≤ s} in (M̄, ḡ). Let D_{q}(s) = B_{q}(s) ∩ ∂M, so that D_{q}(s) is the metric s-ball in (∂M, γ), since ∂M is totally geodesic. Observe that there are constants μ_{0} > 0 and μ_{1} < ∞, which depend only on the C^{0} geometry of the boundary metric γ, such that

(3.26)  μ_{0} ≤ \frac{vol_{γ}(D_{q}(s))}{s^{n}} ≤ μ_{1}.

Let D_{q}(s, t) = {x ∈ M : x = Ē(y, t), for some y ∈ D_{q}(s)}, so that D_{q}(s, t) ⊂ S(t). It follows immediately from (3.25) by integration over (a domain in) ∂M that, for any fixed s,

(3.27)  \frac{volḡD(s, t)}{(sτ̄)^{n}(1 − (\frac{t}{τ̄})^{2})^{n}} ↑, and \frac{volḡS(t)}{τ̄^{n}(1 − (\frac{t}{τ̄})^{2})^{n}} ↑,

for τ̄ as in (3.19). Now let t_{1} = τ̄/2, so that (3.27) implies in particular that

(3.28)  volḡS(t_{1}) ≥ (\frac{3}{4})^{n}vol_{γ}∂M,

and hence, with respect to (M, g),

(3.29)  vol_{g}S(t_{1}) ≥ (\frac{3}{4})^{n}t_{1}^{−n}vol_{γ}∂M.

This leads easily to the following lower bound on volumes of balls.

Corollary 3.9. Let (M, g) be an AH Einstein metric, satisfying (3.19)-(3.20), and choose a point x satisfying (3.22), i.e. d ≤ distḡ(x, ∂M) ≤ D. Then

(3.30)  vol_{g}B_{x}(1) ≥ ν_{0} > 0,

where ν_{0} depends only on d, D, τ̄, T and the C^{0} geometry of the boundary metric γ.

Proof: Again, by the Bishop-Gromov volume comparison theorem on (M, g), one has for any R ≥ r,

(3.31)  \frac{volB_{x}(r)}{sinh^{n}(r)} ≥ \frac{volB_{x}(R)}{sinh^{n}(R)}.

Suppose r(x) = D_{1}, so that D_{1} ∈ [− log D, − log d]. Then for any y ∈ S(t_{1}), the triangle inequality and (3.20) imply that

dist_{g}(x, y) ≤ |D_{1}| + t_{1}^{−1} · T ≡ D_{2}.

Hence, choose R so that R = D_{2} + 1, which implies that S(t_{1}) ⊂ B_{x}(R − 1). But volB_{x}(R) ≥ volA(R−2, R) ≥ \frac{1}{2}volS(t_{1}), where the second inequality follows from the coarea formula, (changing t_{1} slightly if necessary). Combining this with the comparison estimate above implies

volB_{x}(1) ≥ c_{2}volS(t_{1}) sinh^{−n}(D_{2}) ≥ c_{3}t_{1}^{−n} sinh^{−n}(D_{2})vol_{γ}∂M,

where the last inequality follows from (3.29). This gives (3.30).

Corollary 3.9 gives a uniform lower bound on the volume radius of each g_{i} at x_{i} satisfying (3.22) and so there is no possibility of collapse. Theorem 3.5 and Remark 3.6 then imply that (M, g_{i}, x_{i}) has a subsequence converging in the Gromov-Hausdorff topology either to a complete Einstein manifold (N, g, x_{∞}) or to a complete Einstein orbifold (V, g, x_{∞}).

Next, we use the hypothesis (3.17) to rule out orbifold limits. Thus, suppose the second alternative above holds, so that (M, g_{i}) converges in the Gromov-Hausdorff topology to a complete Einstein orbifold (X, g). With each orbifold singularity q ∈ X with neighborhood of the form C(S^{3}/Γ), there is associated a (preferred) smooth complete Ricci-flat 4-manifold (E, g_{0}), which is asymptotically locally Euclidean (ALE), in that (E, g) is asymptotic to a flat cone C(S^{3}/Γ_{E}), Γ_{E} ≠ {e}, cf. [1,§5], [3,§3]. The complete manifold (E, g_{o}) is obtained as a limit of blow-ups or rescalings of the metrics g_{i} restricted to small balls B_{y_{i}}(δ_{i}) at the maximal curvature scale; thus one does not necessarily have Γ = Γ_{E}, since there may be other Ricci-flat ALE orbifolds arising from blow-ups at other curvature scales. In any case, the manifold E is embedded in the ambient manifold M; in fact for any δ > 0 and points y_{i} → q, E is topologically embedded in (B_{y_{i}}(δ), g_{i}). The Einstein metrics g_{i} crush the topology of E to a point in that E ⊂ (B_{y_{i}}(δ), g_{i}) and B_{y_{i}}(δ) converges to the cone C(S^{3}/Γ), in the Gromov-Hausdorff topology for any fixed δ > 0 sufficiently small.

Next we point out that any such ALE space E has non-trivial topology. This result corrects a minor inaccuracy in [1, Lemma 6.3].

Proposition 3.10. Let (E, g_{0}) be a complete, non-flat, oriented Ricci-flat ALE manifold. Then

(3.32)  H_{2}(E, 𝔽) ≠ 0,

for some field 𝔽. Moreover, if E is simply connected, then H_{2}(E, ℤ) is torsion-free, so that H_{2}(E, ℝ) ≠ 0, while H_{2}(E, 𝔽) = 0, for any finite field 𝔽.

Proof: It is well-known that |π_{1}(E)| < ∞, so that b_{1}(E) = 0, and also b_{3}(E) = 0, (cf. [1] for example). Hence

χ(E) = 1 + b_{2}(E) ≥ 1.

If b_{2}(E) > 0, then of course (3.32) follows, so suppose b_{2}(E) = 0, so that χ(E) = 1. The Euler characteristic may be computed with respect to homology with coefficients in any field 𝔽, so that

χ(E) = 1 − H_{1}(E, 𝔽) + H_{2}(E, 𝔽) = 1,

since again H_{3}(E, 𝔽) = H_{4}(E, 𝔽) = 0. Now if (3.32) does not hold, then it follows that H_{1}(E, 𝔽) = 0, for all 𝔽. By the universal coefficient theorem, this implies H_{1}(E, ℤ) = H_{2}(E, ℤ) = 0.

Thus, E is an integral homology ball, with finite π_{1}. It then follows by the arguments in [1, Lemma 6.3] that E is flat, giving a contradiction. This proves (3.32). The proof of the second statement is an immediate consequence of the universal coefficient theorem.

Next, observe that there is an injection

(3.33)  0 → H_{2}(E, 𝔽) → H_{2}(M, 𝔽).

To see this, the Mayer-Vietoris sequence for (a thickening of) the pair (E, M \ E) gives

H_{2}(S^{3}/Γ, 𝔽) → H_{2}(E, 𝔽) ⊕ H_{2}(M \ E, 𝔽) → H_{2}(M, 𝔽).

Thus, it suffices to show that i_{∗} : H_{2}(S^{3}/Γ, 𝔽) → H_{2}(E, 𝔽) is the zero-map. If E is simply connected, this follows immediately from Proposition 3.10, since H_{2}(S^{3}/Γ, ℤ) is torsion, while H_{2}(E, ℤ) is torsion-free. In general, let Ẽ be the universal cover of E, so that the covering map π : Ẽ → E is finite-to-one. One has ∂Ẽ = S^{3}/Γ̃, for some finite group Γ̃. Now any 2-cycle in S^{3}/Γ with coefficients in 𝔽 may be represented by a map, (or more precisely a collection of maps), f : S^{2} → S^{3}/Γ. This lifts to a map f̃: S^{2} → S^{3}/Γ̃ = ∂Ẽ. As noted above, the 2-cycle f̃ bounds a 3-chain F in Ẽ. Composing with the projection map π, it follows that f bounds a 3-chain in E, which proves the claim.

Now returning to (3.17) and orbifold limits, let Σ be any non-zero 2-cycle in H_{2}(E, 𝔽); (Σ exists by (3.32)). By (3.17), there is a 2-cycle Σ_{∞} in H_{2}(∂M, 𝔽) homologous to Σ, so that there is a 3-chain W in M such that ∂W = Σ − Σ_{∞}. But Z = W ∩ (S^{3}/Γ) ≠ ∅ and so Z represents a 2-cycle in H_{2}(S^{3}/Γ, 𝔽). By the preceding argument, Z bounds a 3-chain W′ in E. Hence (W ∩ E) + W′ is a 3-chain in E, with boundary Σ, i.e. [Σ] = 0 in H_{2}(E, 𝔽), giving a contradiction.

Thus, the hypothesis (3.17) rules out any possible orbifold degeneration of the sequence (M, g_{i}). Having ruled out the possibilities (ii) and (iii), it follows that (i) must hold. The fact that N is weakly embedded in M follows immediately from Remark 3.6 and the definition of smooth convergence preceding (3.8). This completes the proof of Theorem 3.7.

Remark 3.11. (i). Note that Theorem 3.7 does not imply that the limit manifold N topologically equals M. For instance, it is possible at this stage that some of the topology of M escapes to infinity, and is lost in the limit N. All of the results and discussion in §3.2 hold without change if the 4manifold M is replaced by a sequence of 4-manifolds M_{i}, with a common boundary ∂M_{i} = ∂M, so that one has (M_{i}, g_{i}) in place of (M, g_{i}).

(ii). We also observe that the proof shows that there is a uniform bound Λ = Λ(Λ_{0}, d, D, T, {γ_{i}}) < ∞ such that, for x satisfying (3.22),

(3.34)  |R_{g_{i}}|(x) ≤ Λ,

i.e. there is a uniform bound on the sectional curvatures of {g_{i}} in this region.

(iii). The existence and behavior of (M, g_{i}) at base points x_{i} where distḡ_{i}(x_{i}, ∂M) → ∞ will be discussed at the end of §4.

4. Compactness II: Boundary Behavior.

Theorem 3.7 essentially corresponds to a uniform interior regularity result in that one has uniform control of the AHE metrics g_{i} in the interior of the 4-manifold M, i.e. on compact subsets of M. In this subsection, we extend this to similar control in a neighborhood of definite size about the boundary. The Chern-Gauss-Bonnet theorem in (3.12), cf. also (3.21), plays a crucial role in obtaining control in the interior. A significant point in this section is to prove that such an apriori L^{2} bound on the curvature is not necessary to obtain control near the boundary; thus we show that control of the boundary metric itself gives control of the Einstein metric in a neighborhood of the boundary of definite size, cf. Corollary 4.10. Much of the proof of this result holds in fact in all even dimensions and conjecturally in all dimensions.

We begin with the following result, most of which was proved in [6], cf. also [7]. We fix p > 4 and denote the L^{p} curvature radius by ρ instead of ρ^{p}.

Proposition 4.1. On a fixed 4-manifold M, let g ∈ E_{AH}^{m,α}, m ≥ 4, with boundary metric γ satisfying ||γ||_{C^{m,α}} ≤ K in a fixed coordinate atlas for ∂M. Let U_{δ} = {x ∈ M̄ : distḡ(x, ∂M) ≤ δ} where ḡ is the geodesic compactification of (M, g) with boundary metric γ. Suppose the L^{p} curvature radius of ḡ satisfies

(4.1)  ρ(x) ≥ ρ_{0},

for some constant ρ_{0} > 0, for all x ∈ U_{δ_{0}}, some δ_{0} > 0.

Then there is a δ_{1} = δ_{1}(ρ_{0}, δ_{0}, K) > 0 such that the C^{m−1,α} geometry of the geodesic compactification ḡ is uniformly controlled in U_{δ_{1}}, in that

(4.2)  r_{h}^{m−1,α} ≥ r_{0},

where r_{0} depends only on ρ_{0}, δ_{0}, and K. The same statement holds if m = ∞ or m = ω.

Proof: By the discussion in §2, boundary regularity implies that (M, g) has a C^{m,α} conformal compactification g̃, with boundary metric γ. The metric g̃ may be chosen to be of constant scalar curvature s̃ = const and g̃ is C^{m,α} in local harmonic coordinates. Again as noted in §2, the geodesic compactification ḡ of g with boundary metric γ is then at least C^{m−1,α}. Moreover, by [6, Prop. 2.1, Thm. 2.4 and Prop. 2.7], the C^{m−1,α} geometry of the metric ḡ is uniformly controlled in U_{δ_{1}}, i.e. (4.2) holds, for δ_{1} = δ_{1}(ρ_{o}, γ), provided the C^{m,α} geometry of the boundary metric γ is uniformly bounded, (4.1) holds, and provided there is a uniform lower bound on the volume radius of ḡ in U_{δ_{1}}.

The control on the boundary metric and (4.1) are given by hypothesis. To obtain a lower bound on the volume radius ν, observe that Lemma 3.8, cf. also (3.27), gives a lower bound on the volume ratios of balls, volB_{q}(s)/s^{4} centered at points q ∈ ∂M, depending only on the C^{0} geometry of the boundary metric. In particular, there is a constant ν_{0} > 0 such that volB_{q}(ρ_{0})/ρ_{0}^{4} ≥ ν_{0}, q ∈ ∂M, for all (M, ḡ_{i}). Within the L^{p} curvature radius, i.e. for balls B_{x}(r) ⊂ B_{q}(ρ_{0}), standard volume comparison results imply then a lower bound volB_{x}(r)/r^{4} ≥ ν_{1}, where ν_{1} depends only on ν_{0}, cf. [4,§3]. This gives a uniform lower bound on the volume radius within U_{ρ_{0}}, and hence the result follows.

Proposition 4.1 also holds locally, in that if (4.1) holds at all points in an open set V containing a domain of fixed size in ∂M, then the conclusion holds in V ∩ U_{δ_{1}}. Also, this result holds for m ≥ 3. Of course the arguments concerning the lower bound on the volume radius hold in all dimensions.

Proposition 4.1 should be understood as a natural strengthening of the boundary regularity property that AH Einstein metrics with C^{m,α} boundary metric have a C^{m,α} conformal compactification. Thus, it gives uniform control or stability of the metric g ∈ E_{AH}^{m,α} near the boundary, depending only on the boundary metric and a lower bound on ρ as in (4.1). The main result of this section, Theorem 4.4 below, removes the assumption (4.1) and proves the estimate (4.2) independent of ρ_{0} and δ_{0}. Of course this could not be true locally; Theorem 4.4 is necessarily a global result.

The weak hypothesis (4.1) on the curvature radius may obviously be replaced by a stronger assumption. Thus, Proposition 4.1 implies, for instance, that in the geodesic compactification,

(4.3)  r_{h}^{k,β} ≥ r_{0} ⇒ r_{h}^{m−1,α} ≥ r_{1},

for any 2 < k + β < m − 1 + α with r_{1} = r_{1}(r_{0}, δ_{0}, K). Thus, control of the metric in a weak norm, such as L^{k,p} or C^{k,β} implies control of the metric in a stronger norm, governed by the regularity of the metric at the boundary. We formalize this as follows:

Definition 4.2. In any dimension, an AH Einstein metric g ∈ E_{AH} satisfies the strong control property if (4.3) holds, for some (k, β) and (m, α) with k + β < m − 1 + α, and m − 1 ≥ n.

The proof of boundary regularity in [6], (cf. also [7]), uses the fact that AH Einstein metrics and their conformal compactifications satisfy the conformally invariant Bach equation in dimension 4. In a suitable gauge, this is a non-linear elliptic system of PDE and boundary regularity follows essentially from standard boundary regularity for such elliptic systems. The strong control property (4.3), (or its original version in Proposition 4.1), then follows from the fact that concurrent with boundary regularity, one has uniform elliptic estimates for solutions of the Bach equations near and up to the boundary, given weak control in such a region.

Remark 4.3. It will be noted in the proofs of the results below that all of the results of this section hold in all dimensions in which the strong control property holds. A recent result of Helliwell [25] proves boundary regularity for AH Einstein metrics in all even dimensions, by generalizing the proof in dimension 4 in [6], [7]. The idea is to use the Fefferman-Graham obstruction tensor in place of the Bach equations in higher dimensions; again in a suitable gauge, this gives a non-linear elliptic system. The proof given in [25] requires that g be C^{n,α} conformally compact, although we suspect that it suffices for g to be C^{2,α} conformally compact. In any case, it is straightforward to show from [25] that the strong control property (4.3) holds in all even dimensions with n < k + β < m − 1 + α.

In odd dimensions, conformal compactifications of AH Einstein metrics are not C^{n} up to the boundary, exactly due to the presence of the obstruction tensor H. Nevertheless, one can define the spaces C^{m,α}, m ≥ n in a polyhomogeneous sense. We conjecture that with such a suitable modification of the definition of harmonic radius, the strong control property holds in all odd dimensions; this is an interesting open question.

The main result of this section is the following:

Theorem 4.4. Let (M, g) be an AH Einstein metric on a 4-manifold with a L^{2,p} conformal compactification, p > 4, and boundary metric γ. Suppose that γ ∈ C^{m,α}, m ≥ 4 and that, in a fixed coordinate system for ∂M, ||γ||_{C^{m,α}} ≤ K, for some α > 0. If ḡ is the associated C^{m−1,α} geodesic compactification of g determined by γ, then there are constants μ_{0} > 0 and ρ_{0} > 0, depending only on K, α, and p, such that

(4.4)  ρ(x) ≥ ρ_{0}, for all x with t(x) ≤ μ_{0}.

The same result holds in all dimensions in which the strong control property holds, with m ≥ n+1.

The proof is rather long, and so is broken into several Propositions and Lemmas. Before beginning, it is worth pointing out that the curvature of ḡ at ∂M is uniformly bounded by the C^{2} geometry of the boundary metric γ, cf. (A.8)-(A.10) in the Appendix. Thus, the geometry of g at ∂M is well-controlled to 2^{nd} order; one needs to extend this to control of g near ∂M.

We will give the proof in all dimensions, assuming the strong control property holds. Also, we work with the L^{p} curvature radius ρ = ρ^{p}, p > n, but one may work equally well with the L^{k−2,q} curvature or L^{k,q} harmonic radius; the arguments are exactly the same for these radii.

Let τ_{0} be the distance to the cutlocus of the normal exponential map from ∂M; τ_{0}(x) = distḡ(x, C̄), where C̄ is the cutlocus, as in (3.19). The first step is to prove that the function ρ is bounded below by τ_{0} near ∂M. The second, more difficult, step is to prove that τ_{0} itself is uniformly bounded below near ∂M.

We begin with the following result.

Proposition 4.5. For (M, g) as in Theorem 4.4, there is a constant c_{0} > 0, depending only on K, α and p, such that

(4.5)  ρ(x) ≥ c_{0}τ_{0}(x), for all x with t(x) ≤ 1.

Proof: The proof is by contradiction. If (4.5) is false, then there must exist a sequence of AH Einstein metrics g_{i} on M_{i}, with C^{n,α} (polyhomogeneous) geodesic compactifications ḡ_{i}, for which the boundary metrics γ_{i} satisfy

(4.6)  ||γ_{i}||_{C^{n,α}} ≤ K,

but for which the L^{p} curvature radius ρ = ρ_{i} of ḡ_{i} satisfies

(4.7)  ρ(x_{i})/τ_{0}(x_{i}) → 0 as i → ∞,

on some sequence of points x_{i} ∈ (M̄_{i}, ḡ_{i}). Note that the ratio in (4.7) is scale-invariant. Without loss of generality, assume that x_{i} realizes the minimal value of ρ/τ_{0} on (M_{i}, ḡ_{i}). Of course x_{i} may occur at ∂M.

Now blow-up or rescale the metrics ḡ_{i} at x_{i} to make ρ(x_{i}) = 1, i.e. set

(4.8)  g′_{i} = λ_{i}^{2}ḡ_{i},

where λ_{i} = ρ(x_{i})^{−1} → ∞. Let ρ′ = λ_{i}ρ and τ′_{0} = λ_{i}τ_{0} be the L^{p} curvature radius and distance to the cutlocus, with respect to g′_{i}. Then

(4.9)  ρ′(x_{i}) = 1,

and by the minimality property of ρ(x_{i}),

(4.10)  ρ′(y_{i}) ≥ ρ′(x_{i})\frac{τ′_{0}(y_{i})}{τ_{0}(x_{i})} = \frac{τ′_{0}(y_{i})}{τ_{0}(x_{i})}.

for any y_{i} ∈ M̄. Since (4.7) implies τ′_{0}(x_{i}) → ∞, it follows that

(4.11)  ρ′(y_{i}) ≥ \frac{1}{2},

for all y_{i} with dist_{g′_{i}}(x_{i}, y_{i}) ≤ D, for any given D, provided i is sufficiently large. Of course one also has τ′_{0}(y_{i}) → ∞ for any such y_{i}.

It follows that the metrics g′_{i} have uniformly bounded curvature, on the average in L^{p} on all unit balls of g′_{i}-bounded distance to x_{i}. For clarity, we divide the proof into two cases, according to whether dist_{g′_{i}}(x_{i}, ∂M) remains bounded or not.

Case I. Bounded distance.

Suppose dist_{g′_{i}}(x_{i}, ∂M) ≤ D, for some D < ∞. We may then apply Proposition 4.1, or more precisely its local version as following the statement of Proposition 4.1, to conclude that a subsequence of the metrics (M_{i}, g′_{i}, x_{i}) converges in the C^{n,α′} (or stronger) topology to a limit (N, g′, x_{∞}). (Of course the convergence is modulo diffeomorphisms). Since τ′_{0}(y_{i}) → ∞ for any y_{i} within bounded g′_{i}-distance to x_{i}, the limit (N, g′) is a complete manifold with boundary. Moreover, since the unscaled boundary metrics γ_{i} are uniformly bounded in C^{n,α}, the limit (∂N, γ′) = (ℝ^{n}, δ), where δ is the flat metric.

Since the L^{p} curvature radius is continuous in the C^{n,α′} topology, it follows from (4.9) that

(4.12)  ρ′(x_{∞}) = 1.

The rest of the proof in this case is to prove that (N, g′) is flat. This clearly contradicts (4.12), and so will complete the proof.

Let

(4.13)  t′(x) = dist_{g′}(x, ∂N) = lim_{i→∞} t′_{i}(x_{i}),

where t′_{i} is the geodesic defining function for (∂M_{i}, g′_{i}) and x_{i} → x. Since ∂M is totally geodesic in (M, g′_{i}), the smooth convergence implies ∂N = ℝ^{n} is totally geodesic in (N, g′). Since τ′_{0} = ∞ on N, t′ is globally defined and smooth on N.

We now use some of the curvature properties of geodesic compactifications given in the Appendix. The equations (A.1)-(A.3) for the curvatures R′, Ric′ and s′ also hold on (N, g′), with t′ in place of t. We note that the Hessian D^{2}t′ = A, where A is the 2^{nd} fundamental form of the level sets of t′, and Δt′ = H, the mean curvature of the level sets.

By (A.3), the scalar curvature s′ of (N, g′) is given by

(4.14)  s′ = −2n\frac{Δt′}{t′} ,

and by (A.11) satisfies

(4.15)  ṡ′ ≥ \frac{1}{2n^{2}} t′(s′)^{2},

where ṡ′ = ∂s′/∂t′. Since s_{γ′} = 0, (A.8) implies that s′ = 0 on ∂N, and hence by (4.15), s′ ≥ 0 everywhere on N. Moreover, elementary integration of (4.15) implies that if, along a geodesic σ = σ(t′) normal to ∂N, s′(t_{0}) > 0, for some t_{0} > 0, then (t′)^{2} ≤ t_{0}^{2} + 4n^{2}/(s′(t_{0})), which is impossible, since t′ → ∞ along all geodesics normal to ∂N. It follows that one must have

(4.16)  s′ ≡ 0 on (N, g′).

From (4.14) and (4.16), we then have

(4.17)  Δt′ = 0,

so that t′ is a smooth harmonic function on N, with |∇t′| = 1.

Now the Riccati equation (A.7) for the t′ geodesics on (N, g′) gives

(4.18)  |A|^{2} + Ric(∇t′, ∇t′) = 0,

On the other hand, the formula (A.2) for the Ricci curvature holds, so that on (N, g′),

(4.19)  Ric = −(n − 1)t^{−1}D^{2}t − t^{−1}(Δt)g′ = −(n − 1)t^{−1}D^{2}t.

Here and below, we drop the prime from the notation. Clearly D^{2}t(∇t, ∇t) ≡ 0 on N, and so by (4.18), |A|^{2} = 0, i.e. all the level sets are totally geodesic. Since A = D^{2}t, (4.19) implies that Ric ≡ 0, so that (N, g′) is Ricci-flat. The vector field ∇t is thus a parallel vector field, and so (N, g′) splits as a product along the flow lines of ∇t. Since ∂N = {t = 0} is flat ℝ^{n}, it follows that (N, g′) is flat, as claimed.

Case II. Unbounded distance.

Suppose t′_{i}(x_{i}) = dist_{g′_{i}}(x_{i}, ∂M) → ∞, as i → ∞. Using (4.7) again, exactly the same arguments as in (4.14)-(4.19) applied to (M_{i}, g′_{i}, x_{i}) show that, for i sufficiently large, (M_{i}, g′_{i}) almost splits in regions of bounded diameter about x_{i}, in that such regions are topologically products and the flow by the integral curves of t′_{i} are almost isometries of g′_{i}.

More precisely, suppose first that (M_{i}, g′_{i}, x_{i}) does not collapse near x_{i}. Then, exactly as in Case I, by the strong control property, (M_{i}, g′_{i}, x_{i}) is close in the C^{n,α′} topology, to a product metric in regions of g′_{i}-bounded diameter about x_{i}. Suppose instead (M_{i}, g′_{i}, x_{i}) collapses near x_{i}. The strong control property implies the collapse is with bounded curvature. Clearly the collapse, which is caused by short geodesic loops, is transverse to the flow lines of t′_{i}. By the general collapse theory, cf. [15], the collapse may be unwrapped in local covering spaces. Thus, there are covering spaces of balls of fixed but small diameter about x_{i} such that the lifted metrics on the covers do not collapse, and so converge in C^{n,α′} to a local limit. All of the estimates (4.14)-(4.19) remain valid on such local covers, and since these are pointwise estimates transverse to the collapsing directions, it follows again that the original metrics (M_{i}, g′_{i}, x_{i}) are close in C^{n,α′} to a product metric in regions of g′_{i}-bounded diameter about x_{i}.

This shows that ρ′_{i} is almost constant in such regions, and, (again by the strong control property),

(4.20)  \frac{∂ρ′_{i}}{∂t′_{i}} ∼ 0.

Note that ∂ρ/∂t is scale-invariant, (as are the ratios ρ/t and ρ/τ_{0}). The argument above was carried out at the base points x_{i}. However, exactly the same reasoning shows that (4.20) holds at all points p_{i} where (ρ/τ_{0})(p_{i}) << 1. Now integrate (4.20) along integral curves σ_{i} of t′_{i} starting at points y_{i} of bounded g′_{i}-distance to x_{i} in the direction toward ∂M. Since ρ′_{i}(y_{i}) ∼ 1 and t′_{i}(y_{i}) → ∞, this implies in particular that the scale-invariant ratio ρ/t on (M_{i}, g′_{i}) satisfies

(4.21)  \frac{ρ′_{i}}{t′_{i}} << 1,

along such curves, all the way to ∂M, since ρ/τ_{0} << 1 along such curves. However, this is clearly impossible, since again the arguments in Case I imply that (M_{i}, g′_{i}) is almost flat within bounded g′_{i}-distance to ∂M, so that ρ′_{i} ≥ (1 − δ)t′_{i} in such regions, where δ is small for i large. In other words, (4.20) and (4.21) prevent the ratio ρ/t, which is arbitrarily small around x_{i}, from increasing to near 1 near ∂M. This contradiction completes the proof.

Note that the opposite inequality to (4.5) also holds, i.e.

(4.22)  ρ(x) ≤ τ_{0}(x).

This is essentially a tautology, since the metric ḡ is singular at the cutlocus and so ρ-balls are not defined past the cutlocus. The proof of Proposition 4.5 does not use the strong control property with m − 1 ≥ n. Basically, it suffices to have this property in any norm stronger than the L^{2,p} norm, so that the L^{p} curvature radius is continuous. However, the condition m − 1 > n will be crucial in the next result.

The next step in the proof of Theorem 4.3 is to prove that τ_{0} is uniformly bounded below near ∂M.

Proposition 4.6. For (M, g) as in Theorem 4.4, there is a constant μ_{1} > 0, depending only on K, α and p, such that, for any x ∈ ∂M,

(4.23)  τ_{0}(x) ≥ μ_{1}.

Proof: The proof of this estimate is much more subtle than that of Proposition 4.5. Propositions 4.1 and 4.5 are essentially local results; they hold for AH Einstein metrics defined only in a geodesic collar neighborhood of ∂M, t ∈ [0, ε_{0}], for some ε_{0} > 0. However, Proposition 4.6, (and hence Theorem 4.4), is global; it requires g to be an AH Einstein metric defined on a compact manifold M, with conformal boundary ∂M at infinity.

We first prove some preliminary lemmas, which are basically straightforward consequences of the strong control property. The actual proof begins after the proof of Lemma 4.9; on a first reading, one might want to start at this point to understand how the Lemmas are used. The crux of the proof is the use of the relation (2.22) in Proposition 2.4. It is this relation in particular, (as well as Lemma 4.7), which requires (M, g) is global.

To begin, we examine the geometry of the geodesic compactification ḡ in regions where τ_{0} is very small, in fact possibly arbitrarily small. Thus suppose (M_{i}, g_{i}) is a sequence of AH Einstein metrics with boundary metrics γ_{i} on ∂M_{i} = ∂M uniformly controlled in C^{m,α}, m ≥ n + 1. Hence, a subsequence of γ_{i} converges in C^{m,α′} to a limit C^{m,α} boundary metric γ. Let x_{i} be any sequence of points in M_{i} such that

(4.24)  τ_{0}(x_{i}) → 0.

Rescale the compactified metrics ḡ_{i} to

(4.25)  g′_{i} = (τ_{0}(x_{i}))^{−2}ḡ_{i},

so that

(4.26)  τ′_{0}(x_{i}) = 1.

By Proposition 4.5 and (4.22), ρ′(x_{i}) is uniformly bounded above and below. Hence as in the proof of Proposition 4.5, a subsequence of (M, g′_{i}, x_{i}) converges in the C^{n,α′} topology (modulo diffeomorphisms), to a maximal limit (N, g′, x_{∞}); maximal here means the maximal connected domain on which the convergence is C^{n,α′}. The manifold N has boundary-at-infinity ∂N a domain in ℝ^{n}, with boundary metric the flat metric γ′ = δ, and containing the ball B_{x_{∞}}(1) in ℝ^{n}. However, in this case, N is not complete away from its boundary, since τ′_{0}(x_{∞}) ≤ 1.

Consider now also the sequence of AH Einstein metrics g_{i} on M_{i}, with conformal compactification ḡ_{i}, with base points x_{i} chosen above. Of course distḡ_{i}(x_{i}, ∂M) → 0. The metrics ḡ_{i} are rescaled up to g′_{i} = λ_{i}^{2}ḡ_{i}, λ_{i} = (τ_{0}(x_{i}))^{−1}, and converge, (in a subsequence), in C^{n,α′} to the maximal limit (N, g′, x). Now the rescaling of ḡ_{i} just corresponds to changing the defining function t_{i} to t′_{i} = λ_{i}t_{i}. Thus it does not change g_{i} itself; in fact g_{i} = (t′_{i})^{−2}g′_{i}. Since t′_{i} → t′ and g′_{i} → g′ on N, it follows that the pointed sequence of AH Einstein metrics (M_{i}, g_{i}, x_{i}) converges to a limit AH Einstein metric (N , g_{∞}, x_{∞}). Again here N is the maximal connected domain containing x_{∞} on which the convergence is C^{n,α′}; in fact the convergence is now C^{∞} smooth on compact subsets, by regularity properties of Einstein metrics. The limit (N , g_{∞}) is of course not conformally compact in general; the “compactification” of (N , g_{∞}) by t gives the manifold (N, g′), i.e.

(4.27)  g_{∞} = t^{−2}g′,

where N ⊂ N is the region where t is smooth.

The first result is an analogue of (4.12) in this setting.

Lemma 4.7. Any blow-up limit manifold (N, g′, x_{∞}) is not flat.

Proof: If (N, g′) is flat, then by (4.27), (N , g_{∞}) is hyperbolic, i.e. of constant curvature -1, and so locally embeds in the hyperbolic space ℍ^{n+1}(−1). By construction, the maximal limit (N , g_{∞}) is not complete; it has a “boundary” corresponding to the points where ρ_{g_{i}} → 0. However, the pointed sequence (M_{i}, g_{i}, x_{i}) of complete manifolds has a uniform lower bound on Ricci curvature, since the metrics are Einstein, and is non-collapsing at points within bounded distance to x_{i}. By the Gromov weak compactness theorem [22], a subsequence of (M_{i}, g_{i}, x_{i}) converges in the pointed Gromov-Hausdorff topology to a limit (N̄ , d, x), where (N̄ , d) is a complete, non-compact, length space. In particular, (N̄ , d) is geodesically complete; any two points may be joined by a minimizing geodesic and all geodesic balls B_{x}(r) in N̄ have compact closure strictly contained in N̄.

Clearly the smooth domain (N , g_{∞}) embeds in (N̄ , d). By a result of Cheeger-Colding [13], the singular set ∂N = N̄ \ N of N̄ is of codimension 2 in N̄. In particular the metric boundary ∂N is not a topological boundary, (which would have codimension 1).

By the volume comparison theorem (3.31) on (M_{i}, g_{i}), the ratio volB_{z}^{i}(r)/volB_{−1}(r) is monotone non-increasing in r, where B_{z}^{i}(r) is the geodesic r-ball about any z in (M_{i}, g_{i}) and volB_{−1}(r) is the volume of the geodesic r-ball in ℍ^{n+1}(−1). Further, for any given i, lim_{r→0} volB_{z}^{i}(r)/volB_{−1}(r) = 1. A fundamental result of Colding [14], cf. also [13], shows that the volume of geodesic balls is continuous under Gromov-Hausdorff limits so that in (N̄ , d), volB_{z}(r)/volB_{−1}(r) is also monotone non-decreasing, and hence lim_{r→0} volB_{z}(r)/volB_{−1}(r) ≤ 1, with equality for z ∈ N.

Now suppose first for simplicity of argument that N is simply connected. Since N is of constant curvature −1, the developing map based at any point in N gives an isometric immersion F : N → H^{n+1}(−1). Observe that the image of F is of full measure in ℍ^{n+1}(−1); this is because N̄ is complete, so that geodesics do not terminate, and ∂N is of codimension 2 and so of measure 0. In fact F extends to a continuous map of N̄ onto ℍ^{n+1}(−1).

It follows that F maps B_{z}(r) onto B_{F (z)}(r) ⊂ ℍ^{n+1}(−1) modulo sets of measure 0. Moreover, F preserves the volumes of these balls when counting multiplicities of the image, so that

(4.28)  \frac{volB_{z}(r)}{volB_{−1}(r)} ≥ \frac{volF (B_{z}(r))}{volB_{−1}(r)} = 1.

Since by the monotonicity above one has

(4.29)  \frac{volB_{z}(r)}{volB_{−1}(r)} ≤ 1,

for all r, it is then immediate that

(4.30)  \frac{volB_{z}(r)}{volB_{−1}(r)} = 1,

for all r. This of course implies that N̄ is isometric to ℍ^{n+1}(−1), by the volume rigidity theorem for Ricci curvature. Moreover, by [2, Thm. 3.2], the sequence (M_{i}, g_{i}, x_{i}) now converges smoothly to (N̄ , g_{∞}, x_{∞}) everywhere. In particular, τ_{0} = ∞, contradicting the fact that τ_{0}(x_{∞}) = 1 from (4.26).

The proof is similar when N is not simply connected. Thus, let Ñ be the universal cover of N. (For illustration, it is useful to picture Ñ as a branched cover of N̄ branched over the singular locus ∂N). Then the developing map F is an isometric immersion of Ñ into ℍ^{n+1}(−1). Let D be a Dirichlet fundamental domain for the action of π_{1}(N) on Ñ, based at a point z ∈ N. Modulo sets of measure 0, D may be identified with N, and F gives an isometric immersion F : D → ℍ^{n+1}(−1). Now however ∂D is of codimension 1 and D, (or D̄), is not geodesically complete; the action of π_{1}(N) identifies subsets of ∂D to obtain the manifold N or its completion N̄. Nevertheless, letting S_{z}(r) be the geodesic r-sphere about z in N, or equivalently in D modulo sets of measure 0, and letting S_{F (z)}(r) be the corresponding sphere in ℍ^{n+1}(−1), we claim that

(4.31)  lim sup_{r→∞} \frac{volS_{z}(r)}{vol_{−1}(S_{F (z)}(r))} ≥ 1.

To prove (4.31), recall the construction of the blow-up limit N and its compactification N with conformal infinity given by the domain ∂N ⊂ ℝ^{n}. View the codimension 2 singular set S = ∂N as a subset of the compactification N̄, and suppose first for clarity that S intersects the full conformal boundary ℝ^{n} in a closed set S_{∞} also of codimension 2 in ℝ^{n}. Then ∂N = ℝ^{n} \ S_{∞}, and so in particular ∂N is of full measure in ℝ^{n}. As r → ∞, the geodesic spheres S_{z}(r) in N, (or D), tend to ∂N. Then, as in the proof of (4.28), (4.31) follows from the fact that ∂N is a set of full measure in the conformal compactification S^{n} = ℝ^{n} ∪ {∞} of ℍ^{n+1}(−1).

Now the same argument holds regardless of the exact structure of the singular set S_{∞} ⊂ ℝ^{n}, since it is still the case that S_{z}(r) ⊂ N is of full measure in S̄_{z}(r) ⊂ N̄ and lim_{r→∞} S̄_{z}(r) = ℝ^{n}.

By integration, the estimate (4.31) also holds for balls B_{z}(r) and the relation (4.30) then follows from (4.29) as before.

We point out that the same proof shows that (N, g′, x_{∞}) cannot be conformally flat. This is because (4.27) would then imply that the metric (N , g_{∞}) is conformally flat and Einstein, and hence again hyperbolic; the proof then proceeds just as before.

In the following, we write

(4.32)  ϕ ∼ 1,

if there is a constant C < ∞ such that C^{−1} ≤ ϕ ≤ C. The data ϕ will be determined by (M, g) as in Theorem 4.4, but C is required to be independent of (M, g). The next result, which is of independent interest, relates the curvature radius ρ with the stress-energy term τ_{(n)} in (2.12).

Lemma 4.8. For (M, g) as in Theorem 4.4, and for any x ∈ ∂M with τ_{0}(x) sufficiently small, one has the estimate

(4.33)  ρ^{n}(x) · sup_{y∈B_{x}( \frac{1}{2} ρ(x))} |τ_{(n)}|(y) ∼ 1.

The same estimate holds for g_{(n)} in place of τ_{(n)}.

Proof: Observe that the product in (4.33) is scale invariant, cf. also (2.14). Suppose first that there exists a sequence of metrics (M_{i}, g_{i}) and points x_{i} ∈ ∂M_{i}, such that

(4.34)  ρ^{n}(x_{i}) · sup_{y∈B_{x_{i}}( \frac{1}{2} ρ(x_{i}))} |τ_{(n,i)}|(y) → 0.

We work in the scale ĝ_{i} = ρ(x_{i})^{−2}ḡ_{i} where

(4.35)  ρ̂(x_{i}) = 1,

so that τ̂_{0}(x_{i}) ∼ 1. The estimate (4.34) implies |τ̂_{(n,i)}|(y) << 1, for all y ∈ B̂_{x_{i}}(\frac{1}{2}). By Proposition 4.1, (or the strong control property), the rescaled metrics ĝ_{i} converge (in a subsequence) in the C^{n,α′} topology to a limit metric ĝ on a maximal connected domain (U, x_{∞}) where ρ̂ does not converge to 0, (i.e. the region where the curvature of ĝ_{i} does not blow-up in L^{p}). Of course B_{x_{∞}}(1) ⊂ U. As noted prior to Lemma 4.7, the blow-up limit of the metric γ on ∂M is the flat metric on ℝ^{n}.

Now the C^{n,α′} convergence implies that

(4.36)  τ_{(n)} = 0,

on ∂U. Since the boundary metric is flat, this implies that g_{(n)} = 0, cf. [18], [33]. The unique continuation property, Proposition 2.1, then implies that (U, ĝ) is flat, which however contradicts Lemma 4.7.

The proof of the opposite inequality in (4.33) is similar. Thus, suppose there exist (M_{i}, g_{i}) and points x_{i} ∈ ∂M such that

ρ^{n}(x_{i}) · sup_{y∈B_{x_{i}}( \frac{1}{2} ρ(x_{i}))} |τ_{(n,i)}|(y) >> 1.

Now work in the scale where sup_{y∈B_{x_{i}}( \frac{1}{2} ρ(x_{i}))} |τ_{(n,i)}|(y) = 1, and at points y_{i} realizing this supremum. In this scale, (using the fact that ρ is Lipschitz, with Lipschitz constant 1), ρ(y_{i}) >> 1. Hence, via (4.22), τ_{0}(y_{i}) >> 1 in this scale. Then the same arguments as in (4.14)-(4.19) together with the strong control property imply that the metric is C^{n,α′} close to the flat metric, in bounded domains about y_{i}. It follows that τ_{(n)} and g_{(n)} are (arbitrarily) small in bounded domains about y_{i} for i sufficiently large, which gives a contradiction to the scale normalization above.

Note that the strong control property implies further that the product in (4.33) is bounded away from 0 and ∞ in C^{α′}.

Now consider a conformal rescaling of the metric ḡ in place of the constant rescalings used before. Thus, set

(4.37)  g̃ = ρ(x)^{−2}ḡ.

Apriori, the function ρ may not be smooth in x, and so (4.37) should be replaced by the expression g̃ = ρ_{s}(x)^{−2}ḡ, where ρ_{s} is a C^{∞} smoothing of ρ satisfying \frac{1}{2}ρ ≤ ρ_{s} ≤ ρ. Thus, in the following, (4.37) is understood with respect to ρ_{s} in place of ρ; however, to keep the notation reasonable, and also because this difference is only of minor significance, we continue to work the notation ρ.

If one sets ĝ = τ_{0}^{−2}(x)ḡ, then by (4.5) and (4.22) the two metrics g̃ and ĝ are C-quasi-isometric, i.e.

(4.38)  ĝ ∼ g̃,

in the sense of (4.32).

Let ρ̃ be the L^{p} curvature radius of (M, g̃) and similarly for r_{h}^{m−1,α}. Also, let g̃_{(n)} be the n^{th} term in the Fefferman-Graham expansion (2.9)-(2.10) for the geodesic compactification of (M, g) determined by the boundary metric γ̃ = g̃|_{∂M}, and similarly for the stress-energy term τ̃_{(n)}.

Lemma 4.9. Let (M, g) be as in Theorem 4.4. Then on (M, g̃), for all x ∈ ∂M, one has the estimates

(4.39)  ρ̃(x) ∼ 1, and r_{h}^{m−1,α}(x) ∼ 1.

These estimates also hold for the geodesic compactification determined by γ̃. Further, for ρ(x) sufficiently small,

(4.40)  sup_{y∈B_{x}( \frac{1}{2} )} τ̃_{(n)}(y) ∼ 1.

Proof: We first prove ρ̃ is bounded below on ∂M. To do this, standard formulas for the behavior of the curvature under conformal changes give

R̃ = ρ^{−2}[R − g ∧ (D^{2} log ρ^{−1} − (d log ρ)^{2} + \frac{1}{2}|d log ρ|^{2}g)],

where ∧ denotes the Kulkarni-Nomizu product, cf. [10]. Hence,

(4.41)  |R̃|g̃ ≤ c(ρ^{2}|R| + ρ|D^{2}ρ| + |dρ|^{2}),

where the right side is taken with respect to ḡ and we recall the remark on smoothing following (4.37). Note also that the right side of (4.41) is scale invariant. Thus, it suffices to obtain an L^{p} or L^{∞} bound of the terms on the right in (4.41), in the scale where ρ = 1, on balls of radius r_{0}, for a uniform r_{0} > 0. Since the function ρ is Lipschitz, with Lipschitz constant 1, the last term is bounded in L^{∞}. By the strong control property, the first two terms are also bounded in L^{∞} on the ball of radius \frac{1}{2}. This gives a uniform upper bound on |R̃|g̃ on the ball of radius \frac{1}{2} and hence

ρ̃ ≥ ρ_{0},

for a fixed ρ_{0}, (depending only on the constant c_{0} in (3.4)). Also, since (M, ḡ) satisfies the strong control property, in the scale where ρ(x) = 1 one has a uniform lower bound on r_{h}^{m−1,α}, r_{h}^{m−1,α}(x) ≥ r_{0}, which again via (4.41) gives a lower bound on r_{h}^{m−1,α} on ∂M.

The opposite estimate ρ̃(x) ≤ ρ_{1}ρ(x), for a fixed ρ_{1} < ∞, also holds. For if, on some sequence (M_{i}, ḡ_{i}, x_{i}), ρ̃(x_{i}) → ∞ while ρ(x_{i}) = 1, then the metric g̃_{i} is almost flat in L^{p} in large balls B_{x_{i}}((1 − δ)ρ̃(x_{i})), for any fixed δ > 0. It follows that any maximal limit (N, g′_{∞}, x_{∞}), as following (4.25), is conformally flat. As noted following Lemma 4.7, this contradicts (the proof of) Lemma 4.7. Combining the arguments above proves (4.39).

Next, the estimate (4.39) gives uniform control in C^{m−1,α} of the conformal factor relating the geodesic compactification with respect to g̃ and the rescaling of ḡ by the constant factor ρ(x)^{−2}, for any fixed x. This implies that (4.39) also holds for the geodesic compactification with respect to g̃. In particular, this gives an upper bound on the term τ̃_{(n)} in (4.40).

Regarding the lower bound, suppose first n is odd. Then the lower bound on τ̃_{(n)} in (4.40) follows immediately from (4.33) and the conformal transformation rule (2.14). A similar argument holds also for n even. Thus, although (2.14) does not hold exactly, it does hold modulo terms which involve lower order derivatives of the boundary metric; for ρ(x) sufficiently small, these terms scale at a lower power of ρ(x)^{−1} than g_{(n)}, so that (2.14) holds to leading order.

We are now in position to begin the proof of Proposition 4.6 per se. Suppose then (4.23) is false. Then there exist AH Einstein metrics (M_{i}, g_{i}) with C^{m,α} controlled boundary metrics γ_{i} and points x_{i} such that τ_{0}(x_{i}) → 0, and so

(4.42)  ρ_{i}(x_{i}) → 0,

where ρ_{i} is the L^{p} curvature radius of the geodesic compactification ḡ_{i} of g_{i} with boundary metric γ_{i}. From now on, we work in the conformally rescaled metric (4.37). The Lipschitz property of ρ, together with (4.42) implies that

(4.43)  volγ̃_{i}(∂M) → ∞.

The main point now is to use the relation (2.22) in the γ̃_{i} metric:

(4.44) ∫_{∂M}⟨L_{X}τ_{(n)} + [(1 − \frac{2}{n})divX]τ_{(n)}, h_{(0)}⟩dVγ̃_{i} = ∫_{∂M}⟨σ_{(n)} + \frac{1}{2}trh_{(0)}τ_{(n)}, L̂_{X}γ⟩dVγ̃_{i} + b_{(n)}.

Here we have dropped the i and tilde from the notation; the inner product in (4.44) as well as τ_{(n)} and σ_{(n)} are taken with respect to γ̃_{i}; also the term b_{(n)} is given by the last term in (2.22). Recall from (2.12) that τ_{(n)} is determined by g_{(n)} and the boundary metric. By (2.21) and Lemma 2.2, σ_{(n)} is any solution of the system

(4.45)  δσ_{(n)} = −δ′(τ_{(n)}),
trσ_{(n)} = −tr′(τ_{(n)}) + (a_{(n)})′,

where δ′, tr′ and (a_{(n)})′ are the variations of δ, tr and a_{(n)} at (∂M, γ̃_{i}) in the direction h_{(0)}. Of course solutions σ_{(n)} of (4.45) are invariant under the addition of transverse-traceless terms, as is (4.44). We point out that the system of equations (4.45) is conformally invariant, since these equations define formal solutions to the conformally compactified linearized Einstein equations. In more detail, a conformal change of the boundary metric induces a conformal change of h_{(0)}; as described in and following (2.14), such conformal changes in turn induce a transformation of τ_{(n)}, a_{(n)} and σ_{(n)}. The space of solutions of (4.45) is thus transformed into itself under changes in the conformal compactification of solutions of the linearized Einstein equations.

Consider then the equations (4.45) on (∂M, γ_{i}), and set γ_{i} = γ. We first claim that the system (4.45) is solvable for any variation h_{(0)} of (∂M, γ), and hence, by Proposition 2.4, (4.44) also holds for all such variations h_{(0)}. To see this, by Lemma 2.2 and Proposition 2.3, we know that each equation in (4.45) is individually solvable, and thus need to show that the equations are simultaneously solvable. As is well-known, any symmetric bilinear form σ on (∂M, γ) can be written uniquely as σ = δ^{∗}V + fγ + k, where k is transverse-traceless. Setting −δ′(τ_{(n)}) = ϕ_{1}, −tr′(τ_{(n)}) + a′_{(n)} = ϕ_{2}, the system (4.45) becomes

(4.46)  δδ^{∗}V − \frac{1}{n}dδV = ϕ_{1} + \frac{1}{n}dϕ_{2},
−δV + nf = ϕ_{2}.

By Proposition 2.3, the term ϕ_{1} ∈ Imδ, and similarly dϕ_{2} ∈ Imδ, since for any Killing field X on (∂M, γ), ⟨dϕ_{2}, X⟩ = ⟨ϕ_{2}, δX⟩ = 0. Thus, the first equation is solvable for V, and hence the second equation is solvable for f, which proves the claim. Note that the first equation in (4.46) is an elliptic equation for V.

Next, we claim there is a solution σ_{(n)} = δ^{∗}V + fγ of (4.45) on (∂M, γ), (γ = γ_{i}), such that

(4.47)  ||σ_{(n)}||_{C^{m−n,α}} ≤ C||h_{(0)}||_{C^{m−n,α}}||τ_{(n)}||_{C^{m−n,α}},

where C depends only on K in Theorem 4.4. This follows directly from standard elliptic estimates for V associated to the equation (4.46), (choosing V orthogonal to the kernel of the operator), together with the fact that γ_{i} are uniformly bounded in C^{m,α}.

Transforming then to the conformally related metrics γ̃_{i} and using the fact (4.39) that τ_{(n)} is uniformly controlled in C^{α} then gives

(4.48)  ||σ_{(n)}||_{C^{α}} ≤ C||h_{(0)}||_{C^{α}},

for a fixed constant C, independent of i, where σ_{(n)} and the norms are taken with respect to the γ̃_{i} metric.

Let Z_{i} = {x ∈ ∂M : ρ_{i}(x) ≤ ε_{i}}, for some sequence ε_{i} → 0. For any x_{i} ∈ Z_{i}, by Lemma 4.9 the pointed sequence (∂M, γ̃_{i}, x_{i}) converges in C^{n,α′}, and uniformly on compact sets, to a complete conformally flat metric γ̃_{∞} on a domain V ⊂ ∂M. The metric γ̃_{∞} is conformally flat because one is conformally blowing up the sequence of smoothly controlled metrics γ_{i} on ∂M. Of course, the convergence is modulo diffeomorphisms which blow-up small regions in ∂M to unit size. Thus, there exist embeddings F_{i} : V → ∂M such that (F_{i})^{∗}γ̃_{i} → γ̃_{∞}. For example, if x_{i} realizes the minimal value of ρ_{i} in (4.42), then γ̃_{∞} is a complete conformally flat metric on ℝ^{n}. Other choices of base points may lead to complete conformally flat metrics on ℝ × S^{n−1} for example. In general, since the conformal class [γ_{i}] is controlled in C^{m,α}, the limit domain V is contained in ℝ^{n}, with closure V̄ = ℝ^{n}. Similarly, the forms τ_{(n,i)} and σ_{(n,i)} converge in C^{α′} to limit forms τ_{(n)} and σ_{(n)} on V; recall that the tilde has been dropped from the notation.

For the same reasons, since the metrics g̃_{i} satisfy (4.39), the pointed sequence (M_{i}, g̃_{i}, x_{i}) also converges, in C^{n,α′}, to a maximal connected limit (N, g̃_{∞}, x_{∞}), with boundary data (V, γ̃_{∞}).

Let Y be any C^{2} smooth vector field on ∂M. The vector fields X_{i} = (F_{i})^{∗}Y are blow-ups of Y based at x_{i} and converge to a conformal Killing field X_{∞} on (V, γ̃_{∞}), so that

(4.49)  L̂_{X_{i}} γ̃_{i} → L̂_{X_{∞}} γ̃_{∞} = 0.

Similarly, since the metrics γ_{i} are bounded in C^{n,α}, the terms a_{(n)} are also bounded, and hence, wirh respect to the boundary metrics γ̃_{i}, (a_{(n)})_{i} → 0 pointwise. Combining then (4.44) with (4.43) and (4.48), it thus follows that for i large,

(4.50)  sup_{|Y |≤1} sup_{|h_{(0)}|≤1} | ∫_{∂M}⟨L̃_{X_{i}}τ_{(n,i)}, h_{(0)}⟩dVγ̃_{i}| << volγ̃_{i}(∂M),

where L̃_{X}τ_{(n)} = L_{X}τ_{(n)}+[(1− \frac{2}{n})divX]τ_{(n)}, |h_{(0)}| = ||h_{(0)}||_{C^{α}(γ̃_{i})}, and |Y | = ||Y ||_{C^{2}(γ_{i})} ∼ ||Y ||_{C^{2}(γ_{∞})}. Since h_{(0)} is arbitrary, it follows from (4.50) that there exist (possibly many) regions where ρ_{i} → 0, for which, on a complete, conformally flat limit (V, γ̃_{∞}, x_{∞}) of the boundary, one has

(4.51)  L̃_{X_{∞}}τ_{(n)} = 0,

for any conformal Killing field X_{∞} on (V, γ̃_{∞}). We claim that the only solution of (4.51), (for all conformal Killing X_{∞}), is

(4.52)  τ_{(n)} = 0.

To see this, a well-known relation gives L_{X}τ_{(n)} = ∇_{X}τ_{(n)} + 2δ^{∗}X ∘ τ_{(n)}, so that if X = X_{∞} is conformal Killing, then L_{X_{∞}}τ_{(n)} = ∇_{X_{∞}}τ_{(n)} + \frac{2}{n}(divX_{∞})τ_{(n)}. Hence (4.51) gives

(4.53)  ∇_{X_{∞}}τ_{(n)} = −(divX_{∞})τ_{(n)},

for all conformal Killing fields X_{∞}. Pairing (4.53) with τ_{(n)} gives

(4.54)  \frac{1}{2}X_{∞}(|τ_{(n)}|^{2}) = −(divX_{∞})|τ_{(n)}|^{2}.

Hence, if τ_{(n)} ≠ 0, then X_{∞}(log |τ_{(n)}|) = −divX_{∞}, so that by (4.53),

∇_{X_{∞}}(|τ_{(n)}|^{−1}τ_{(n)}) = 0.

Since (V, γ̃_{∞}) is conformally flat and X_{∞} is an arbitrary conformal Killing field, this easily implies that |τ_{(n)}| is constant, which from (4.54) implies (4.52).

Since the convergence of (M_{i}, g̃_{i}, x_{i}) to (N, g̃_{∞}, x_{∞}) is C^{n,α′}, (4.52) contradicts (4.39), which proves Proposition 4.6. (Equivalently, (4.52) can be shown to imply that the the blow-up limit (N, g′, x_{∞}) of (M_{i}, ḡ_{i}, x_{i}) as preceding Lemma 4.7 is flat, contradicting Lemma 4.7). Propositions 4.5 and 4.6 together imply Theorem 4.4, which also completes the proof of this result.

Combining Proposition 4.1 and Theorem 4.4 gives the following main result of this section.

Corollary 4.10. On a fixed 4-manifold M, suppose g ∈ E_{AH}^{m,α}, m ≥ 4, has boundary metric γ satisfying ||γ||_{C^{m,α}} ≤ K in a fixed coordinate atlas for ∂M. Then there exists δ_{1} = δ_{1}(K) > 0 such that the C^{m−1,α} geometry of the geodesic compactifications ḡ is uniformly controlled in U_{δ_{1}}, in that (4.2) holds, with r_{0} depending only on K.

The same result holds in all dimensions in which the strong control property holds.

5. Properness of the Boundary Map.

In this section, we combine the results of §3 and §4 to prove Theorem A, i.e. the properness of Π. Following this, in §5.2, we also analyse the behavior when Π is not proper and prove this is solely due to the formation of Einstein cusp metrics, cf. Theorem 5.4 and the discussion following it.

5.1. Corollary 4.10 gives uniform C^{m−1,α} control of the geodesic compactification of an AH Einstein metric (M, g) in a neighborhood of definite size about ∂M, in terms of C^{m,α} control of the boundary metric γ. Theorem 3.7 addresses control in the interior, away from the boundary. Regarding the hypotheses (3.17)-(3.21) of Theorem 3.7, the assumption (3.19) on a lower bound on the inradius of ḡ and the assumption (3.20) on an upper bound on the diameter T of S(t_{1}) are now immediate consequences of Corollary 4.10. Next we show that there is a global L^{2} bound on W, i.e. (3.21) holds.

Proposition 5.1. Let (M, g) be an AH Einstein metric on a 4-manifold M, with boundary metric γ. Then there is a constant Λ, depending only on the topology of M and the C^{4,α} norm of γ, such that

(5.1)  ∫_{M} |W_{g}|^{2}dV_{g} ≤ Λ.

Proof: The integral (5.1) is conformally invariant. In the tubular neighborhood U = U_{μ_{1}} about ∂M, (cf. (4.23)), we compute the L^{2} integral (5.1) with respect to the compactification ḡ, while in M \ U, the integral is computed with respect to the Einstein metric g.

Thus, by Corollary 4.10, the curvature R̄ of ḡ is uniformly bounded in U. Since vol_{γ}∂M is also uniformly bounded, the curvature bound also gives a uniform upper bound on the volume of (U, ḡ). Hence, by conformal invariance, the integral over U in (5.1) is uniformly bounded.

For the integral over M \ U, we use the Chern-Gauss-Bonnet theorem for manifolds with boundary, as in [5]. Let S(t_{0}) = ∂U_{μ_{0}}, μ_{0} = μ_{1}/2, viewed as a boundary in (M, g), and let Ω = M \ U_{μ_{0}}. Since (Ω, g) is Einstein, one has,

\frac{1}{8π^{2}} ∫_{Ω} |W |^{2}dV = \frac{1}{8π^{2}} ∫_{Ω}(|R|^{2} − 6) = χ(Ω) − \frac{3}{4π^{2}} volΩ + ∫_{S(t_{1})} B(R, A) ≤ χ(M) + ∫_{S(t_{1})} B(R, A);

compare also with (3.16). Here B(R, A) is a boundary term, depending on the curvature R and second fundamental form A of S(t_{0}) in (M, g). But again by Corollary 4.10, R and A are uniformly controlled on S(t_{0}) ⊂ (M, g), as is vol_{g}S(t_{0}). Hence, the L^{2} norm of W over Ω is uniformly bounded and so (5.1) follows.

We now assemble the work above to obtain the following result, which represents the major part of Theorem A. Recall from [6] that if (M, g) is an AH Einstein metric and ḡ is a geodesic compactification, then its width W idḡM is defined by

(5.2)  W idḡM = sup{t(x) : x ∈ M}.

The width depends on a choice of the boundary metric γ for the conformal infinity. However, if γ and γ′ are representatives in [γ], then

C^{−1}W idḡM ≤ W id_{}_{ḡ′}M ≤ CW idḡM,

where C depends only on the C^{1} norm of γ^{−1}γ′ and (γ′)^{−1}γ in a fixed coordinate system on ∂M.

Theorem 5.2. Let {g_{i}} be a sequence of AH Einstein metrics on M, with boundary metrics γ_{i} and suppose γ_{i} → γ in the C^{m,α′} topology on ∂M, m ≥ 4. Suppose further that

(5.3)  H_{2}(∂M, ℝ) → H_{2}(M̄, ℝ) → 0,

and there is a constant D < ∞ such that

(5.4)  W idḡ_{i}M ≤ D.

Then a subsequence of {g_{i}} converges smoothly and uniformly on compact subsets to an AH Einstein metric g on M with boundary metric γ. The geodesic compactifications ḡ_{i} converge in the C^{m−1,α′} topology to the geodesic compactification ḡ of g, within a fixed collar neighborhood U of ∂M.

Proof: Corollary 4.10 proves the last statement. As noted above, together with Proposition 5.1, it follows that the all the hypotheses (3.17)-(3.21) of Theorem 3.7 are satisfied. Thus if one chooses base points x_{i} satisfying (3.22), then a subsequence of (M, g_{i}, x_{i}) converges smoothly and uniformly on compact sets, to a limit AH Einstein metric (N, g, x).

For a fixed d > 0 small, let

(5.5)  (M_{i})_{d} = {x ∈ (M, g_{i}) : t_{i}(x) ≤ d},

and define N_{d} in the same way. Then the smooth convergence of the compactifications ḡ_{i} implies that (M_{i})_{2d} is diffeomorphic to N_{2d} and each is a collar neighborhood of ∂M. Further, the limit metric g on N_{2d} is AH, with boundary metric γ.

On the other hand, (5.4) implies that the complementary domains

(5.6)  (M_{i})^{d} = {x ∈ (M, g_{i}) : t_{i}(x) ≥ d}

have uniformly bounded diameter with respect to g_{i}, and so Theorem 3.7 implies that (M_{i})^{d/2} is diffeomorphic to the limit domain N^{d/2} in (N, g). It follows that N = M and so g is an AH Einstein metric on M, with boundary metric γ.

Combining the results above leads easily to the proof of Theorem A.

Proof of Theorem A.

Let γ_{i} be a sequence of boundary metrics in C^{o}, with γ_{i} → γ ∈ C^{o} in the C^{m,α′} topology on ∂M, m ≥ 4, with Π(g_{i}) = [γ_{i}]. Since only the conformal classes are uniquely determined, one may choose for instance γ_{i} to be metrics of constant scalar curvature. To prove Π^{o} is proper, one needs to show that {g_{i}} has a convergent subsequence in E_{AH} to a limit metric g ∈ E_{AH} with Π[g] = [γ].

Suppose first there is a constant s_{0} > 0 such that

s_{γ_{i}} ≥ s_{0}.

where s_{γ_{i}} is the (intrinsic) scalar curvature of the boundary metric γ_{i}. It is then proved in [6, Prop.5.1] that

(5.7)  W idḡ_{i}M ≤ \sqrt{3}π/\sqrt{s_{0}},

cf. also (A.12). Hence, in this case Theorem A follows directly from Theorem 5.2.

Next, suppose only s_{γ_{i}} ≥ 0. If there is some constant D < ∞ such that W idḡ_{i}M ≤ D, then again Theorem 5.2 proves the result. Suppose instead

(5.8)  W idḡ_{i}M → ∞.

In this case, there is a rigidity result associated with the limiting case of (5.7) as s_{0} → 0, proved in [6, Rmk.5.2, Lem.5.5]. Namely, s_{γ_{i}} ≥ 0 and (5.8) imply that the sequence (M, g_{i}, x_{i}), for x_{i} as in (3.22), converges in the Gromov-Hausdorff topology to a hyperbolic cusp metric

(5.9)  g_{C} = dr^{2} + r^{2}g_{F} ,

where g_{F} is a flat metric on ∂M. It follows from the proof of Theorem 5.2, as following (5.5), that γ = g_{F}, so that, by definition, γ ∉ C^{o}. This implies that necessarily W idḡ_{i} ≤ D, for some D < ∞, which completes the proof.

Remark 5.3. Theorem A implies that the space E_{AH}^{o} on a given 4-manifold M satisfying (5.3) has only finitely many components whose conformal infinities intersect any given compact set in C^{o}.

A similar result holds if the filling manifold M is also allowed to vary. Thus, there are only finitely many 4-manifolds M_{i} having a common boundary ∂M, which satisfy (5.3) and χ(M_{i}) ≤ C, for some constant C < ∞, for which

∩_{i}Π(E_{AH}^{o}(M_{i})) ≠ ∅.

The proof of this is exactly the same, using Theorem 3.5 and Remark 3.6 with varying manifolds M_{i}.

§5.2. In this section, we characterize the possible degenerations when Π is not proper. Observe first that Theorem 5.2 implies that if the manifold M satisfies (5.3), then the ”enhanced” boundary map

(5.10)  Ψ : E_{AH} → Met(∂M) × ℝ,
Ψ(g) = (Π(g), W id_{Π(g)}M)

is proper. Thus, degenerations of AH Einstein metrics with controlled conformal infinity can only occur when the width diverges to ∞. On the other hand, as indicated in the Introduction, in general the boundary map Π is not proper.

Define an AH Einstein metric with cusps (N, g) to be a complete Einstein metric g on a 4manifold N which has two types of ends, namely AH ends and cusp ends. A cusp end of (N, g) is an end E such that vol_{g}E < ∞. Thus, N has a compact (possibly disconnected) hypersurface H, disconnecting N into two non-compact connected components N = N_{1} ∪ N_{2} where (N_{1}, g) is an AH Einstein metric with boundary H and (N_{2}, g) has finite volume, so that each end of N_{2} is a cusp end. A natural choice for H is the level set t^{−1}(1), where t is a geodesic defining function for the conformally compact boundary ∂_{AH}N of N. Then N_{1} = {x ∈ N : t(x) ≤ 1}, N_{2} = {x ∈ N : t(x) ≥ 1}.

Note that any cusp end E is not conformally compact. As one diverges to infinity in E, the metric g is collapsing, in that vol_{g}B_{x}(1) → 0 as x → ∞ in E, and hence the injectivity radius satisfies inj_{g}(x) → 0, as x → ∞ in E, see §3.1.

Theorem 5.4. Let M be a 4-manifold satisfying (1.4), and let g_{i} ∈ E_{AH} be AH Einstein metrics with boundary metrics γ_{i} ∈ C^{m,α}, with γ_{i} → γ in the C^{m,α′} topology on ∂M, m ≥ 4. Let t_{i} be the geodesic defining function associated with γ_{i} and choose base points x_{i} ∈ H_{i} = t_{i}^{−1}(1).

Then a subsequence of {g_{i}} converges, modulo diffeomorphisms in D_{1}, either to an AH Einstein metric g on M, or to an AH Einstein metric with cusps (N, g, x), x = lim x_{i}. The convergence is smooth and uniform on compact subsets of M, N respectively. In both cases, the conformal infinity is given by (∂M, [γ]). Further, the manifold N weakly embeds in M, as in (3.23).

Proof: For any D < ∞, let (M_{i})_{D} = {x ∈ (M, g_{i}) : t_{i}(x) ≤ D}, as in (5.5). Theorem 5.2 implies that a subsequence of ((M_{i})_{D}, g_{i}, x_{i}) converges smoothly to a limit AH Einstein metric g on a domain N_{D}, with N_{D} diffeomorphic to (M_{i})_{D}, and with conformal infinity of N_{D} given by (∂M, [γ]).

If there is a fixed D < ∞ such that (5.4) holds, then the result follows from the proof of Theorem 5.2 or Theorem A. Thus, we may suppose

(5.11)  W idḡ_{i}M → ∞.

In this case, it follows from [6, Lemma 5.4] that there is a constant V_{0} < ∞, depending only on {γ_{i}} and the Euler characteristic χ(M), such that

(5.12)  vol_{g_{i}}(M_{i})^{1} ≤ V_{0},

where (M_{i})^{1} = {x ∈ (M, g_{i}) : t_{i}(x) ≥ 1} is the complementary domain to (M_{i})_{1}. (The estimate (5.12) is a straightforward consequence of (2.18), or more precisely the bound V ≤ \frac{4π^{2}}{3} χ(M), given uniform control of the metrics g_{i} and ḡ_{i} on (M_{i})_{1}).

By Theorem 3.5 and Remark 3.6, the pointed manifolds (M, g_{i}, x_{i}), for x_{i} base points in S_{i}(1) = t_{i}^{−1}(1), converge in a subsequence and in the Gromov-Hausdorff topology to a complete Einstein manifold (N, g, x), x = lim x_{i}. The convergence is also in the C^{∞} topology, uniform on compact sets. The domain N^{1} = {x ∈ (N, g) : t(x) ≥ 1}, with t(x) = lim t_{i}(x), is the limit of the domains (M_{i})^{1}.

The bound (5.12) implies that N^{1} is of finite volume while (5.11) implies that N^{1} is non-compact. It follows that the full limit N = N^{1} ∪ N_{1} with limit metric g is a complete AH Einstein manifold with conformal infinity (∂M, [γ]) and with a non-empty collection of cusp ends. The fact that N weakly embeds in M follows exactly as in the proof of Theorem 3.7.

The fact that the AH cusp metric (N, g) has conformal infinity (∂M, [γ]) and that it weakly embeds in M implies that there is a sequence t_{j} → ∞ such that N_{t_{j}} ⊂ N embeds in M, for N_{t_{j}} as above. Hence, for any j large, the manifold M may be decomposed as

(5.13)  M = N_{t_{j}} ∪ (M \ N_{t_{j}}).

With respect to a suitable diagonal subsequence j = j_{i}, the metrics g_{i} on M push the region M \ N_{t_{j_{i}}} off to infinity as i → ∞ and t_{j_{i}} → ∞, giving rise to cusp ends in the limit (N, g).

Theorem 5.4 suggests the construction of a natural completion Ē_{AH} of E_{AH}. Thus, for any R < ∞ large, let C(R) denote the space of conformal classes [γ] on ∂M which contain a representative γ satisfying ||γ||_{C^{4,α}} ≤ R, with respect to some fixed coordinate system for ∂M. Let E_{AH}(R) = Π^{−1}(R) and for any g ∈ E_{AH}(R), choose a base point x ∈ t^{−1}(1) ≡ H_{g}, where t is the geodesic defining function associated to the boundary metric γ.

Then Theorem 5.4 implies that the completion Ē_{AH}(R) of E_{AH}(R) in the pointed Gromov-Hausdorff topology based at points x ∈ H_{g} is the set of AH Einstein metrics on M, together with AH Einstein metrics with cusps (N, g). If γ_{i} are metrics in C(R), then a subsequence of γ_{i} converges in C^{m,α′} to a limit metric γ ∈ C(R). If g_{i} ∈ E_{AH}(R) satisfy Π(g_{i}) = γ_{i}, then the corresponding subsequence of g_{i} converges in the Gromov-Hausdorff topology based at x_{i} to (N, g) ∈ Ē_{AH}(R), and the conformal infinity of (N, g) is γ = lim γ_{i}. The convergence is also in the C^{∞} topology, uniform on compact sets. Of course one may have N = M, in which case the limit is an AH Einstein metric on M.

For R < R′, E_{AH}(R) ⊂ E_{AH}(R′) and so we may form the union Ē_{AH} = ∪_{R}Ē_{AH}(R) with the induced topology. It is clear that in this topology, the boundary map Π extends to a continuous map

(5.14)  Π̄ : Ē_{AH} → C.

Further, the following corollary is an immediate consequence of Theorem 5.4.

Corollary 5.5. Let M be a 4-manifold satisfying (1.4). Then the extended map Π̄ : Ē_{AH} → C is proper.

It follows in particular that the image

Π̄(Ē_{AH}) ⊂ C

is a closed subset of C. However, it is not known if Ē_{AH} is a Banach manifold. Even if Ē_{AH} is not a Banach manifold, it would be interesting to understand the ’size’ of the set of AH Einstein metrics with cusps. The set of such metrics corresponds of course exactly to

∂Ē_{AH} = Ē_{AH} \ E_{AH}.

In particular, with regard to the work to follow in §6-§7, one would like to know if the image Π̄(∂E_{AH}) disconnects C or not, cf. also Remark 7.8 below. Again the fact that Π̄ is proper implies that Π̄(∂E_{AH}) is a closed subset of C.

Let

(5.15)  Ĉ = C \ Π̄(∂E_{AH});

this is the space of conformal classes on ∂M which are not the boundaries of AH Einstein metrics with cusps associated to the manifold M. Also let Ê_{AH} = Π̄^{−1}(Ĉ), so that Ê_{AH} is the class of AH Einstein metrics on M whose conformal infinity is not the conformal infinity of any AH Einstein metric with cusps associated to M. If one gives Ĉ the relative topology, as a subset of C, then the following result is also an immediate consequence of the results above.

Corollary 5.6. Let M be a 4-manifold satisfying (1.4). Then the map

(5.16)  Π̂ : Ê_{AH} → Ĉ

is proper.

Remark 5.7. The first examples where Π is not proper, for example Π^{−1}(pt) is non-compact in E_{AH}, is the sequence of AH Einstein metrics g_{i} on ℝ^{2} × T^{n−1}, converging to the hyperbolic cusp metric on ℝ × T^{n}, discussed in Remark 2.6; cf. [6, Prop.4.4] for the explicit construction of {g_{i}}.

However, these metrics lie in distinct components of the moduli space E_{AH} on ℝ^{2} × T^{2}, and so cannot be connected by a curve of metrics on ℝ^{2} × T^{2}. Thus E_{AH} has infinitely many components, and this is the cause of Π being non-proper. Currently, there are no known examples where Π restricted to a component of E_{AH} is not proper. On the other hand, if one passes to the quotient of E_{AH} by the full diffeomorphism group, then the metrics g_{i} above may be joined by a curve.

More recently, G. Craig [17] has constructed infinite sequences of AH Einstein metrics (M_{i}, g_{i}) on non-diffeomorphic manifolds M_{i}, which all have the same conformal infinity (∂M, [γ_{0}]). These limit on a complete hyperbolic manifold (N, g_{−1}), with conformal infinity given by (∂M, [γ_{0}]), but with additional cusp, i.e. parabolic, ends analogous to the hyperbolic cusp end above. In fact, the Einstein metrics are constructed by Dehn filling each cusp end of N.

6. Degree of the Boundary Map.

In this section, we prove that the boundary map Π^{o} has a well-defined ℤ-valued degree, following Smale [34] and White [37], cf. also [34]. Throughout this section, we work componentwise on E_{AH}^{o} and C^{o}, but we will not distinguish components with extra notation. Thus, we assume

Π^{o} : E_{AH}^{o} → C^{o}

where E_{AH}^{o} and C^{o} are connected, that is connected components of the full spaces. The results of this section also hold for the restricted boundary map Π̂ : Ê_{AH} → Ĉ in (5.16), but since there is no intrinsic characterization of Ĉ, we work with C^{o}; see also Remark 7.8.

Since Π^{o} is a proper Fredholm map of index 0, the Sard-Smale theorem [34] implies that the regular values of Π^{o} are open and dense in C^{o}. For γ a regular value, the fiber (Π^{o})^{−1}(γ) thus consists of a finite number of points, i.e. (equivalence classes of) AH Einstein metrics on M. By [34]

(6.1)  deg_{2}Π^{o} = #(Π^{o})^{−1}([γ]) mod 2,

is well-defined, for any regular value [γ] in C^{o}. We recall that if [γ] ∉ ImΠ^{o}, then [γ] is tautologically a regular value of Π^{o}.

Next, we show that Π^{o} has a well-defined degree in ℤ, essentially following [37]. Thus, for [γ] ∈ ImΠ^{o}, let g be any AH Einstein metric on M, with Π^{o}(g) = [γ]. Consider the linearization of the Einstein equations, i.e. as in (2.4), the elliptic operator

L = \frac{1}{2}D^{∗}D − R,

acting on L^{2,2}(M, g). The operator L is bounded below on L^{2,2} and as in the theory of geodesics or minimal surfaces, let

(6.2)  ind_{g} ∈ ℤ,

be the L^{2} index of the operator L at (M, g), i.e. the maximal dimension of the subspace of L^{2,2}(M, g) on which L is a negative definite bilinear form, with respect to the L^{2} inner product. The nullity of (M, g) is the dimension of the L^{2} kernel K.

The main result of this section is the following:

Theorem 6.1. Let γ be a regular value of Π^{o} on (M, g) and define

(6.3)  degΠ^{o} = ∑_{g_{i}∈(Π^{o})^{−1}([γ])}(−1)^{ind_{g_{i}}} .

Then deg Π^{o} is well-defined, i.e. independent of the choice of [γ] among regular values of Π^{o}.

Proof: In [37], White presents general results guaranteeing the existence of a ℤ-valued degree, and we will show that the current situation is covered by these results. Thus, we refer to [37] for some further details.

Let [γ_{1}] and [γ_{2}] be regular values of Π^{o} and let [σ̄(t)], t ∈ [0, 1] be an oriented curve in C^{o} joining them. Choose representatives γ_{1} ∈ [γ_{1}] and γ_{2} ∈ [γ_{2}] and let σ̄(t) be a curve in Met(∂M) joining γ_{1} to γ_{2}. By [34], we may assume that σ̄ is transverse to Π, so that the lift σ = Π^{−1}(σ̄) is a collection of curves in E_{AH}, with boundary in the fibers over γ_{1} and γ_{2}. Define an orientation on σ by declaring that Π is orientation preserving at any regular point of σ which has even index, while Π is orientation reversing at regular points of σ of odd index. Thus, provided this orientation is well-defined, the map Π|_{σ} : σ → σ̄ has a well-defined mapping degree, as a map of 1-manifolds. By construction, this 1-dimensional degree is given by (6.3) at any regular point of σ and hence it follows that (6.3) is well-defined. Thus, it suffices to prove that the orientation constructed above is well-defined.

If Π_{∗}(dσ/dt) ≠ 0 for all t, so that all points of σ are regular, then the index of σ(t) is constant, and so there is nothing more to prove. Suppose instead that Π_{∗}(σ′(t_{0})) = 0, so that σ(t_{0}) is a critical point of Π; (without loss of generality, from here on assume σ is connected). Hence σ′(t_{0}) = κ_{0} ∈ K, and for t near t_{0}, σ(t) = σ(t_{0}) + (t − t_{0})κ_{0} + O((t − t_{0})^{2}). Without loss of generality, (cf. [37]), one may assume that K = ⟨κ_{0}⟩ is the span of κ_{0}. Thus, σ(t) may be viewed as a graph over K_{0}.

Let S_{AH}^{m,α} be the space of complete metrics g on M which have a C^{m,α} conformal compactification and which satisfy the slice condition β_{σ(t_{0})}(g) = 0. It is proved in [7, Thm.4.1] that the map

(6.4)  S_{AH}^{m,α} → 𝕊^{m−2,α}, g → Ric_{g} + 3g,

is a submersion at any g ∈ E_{AH}^{m,α}. Since Ric_{g} + 3g = 0 on the curve σ(t), the implicit function theorem implies that there is a 2-parameter family σ(t, s) ∈ S_{AH}^{m,α}, with Π(σ(t, s)) = Π(σ(t)) = γ(t), for t near t_{0} and s near 0, such that

(6.5)  Ric_{σ(t,s)} + 3σ(t, s) ∈ K.

It is well-known that forms κ ∈ K satisfy |κ| = O(t^{n}), i.e. O(t^{3}) in the case at hand, cf. [11], [30]. This, together with the fact that such κ are transverse-traceless, (cf. (2.6)), implies that the metrics σ(t, s) are Einstein to order 3 at conformal infinity ∂M, in that \frac{dσ}{ds} ∈ T̃_{σ(t,s)}E_{AH}, cf. (2.20).

By Lemma 2.2, the renormalized action I_{EH}^{ren} is then well-defined on the family σ(t, s). When viewed as a function of t, s, one has

(6.6)  \frac{d}{ds}I_{EH}^{ren}(σ(t, s)) = ∫_{M}⟨Ric_{σ(t,s)} + 3σ(t, s), \frac{dσ}{ds} ⟩dV,

since dσ/ds has 0 boundary values on ∂M. (The Euler-Lagrange equations for I_{EH}^{ren} are the same as those for the usual Einstein-Hilbert action). By (2.6) and (6.5),

\frac{d}{ds}(Ric_{σ(t,s)} + 3σ(t, s)) = L(\frac{dσ(t, s)}{ds}) ∈ K,

which implies that \frac{d}{ds}σ(t, s) ∈ K. From this, it follows that the integral in (6.6) vanishes only on the curve σ(t); thus the curve σ(t) in the plane P spanned by t, s is the set where dI_{EH}^{ren}/ds = 0. (The functional I_{EH}^{ren} plays the role of g in [37]). It follows that σ(t) is (locally) the boundary of the open set {dI_{EH}^{ren}/ds > 0} in P:

(6.7)  σ(t) = ∂{dI_{EH}^{ren}/ds > 0}.

If ind_{σ(t)} is even, for t near t_{0}, t ≠ t_{0}, give σ(t) the boundary orientation induced by this open domain, while if ind_{σ(t)} is odd for t near t_{0}, t ≠ t_{0}, give σ(t) the reverse orientation. It is now straightforward to see that this definition coincides with the orientation defined at the beginning of the proof. Thus, the point t_{0} → σ(t_{0}) is a critical point for the map π_{1} ∘ σ, where π_{1} : (t, s) → t is projection onto the first factor. If this critical point is a folding singularity for π_{1} ∘ σ, then the index of σ(t) changes by 1 in passing through σ(t_{0}) and reverses the orientation of λ(t) = π_{1}(σ(t)), (exactly as is the case with the standard folding singularity x → x^{2}). On the other hand, if π_{1} ∘ σ does not fold with respect to π_{1}, (so that one has an inflection point), then the index of σ(t) does not change through t_{0} and π_{1} maps σ(t) to λ(t) in an orientation preserving way. We refer to [37] for further details, and also to §7 for concrete examples of such folding behavior.

Remark 6.2. As discussed in §2, the inclusion E_{AH}^{(m′,α′)} ⊂ E_{AH}^{(m,α)}, is dense, for any (m′, α′) ≥ (m, α) and these spaces are diffeomorphic. Hence, the degree of Π^{o} is independent of (m, α) and defined on the spaces E_{AH}^{(m,α)} for all m ≥ 4. Moreover, the degree is also defined on the spaces Ē_{AH} and Ê_{AH} in (5.14) and (5.16), and is independent of (m, α).

7. Computations of the Degree.

We conclude the paper with computations of degΠ^{o} for several interesting examples of 4-manifolds. In all cases, the evaluation of the degree is made possible by symmetry arguments, using Theorem 2.5.

Recall that if degΠ^{o} ≠ 0, then Π is surjective onto C^{o}, and so any conformal class [γ] in C^{o} on ∂M has a filling by an AH Einstein metric g with conformal infinity [γ]. Of course, degΠ^{o} = 0 does not imply that Π cannot be surjective.

We begin with the proof of Theorem B.

Proof of Theorem B.

As seed metric, choose the hyperbolic (Poincaré) metric g_{0} on B^{4}. This has conformal infinity [γ_{0}], where γ_{0} is the round metric on S^{3} = ∂B^{4}. Now the boundary metric γ_{0} admits a large connected group SO(4) of isometries. Theorem 2.5 implies that for any AH Einstein metric g on B^{4}, SO(4) acts effectively by isometries on (B^{4}, g). It is then standard that g must be the Poincaré metric on B^{4}.

It follows that up to isometry, g_{0} is the unique metric with conformal infinity [γ_{0}]. (This also follows from a rigidity theorem in [9, Thm.5.2]). Further, it is well-known that the L^{2} kernel K of g_{0} is trivial, i.e. K = {0}; in fact, this holds for any Einstein metric of negative sectional curvature, cf. [10], [11]. Thus, g_{0} is a regular point of Π and since Π^{−1}[γ_{0}] = g_{0}, the point [γ_{0}] is a regular value of Π. The result then follows from (6.3).

We point out the following immediate consequence of the proof.

Corollary 7.1. Let M be any 4-manifold satisfying π_{1}(M, ∂M) = 0, and (1.4), with ∂M = S^{3} and M ≠ B^{4}. Then on any component of E_{AH}^{o},

(7.1)  degΠ^{o} = 0,

and Π is not surjective. In fact the conformal class of the round metric γ_{0} on S^{3} is not in ImΠ.

Proof: Suppose the class [γ_{0}] ∈ ImΠ, so that there is an AH Einstein metric g on M, with boundary metric γ_{0}. The proof of Theorem B above implies that necessarily g = g_{0}, where g_{0} is the Poincaré metric on B^{4}. Hence, M = B^{4}, a contradiction.

Next, we have:

Proposition 7.2. Let M = ℝ^{2} × S^{2}, so that ∂M = S^{1} × S^{2}. Then

(7.2)  degΠ^{o} = 0,

and Π is not surjective.

Proof: As seed metric(s) in this case, we take the remarkable 1-parameter family of AdS Schwarzschild metrics, discussed in detail in [23], cf. also [38]. Thus, on ℝ^{2} × S^{2}, consider the metric

(7.3)  g_{m} = F^{−1}dr^{2} + F dθ^{2} + r^{2}g_{S^{2}(1)},

where F = F (r) = 1+r^{2} − \frac{2m}{r}. The mass parameter m > 0 and r ∈ [r_{+}, ∞), where r_{+} is the largest root of the equation F (r_{+}) = 0. The locus {r_{+} = 0} is a totally geodesic round 2-sphere S^{2}, of radius r_{+}. Smoothness of the metric at {r_{+} = 0} requires that the circle parameter θ run over the interval [0, β), where β is given by

(7.4)  β = \frac{4πr_{+}}{1 + 3r_{+}^{2}} .

It is easily seen that as m varies from 0 to ∞, r_{+} varies monotonically from 0 to ∞.

The metrics g_{m} are isometrically distinct, for distinct values of m, and form a smooth curve in E_{AH}, with conformal infinity given by the conformal class of the product metric γ_{m} = S^{1}(β)×S^{2}(1). Notice however that the length β has a maximum value as m ranges over (0, ∞), namely

(7.5)  β ≤ β_{max} = 2π/\sqrt{3},

achieved at r_{+} = 1/\sqrt{3}, m = m_{0} = 2/\sqrt{3}. As m → 0 or m → ∞, one has β → 0.

Thus, the boundary map Π on the curve g_{m} is a fold map, folding the ray m ∈ (0, ∞) onto the β-interval (0, β_{max}]. The map Π restricted to the curve g_{m} is a 2-1 map, except at the point g_{m_{0}}. The metric g_{m_{0}} is a critical point of Π, and, (as will be seen below), the tangent vector (dg_{m}/dm)_{m=m_{0}} spans the L^{2} kernel K_{g_{m_{0}}} .

Next, we claim that the metrics g_{m} are the only AH Einstein metrics on ℝ^{2} × S^{2} with conformal infinity given by a product γ_{L} = S^{1}(L) × S^{2}(1). The isometry group of γ_{L} contains G = SO(2) × SU(2) and by Theorem 2.5, any AH Einstein metric g_{L} on M with boundary metric γ_{L} has G acting effectively by isometries. As in (2.7), we may choose a geodesic defining function for g_{L} so that g_{L} has the form

g_{L} = ds^{2} + g_{s},

where g_{s} is a curve of metrics on S^{1} × S^{2} invariant under the G-action. Thus, the Einstein metric g_{L} has cohomogeneity 1. It then follows from the classification given in [31] for instance that the metric g_{L} on ℝ^{2} × S^{2} is isometric to g_{m}, for some m = m(L).

A similar argument shows that the metrics g_{m}, for m ≠ m_{0} are regular points of Π. For suppose there exists κ ∈ K = K_{g_{m}}. Then κ is the tangent vector to a curve σ(s) ∈ E_{AH} with σ(0) = g_{m} and Π_{∗}(κ) = 0, so that the boundary metric of σ(s) is fixed to first order in s. The proof of Theorem 2.5 in [8] then shows that κ inherits the symmetries of g_{m}, i.e. κ is G-invariant. Hence, κ is tangent to the curve g_{m}, so that, without loss of generality, σ(s) = g_{m(s)}. Since Π_{∗}(\frac{d}{dm}g_{m}) ≠ 0, when m ≠ m_{0}, this proves the claim.

We may thus compute degΠ^{o} by evaluating the formula (6.3) on a pair of distinct metrics g_{m_{1}} and g_{m_{2}} with Π(g_{m_{1}}) = Π(g_{m_{2}}). From [23, §3], one has

(7.6)  ind_{g_{m_{1}}} = +1, ind_{g_{m_{2}}} = 0,

for m_{1} < m_{0} < m_{2}, which proves (7.2). Alternately, since Π is a 2-1 fold map on g_{m}, the proof of Theorem 7.1 shows directly that (7.2) holds.

Further, the symmetry argument above implies that the metrics S^{1}(L) × S^{2}(1) are not in ImΠ, whenever

(7.7)  L > β_{max},

and hence Π is not surjective.

This result should be compared with the following:

Proposition 7.3. Let M = S^{1} × ℝ^{3}, so that ∂M = S^{1} × S^{2}. Then

(7.8)  degΠ^{o} = 1,

and Π^{o} is surjective.

Proof: As seed metrics in this situation, we take a family of hyperbolic metrics, namely the metrics ℍ^{4}(−1)/ℤ, where the ℤ-quotient is obtained by a hyperbolic or loxodromic translation of length L along a geodesic in ℍ^{4}(−1). The conformal infinity is the product metric S^{1}(L) × S^{2}(1) in the case of a hyperbolic translation, and the bent product metric S^{1}(L) ×_{α} S^{2}(1) on the same space when the translation is loxodromic; the angle α between the factors S^{1}(L) and S^{2}(1) corresponds to the loxodromic rotation.

As in Proposition 7.2, any AH Einstein metric on S^{1} × ℝ^{3} with boundary metric S^{1}(L) × S^{2}(1) has SO(2) × SO(3) acting effectively by isometries. Again, the classification in [31] implies that, on this manifold, the only such metrics are hyperbolic. Hence the result follows as in the proof of Theorem B.

Next, we turn to non-trivial disc bundles over S^{2}. For the disc bundle of degree 1 over S^{2}, i.e. M = ℂℙ^{2} \ B^{4}, with ∂M = S^{3}, Corollary 7.1 implies that Π is not surjective so that

(7.9)  degΠ^{o} = 0.

Remark 7.4. Actually, to justify this statement one needs to remove the hypothesis (1.4), since in this situation H_{2}(∂M) does not surject onto H_{2}(M̄). Recall that (1.4) was only used to rule out orbifold degenerations in the proof of Theorem 3.7. However, it is sometimes possible to rule out orbifold degenerations in the case of sufficiently low Euler characteristic directly, without the use of (1.4). This situation was treated in [1] in the case of orbifold degenerations on compact manifolds, and the argument for AH metrics is similar; thus we refer to [1], [3] for some further details.

Let M be the disc bundle of degree k over S^{2}, so that χ(M) = 2, and let {g_{i}} be a sequence of AH Einstein metrics on M which converge to an AH Einstein orbifold (X, g), with boundary metrics γ_{i} converging in C^{m,α} to the boundary metric γ for (X, g). By (2.18), one has

\frac{1}{8π^{2}} ∫_{M} |W_{g_{i}}|^{2} = 2 − \frac{3}{4π^{2}} V (g_{i}).

The results of Theorem 3.7, Corollary 4.10 and Proposition 5.1 show that V (g_{i}) → V (g), where V (g) is the renormalized volume of the Einstein orbifold (X, g). Now as discussed following Proposition 3.10, orbifold singularities arise by crushing essential 2-cycles in M to points. Since M has only one essential 2-cycle, (up to multiplicity), given by the zero-section, there can only be one singular point on X. For the same reason, there is only one Ricci-flat ALE space (E, g_{∞}) associated to the singularity. It is then easy to see, cf. [1, §6], [3, §3.3] and references therein, that

lim_{i→∞} ∫_{M} |W_{g_{i}}|^{2} = ∫_{X} |W_{g}|^{2} + ∫_{E} |W_{g_{∞}}|^{2}.

The formula (2.18) for Einstein orbifolds (X, g) gives

\frac{1}{8π^{2}} ∫_{X} |W_{g}|^{2} = χ(X_{0}) + \frac{1}{|Γ_{0}|} − \frac{3}{4π^{2}} V (g),

where X_{0} is the regular set of X and the orbifold singularity of X is C(S^{3}/Γ_{0}), (cf. [1, (6.2)]). Also

∫_{E} |W_{g_{∞}}|^{2} = χ(E) − \frac{1}{|Γ|},

where E is asymptotic to C(S^{3}/Γ) at infinity, (cf. [1, (6.3)]). Combining these equations gives Γ_{0} = Γ = ℤ_{k}, and hence χ(X_{0}) = 0, so that E is diffeomorphic to M. In particular, this rules out orbifold degenerations in the case k = 1, where M = ℂℙ^{2} \ B^{4}, since Γ must be non-trivial, Γ ≠ {e}. This gives (7.9).

For general k, one thus has

(7.10)  ∫_{E} |W_{g_{∞}}|^{2} = 2 − \frac{1}{k} .

On the other hand, the signature formula for ALE metrics, cf. [27] for instance, gives

(7.11)  τ(E) = \frac{1}{12π^{2}} ∫_{E} |W_{+}|^{2} − |W_{−}|^{2} + η(S^{3}/ℤ_{k}),

and

η(S^{3}/ℤ_{k}) = \frac{(k − 1)(k − 2)}{3k}.

Since τ(E) = 1, simple arithmetic as in the Hitchin-Thorpe inequality gives the estimate

(7.12)  4k − 2 ≥ |3k − (k − 1)(k − 2)|.

The estimate (7.12) implies k ≤ 9, so that orbifold degenerations are not possible if k ≥ 10.

We conjecture that there are in fact no orbifold degenerations for k ≥ 3. However, for k = 2, it will be seen below that there are orbifold degenerations. In fact, in this case, (7.12) shows that (E, g_{∞}) must be self-dual, and hence by Kronheimer’s classification [29], (E, g_{∞}) is the Eguchi-Hanson metric.

Example 7.5. It is interesting to compare the result (7.9) with an explicit family of AH Einstein metrics on M = ℂℙ^{2} \ B^{4}, namely the AdS Taub-Bolt family, cf. [24], [31]. This is a 1-parameter family of metrics given by

(7.13)  g_{s} = E_{s}\{(r^{2} − 1)F^{−1}(r)dr^{2} + (r^{2} − 1)^{−1}F (r)(dτ + cos θdϕ)^{2} + (r^{2} − 1)g_{S^{2}( \frac{1}{2} )}\},

where the parameters r, s satisfy r ≥ s and s > 2, the constant E_{s} is given by

(7.14)  E_{s} = \frac{2}{3} \frac{s − 2}{s^{2} − 1},

and the function F (r) = F_{s}(r) is

(7.15)  F_{s}(r) = Er^{4} + (4 − 6E)r^{2} + \{−Es^{3} + (6E − 4)s + \frac{1}{s}(3E − 4)\}r + (4 − 3E).

The parameter τ runs over the vertical S^{1}, and τ ∈ [0, β), with

(7.16)  β = 2π.

The bolt S^{2} \{r = s\} is a round, totally geodesic 2-sphere, of area A_{s} = \frac{2}{3}π(s − 2). These metrics are AH, with conformal infinity given by a Berger (or squashed) S^{3}, with base S^{2}(\frac{1}{2}), and Hopf fiber S^{1} = S^{1}(L) of length L = 2π\sqrt{E}.

In analogy to (7.4)-(7.5), notice that E → 0 as s → 2 or s → ∞, and has a maximal value E_{max} = (2 − \sqrt{3})/3 at s = s_{0} = 2 + \sqrt{3}. Note in particular that since

E_{max} < 1,

the round metric S^{3}(1) is not in Im Π(g_{s}) for any s.

We see that on the curve g_{s}, s ∈ (2, ∞), the boundary map Π has exactly the same 2-1 fold behavior as for the AdS Schwarzschild metric. The use of Theorem 2.5 as in Proposition 7.2 implies that Π is not surjective, and in fact the Berger spheres with Hopf fiber length L, for L > 2π\sqrt{E_{max}} are not in ImΠ.

In contrast, one has the following behavior on more twisted disc bundles over S^{2}.

Proposition 7.6. Let M = M_{k} be the disc bundle of degree k over S^{2}, k ≥ 10, so that ∂M = S^{3}/ℤ_{k}. Then

(7.17)  degΠ^{o} = 1,

and Π^{o} is surjective.

Proof: As seed metrics, choose again the AdS-Taub bolt metrics on M_{k}, cf. [24], [31]. These have exactly the same form as (7.13)-(7.16), except that the parameter s satisfies s > 1, E_{s} in (7.14) is replaced by E_{s,k} of the form

(7.18)  E_{s,k} = \frac{2ks − 4}{3(s^{2} − 1)},

and the period β for τ is given by β = 2π/k. For each k, these metrics are AH, and the conformal infinity on S^{3}/ℤ_{k} is given by the Berger metric, with, as before, Hopf circle fibers of length L = 2π\sqrt{E_{s,k}}/k. When E = 1, conformal infinity is given by the round metric on S^{3}/ℤ_{k}.

Note however from (7.18) that now the function E_{s,k} is a monotone decreasing function of s, as s increases from 1 to ∞. Hence, Π is 1-1 on this curve, and in particular, the metric g_{s_{0}}, s_{0} = \frac{1}{3}(k +\sqrt{k^{2} − 3}) has conformal infinity the constant curvature metric on S^{3}/ℤ_{k}. The symmetry argument using Theorem 2.5 as before, together with [31], implies these metrics are the unique metrics with these boundary values. The metrics g_{s} are regular points for Π, and so (7.17) follows, since by Remark 7.4, orbifold singularities cannot occur when k ≥ 10.

Remark 7.7. We expect Proposition 7.6 holds for all k ≥ 3; the proof above holds for such k, provided there are no orbifold degenerations, cf. Remark 7.4.

However, Proposition 7.6 does not hold when k = 2. In this case, (7.18) becomes

(7.19)  E_{s,2} = \frac{4}{3} \frac{1}{s + 1}.

This is of course monotone decreasing in s ∈ (1, ∞), but it has a finite value at s = 1 with E_{1,2} = \frac{2}{3}. As s → 1, the area of the bolt S^{2} at {r = s} tends to 0, and vanishes at {r = 1} when s = 1. Thus, the Taub-Bolt metric is an orbifold singular metric on C(ℝℙ^{3}) when s = 1. Note this shows that the k = 2 discussion in Remark 7.4 is sharp, in that as s → 1 on the Taub-Bolt curve, the Ricci-flat ALE space associated to the orbifold singularity is necessarily the Eguchi-Hanson metric. In particular, only the values E ∈ (0, \frac{2}{3}) are achieved on the Taub-Bolt curve g_{s}. I am grateful to Michael Singer for pointing out this behavior of the Taub-NUT curve.

Hence the round metric γ_{0} on ℝℙ^{3} is not in ImΠ on the curve g_{s}. Again, Theorem 2.5 and the classification in [31] show that γ_{0} ∉ Π(E), where E is the moduli space of AH Einstein metrics on the disc bundle of degree 2 over S^{2}.

Remark 7.8. We close the paper with some observations on whether the full boundary map

Π : E_{AH} → C

might be surjective, or almost surjective, at least when degΠ^{o} ≠ 0.

First, recall from Corollaries 5.5 and 5.6 that both the extended map Π̄ : Ē_{AH} → C and the restricted map Π̂ : Ê_{AH} → Ĉ are proper. In this regard, it would be very interesting to know if the set of boundary values of AH Einstein metrics with cusps disconnects C or not, that is whether

Π̄(∂Ē_{AH}) ⊂ C

disconnects C, or whether Ĉ = C \ Π̄(∂Ē_{AH}) is path connected, (for instance if Π̄(∂Ē_{AH}) is of codimension at least 2). If Ĉ is path connected, then the degree theory arguments of §6 and §7 hold without any change and give a well-defined degree degΠ on each component of E_{AH}. In particular, if this holds and degΠ ≠ 0, then Π is almost surjective, in that Π surjects onto Ĉ.

On the other hand, if Π̄(∂Ē_{AH}) disconnects C, then Π̄(∂Ē_{AH}) represents a “wall”, past which it may not be possible to fill in boundary metrics with AH Einstein metrics. This would be the case for instance if Ē_{AH} is a Banach manifold with boundary ∂Ē_{AH} and Π̄ maps ∂Ē_{AH} onto a set of codimension 1 in C.

The same issue arises if the condition (1.4) does not hold, so that orbifold singular metrics might arise, as discussed in Remark 7.7.

Finally, it would also be interesting to know if there are topological obstructions to the possible formation of cusps, as is the case with the formation of orbifold singularities as in (1.4). Thus, with respect to the decomposition (5.13), one would like to know for instance if some of the homology of one of the two factors injects into that of the union M.

Appendix

In this appendix, we collect several formulas for the curvature of conformal compactifications of AH Einstein metrics.

Let ḡ = ρ^{2} · g, where ρ is a defining function with respect to ḡ. The curvatures of the metrics g and ḡ are related by the following formulas:

(A.1)  K̄_{ab} = \frac{K_{ab} + | ¯∇ρ|^{2}}{ρ^{2}} − \frac{1}{ρ}\{D̄^{2}ρ(ē_{a}, ē_{a}) + D̄^{2}ρ(ē_{b}, ē_{b})\}.

(A.2)  R̄ic = −(n − 1)\frac{D̄^{2}ρ}{ρ} + (nρ^{−2}(| ¯∇ρ|^{2} − 1) − \frac{Δ̄ρ}{ρ})ḡ,

(A.3)  s̄ = −2n\frac{Δ̄ρ}{ρ} + n(n + 1)ρ^{−2}(| ¯∇ρ|^{2} − 1).

The equation (A.2) is equivalent to the Einstein equation (1.2). Observe that (A.2) implies that if the compactification ḡ is C^{2}, then

(A.4)  | ¯∇ρ| → 1 at ∂M,

and so by (A.1), |K_{ab} + 1| = O(ρ^{2}). For r = − log ρ, a simple calculation gives

(A.5)  | ¯∇ρ| = |∇r|,

where the norm and gradient on the left are with respect to ḡ, and on the right are with respect to g.

A defining function ρ = t is a geodesic defining function if

(A.6)  | ¯∇t| = 1,

in a collar neighborhood U of ∂M. Clearly, the formulas (A.1)-(A.3) simplify considerably in this situation. The function t is the distance function to ∂M on (M, ḡ), and similarly by (A.5), the function r is a (signed) distance function on (M, g). The integral curves of ¯∇t and ∇r are geodesics in (M, ḡ) and (M, g) respectively.

The 2^{nd} fundamental form Ā of the level sets S(t) of t in (M, ḡ) is given by Ā = D^{2}t, with H̄ = Δ̄t giving the mean curvature of S(t). Along t-geodesics of (M, ḡ), one has the standard Riccati equation

(A.7)  H̄′ + |Ā|^{2} + R̄ic( ¯∇t, ¯∇t) = 0,

where H̄′ = dH̄/dt.

The following formulas relating the curvatures of (M, ḡ) at ∂M to the intrinsic curvatures of (∂M, γ) may be found in [6, §1].

Let ḡ be a C^{2} geodesic compactification of (M, g), with C^{2} boundary metric γ. Then at ∂M,

(A.8)  s̄ = 2nR̄ic(N, N) = \frac{n}{n − 1}s_{γ},

where N = ¯∇t is the unit normal to ∂M with respect to ḡ. If X is tangent to ∂M, then

(A.9)  R̄ic(N, X) = 0,

while if T denotes the projection onto T (∂M), then

(A.10)  (R̄ic)^{T} = \frac{1}{n − 2}((n − 1)Ric_{γ} − \frac{s_{γ}}{2(n − 1)}γ).

In particular, the full curvature of ambient metric ḡ at ∂M is determined by the curvature of the boundary metric γ.

Finally, we have the following formula for s̄′ = ds̄/dt:

(A.11)  s̄′ = 2nt^{−1}|D̄^{2}t|^{2} ≥ \frac{1}{2n^{2}} ts̄^{2} ≥ 0.

Hence, if s̄(0) > 0, then

(A.12)  t^{2} < 4n^{2}/s̄(0).

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March, 2008

Department of Mathematics S.U.N.Y. at Stony Brook Stony Brook, N.Y.11794-3651 anderson@math.sunysb.edu