THE STATIC EXTENSION PROBLEM IN GENERAL RELATIVITY

MICHAEL T. ANDERSON AND MARCUS A. KHURI

Abstract. We develop a framework for understanding the existence of asymptotically flat solutions to the static vacuum Einstein equations on M = ℝ^{3} \ B with geometric boundary conditions on ∂M ≃ S^{2}. A partial existence result is obtained, giving a partial resolution of a conjecture of Bartnik on such static vacuum extensions. The existence and uniqueness of such extensions is closely related to Bartnik’s definition of quasi-local mass.

1. Introduction

This paper is concerned with a conjecture of R. Bartnik [B3], [B4] on the existence and uniqueness of static solutions to the vacuum Einstein equations with certain prescribed boundary data. On the physical side, this is closely related to the issue of local mass in general relativity while, on the mathematical side, to the issue of global existence and uniqueness for a rather complicated geometric non-linear system of elliptic boundary value problems.

Let M be a 3-manifold diffeomorphic to ℝ^{3} \ B where B is a 3-ball, so that ∂M ≃ S^{2}. The static vacuum Einstein equations are the equations for a pair (g, u) consisting of a smooth Riemannian metric g on M and a positive potential function u : M → ℝ^{+} given by

(1.1)  uRic_{g} = D^{2}u, Δu = 0,

where the Hessian D^{2} and Laplacian Δ = trD^{2} are taken with respect to g. The equations (1.1) are equivalent to the statement that the 4-dimensional metric

(1.2)  g_{N} = ±u^{2}dt^{2} + g,

on the 4-manifold N = ℝ × M is Ricci-flat, i.e.

(1.3)  Ric_{g_{N}} = 0.

This holds for either choice of sign in (1.2) and since most of the analysis of the paper concerns the Riemannian data (g, u) in (1.1), we will assume g_{N} is Riemannian, and moreover identify t in (1.2) periodically, to obtain a metric on N = S^{1} × M with t replaced by the angular variable θ.

Given (M, g, u) as above, let γ be the Riemannian metric induced on S^{2} = ∂M and let H be the mean curvature of ∂M ⊂ (M, g), (with respect to the inward unit normal into M). Then the Bartnik conjecture [B4] states that, given an arbitrary such pair in C^{∞},

(1.4)  (γ, H) ∈ Met^{∞}(S^{2}) × C_{+}^{∞}(S^{2}), H > 0,

there exists a unique asymptotically flat solution (g, u) to the static vacuum Einstein equations (1.1) inducing the boundary data (γ, H) on ∂M.

This conjecture is a natural outgrowth of Bartnik’s concept of quasi-local mass m_{B}(Ω), [B2], [B3], defined as follows. Let (Ω, g) be a smooth compact 3-manifold with smooth boundary of nonnegative scalar curvature, and define an admissible extension of (Ω, g) to be a complete, asymptotically flat 3-manifold (M̃, g) of non-negative scalar curvature in which (Ω, g) embeds isometrically and is not enclosed by any compact minimal surfaces (horizons). Then

The first author is partially supported by NSF Grant DMS 0905159 while the second author is partially supported by NSF Grant DMS 1007156 and a Sloan Research Fellowship.

2010 Mathematics Subject Classification. Primary 58J23, 83C05; Secondary 35J57, 53C25.

(1.5)  m_{B}(Ω) = inf{m_{ADM}(M̃) : (M̃, g) is an admissible extension of (Ω, g)},

where m_{ADM}(M̃) is the ADM mass of (M̃, g), cf. [B1]. Conjecturally, an extension (M̃, g) realizing the infimum in (1.5) is a solution to the static vacuum Einstein equations (1.1) on M = M̃ \ Ω which is Lipschitz, (but not smooth), across the junction ∂Ω and for which the induced metric and mean curvature at the boundary of the interior and exterior regions agree:

g|_{∂M} = g|_{∂Ω}, H_{∂M} = H_{∂Ω},

leading to the boundary data (1.4). Observe that the boundary data (γ, H) have the character of a mixed Dirichlet-Neumann type boundary value problem for the static equations (1.1), but the potential function u is absent from the boundary data. We point out that more standard Dirichlet or Neumann boundary data are not suitable for the (static) Einstein equations, cf. [A3].

In this paper, we develop a general framework for the Bartnik conjecture and make partial progress on its resolution. To describe the setting, let E_{S} = E_{S}^{m,α} be the moduli space of AF static vacuum solutions (M, g, u) on a given 3-manifold M which are C^{m,α} up to ∂M, m ≥ 3. The exact definition is given in §2, but basically E_{S} is the space of all AF static vacuum metrics on M modulo the action of the group Diff_{1} of diffeomorphisms on M equal to the identity on ∂M. Next, let Met^{m,α}(∂M) be the space of C^{m,α} metrics on ∂M ≃ S^{2} and C^{m−1,α}(∂M) be the space of C^{m−1,α} functions on ∂M. One thus has a natural map, mapping a static vacuum solution to its Bartnik boundary data:

(1.6)  Π_{B} : E_{S}^{m,α} → Met^{m,α}(∂M) × C^{m−1,α}(∂M),
Π_{B}(g) = (γ, H).

Theorem 1.1. The space E_{S}^{m,α} is a smooth (infinite dimensional) Banach manifold, and the map Π_{B} is C^{∞} smooth and Fredholm, of Fredholm index 0.

Theorem 1.1 essentially amounts to the statement that the static vacuum Einstein equations (1.1) with boundary conditions (1.4) form an elliptic boundary value problem, modulo gauge transformations, i.e. diffeomorphisms, and that one has a well-behaved local existence theory for this problem. We note that this boundary value problem also has a variational characterization, cf. Proposition 3.7.

Let E^{+} be the open submanifold of E_{S} for which the mean curvature H is positive, i.e.

E^{+} = (Π_{B})^{−1}(Met^{m,α}(∂M) × C_{+}^{m−1,α}(∂M)).

The Bartnik conjecture above may thus be rephrased to state that the smooth map Π_{B}, restricted to the open submanifold E^{+},

(1.7)  Π_{B} : E^{+} → Met^{m,α}(∂M) × C_{+}^{m−1,α}(∂M),

is surjective and injective, i.e. essentially a smooth diffeomorphism.

However, it is easy to see that Π_{B} in (1.7) cannot be a diffeomorphism. For example, for the flat solution g_{flat} with u = 1, there are boundaries given by embedded spheres (S^{2}, γ_{i}, H_{i}) with H_{i} uniformly positive, which converge smoothly in ℝ^{3} to a limit which is an immersed but not embedded sphere in ℝ^{3}. Such a limit is then at the boundary ∂E^{+}, but the limit boundary data (γ, H) ∈ Met^{m,α}(∂M) × C_{+}^{m−1,α}(∂M). In other words, the condition that H is bounded away from 0 and the metric is uniformly bounded is not sufficient to ensure that one stays within the class of manifolds-with-boundary. As a concrete example, let T^{2} be a torus of revolution embedded in ℝ^{3} with H > 0. One may remove a (small) essential annulus from T^{2} and smoothly attach two embedded discs to obtain a 2-sphere S^{2} with H > 0. This surface may be deformed to obtain a curve (S^{2})_{t}, t ∈ [0, 1], of positive mean curvature spheres which for t < \frac{1}{2} are embedded and for t ≥ \frac{1}{2} are immersed, with a single self-intersection point of the discs at t = \frac{1}{2}. (The same situation holds with any background static vacuum solution and varying boundary ∂M within M). This passage from embedded to immersed behavior also shows that the boundary map Π_{B} on E^{+} is not proper.

The basic issue is in fact that of finding domains within E^{+} on which Π_{B} is proper. Recall that Π_{B} is proper on a domain U ⊂ E^{+} if whenever (γ_{i}, H_{i}) is a sequence of boundary data converging to limit data (γ, H) and (M, g_{i}, u_{i}) are any solutions with Π_{B}(g_{i}, u_{i}) = (γ_{i}, H_{i}), then (M, g_{i}, u_{i}) converges, in a subsequence, to a limit solution (M, g, u) ∈ U. Here convergence is in the topology of the target and domain spaces respectively. In other words, control of the boundary data (γ, H) = Π_{B}(g, u) implies global control of the solution (M, g, u) within U. Equivalently, one needs apriori estimates controlling the behavior of the full solution (M, g, u) in terms of the boundary data (γ, H).

Now let E^{o} ⊂ E^{+} be the domain for which the boundary ∂M is strictly outer-minimizing, i.e. for which

(1.8)  area(Σ) > area(∂M),

for any surface Σ ⊂ M homologous to ∂M with Σ ≠ ∂M. Clearly E^{o} is an open submanifold of E^{+}. It is not known (although likely to be true) that E^{o} is connected. Throughout the following, we thus assume that E^{o} is taken to be the connected component containing the standard flat exterior solution where M is the exterior of the standard unit ball in ℝ^{3}, with ∂M the round S^{2} of radius 1. One then has a natural boundary map

(1.9)  Π^{o} : E^{o} → Met^{m,α}(∂M) × C_{+}^{m−1,α}(∂M).

The second main result of this paper is the following:

Theorem 1.2. The boundary map Π^{o} in (1.9) is ”almost” proper, in the following sense. If (γ_{i}, H_{i}) is a sequence of boundary data converging to limit data (γ, H) in Met^{m,α}(∂M)×C_{+}^{m−1,α}(∂M), and (M, g_{i}, u_{i}) ∈ E^{o} are any solutions with Π^{o}(g_{i}, u_{i}) = (γ_{i}, H_{i}), then (M, g_{i}, u_{i}) converges in E_{S}^{m,α}, in a subsequence, to a limit solution (M, g, u) ∈ E_{S}^{m,α} which is (at least) weakly outer-minimizing, i.e.

(1.10)  area(Σ) ≥ area(∂M),

for Σ as in (1.8).

We point out that the full global property (1.8) is not actually necessary; Theorem 1.2 remains valid if (1.8) holds only in an arbitrarily small neighborhood of ∂M, (depending on (M, g, u)), cf. Remark 4.4.

A smooth proper Fredholm map F : B_{1} → B_{2} between connected Banach manifolds has a ℤ_{2}- valued degree deg_{ℤ_{2}}F, the Smale degree, cf. [Sm]. If the spaces or map have a suitable orientation, this can be extended to a ℤ-valued degree deg_{ℤ}F, cf. [ET] for instance. Essentially an immediate consequence of its definition is that if such a degree is non-zero, say

deg_{ℤ_{2}}F ≠ 0,

then F is surjective.

The definition of degree above may be extended to maps which are almost proper in the sense above, cf. [BFP] for instance. Thus, let ∂E^{o} be the boundary of E^{o} within the space E_{S} of static vacuum solutions. This is the space of solutions in E_{S} satisfying (1.10) but not (1.8). Let Z = Π_{B}(∂E^{o}) ⊂ Met^{m,α}(∂M) × C^{m−1,α}(∂M) be the image of ∂E^{o} under the boundary map Π_{B} and let E^{P} = (Π^{o})^{−1}([Met^{m,α}(∂M) × C^{m−1,α}(∂M)] \ Z) be the corresponding inverse image. Then, as discussed in §5, the induced boundary map (restriction of Π^{o} to E^{P})

Π^{P} : E^{P} → [Met^{m,α}(∂M) × C^{m−1,α}(∂M)] \ Z,

is proper. In particular, E^{P} has a finite number of connected components E^{P_{i}} and the induced boundary map Π^{P_{i}} on E^{P_{i}} has a well-defined ℤ_{2}-valued degree (with respect to a component of the target space). In fact, it is shown in Proposition 5.1, that Π^{P_{i}} has a well-defined ℤ-valued degree, computable effectively in terms of Morse indices.

Let E^{P_{0}} be the component of E^{P} containing the standard round exterior flat solution as following (1.8), and let T_{0} be the component of [Met^{m,α}(∂M)×C_{+}^{m−1,α}(∂M)]\Z containing the corresponding standard boundary data (γ_{+1}, 2). One thus has the boundary map

(1.11)  Π^{P_{0}} : E^{P_{0}} → T_{0}.

A further main result of this paper is the computation of this degree:

Theorem 1.3. For the map Π^{P_{0}} in (1.11), one has

(1.12)  deg_{ℤ}Π^{P_{0}} = 1.

The proof of Theorem 1.3 is based on the well-known black hole uniqueness theorem for the Schwarzschild metric, cf. [I], [R], [BM].

It follows that the boundary map Π^{P_{0}} maps E^{P_{0}} surjectively onto the component T_{0} of the target [Met^{m,α}(∂M) × C_{+}^{m−1,α}(∂M)] \ Z. In particular, the image of Π^{P_{0}} and so the image of Π_{B} has non-empty interior in the target space Met^{m,α}(∂M) × C_{+}^{m−1,α}(∂M); this has not been previously known, cf. Remark 5.4 for further discussion.

Of course, the main issue at this point is what can be said about the structure of the set Z? It is of first category (so non-generic) but its more detailed structure awaits future study. An alternate approach bypassing the issue of the boundary ∂E^{o} would be to find conditions on the boundary data (γ, H) which imply ∂M is outer-minimizing as in (1.8) in any static vacuum extension (M, g, u) of (γ, H). For example, in ℝ^{3} (where u = 1) convexity suffices, which is expressed in terms of boundary data as K_{γ} > 0 where K is the Gauss curvature. It is an open problem whether this condition also suffices for general static vacuum solutions.

The contents of the paper are briefly as follows. In §2, we present background information on the structure of static vacuum solutions and choices of gauge. Section 3 discusses elliptic boundary value problems for the Einstein equations and proves the basic structure theorems on the moduli space E_{S} and the boundary map Π_{B}, including Theorem 1.1. In §4, we then prove the requisite apriori estimates and establish the almost properness of Π^{o} on E^{o}, proving Theorem 1.2. Finally, §5 contains the computation of the degree of Π^{P_{0}} and closes with several related remarks.

We thank Robert Bartnik, Piotr Chruściel, Gerhard Huisken and Xin Zhou for their interest and comments on this work. We are especially grateful to Simon Brendle for pointing out an error in a previous version of the paper.

2. Background Discussion

Let M be a 3-manifold with compact boundary ∂M, and with a single open end E. (All of the results of this section and of §3 hold in all dimensions, but for simplicity, we restrict to dimension 3). Apriori, ∂M need not be connected, although this will be assumed later on. As following (1.2)-(1.3), we let N = S^{1} × M.

Let Met_{S}(N) = Met_{S}^{m,α}(N) be the space of complete C^{m,α} static metrics on N, i.e. metrics of the form (1.1), m ≥ 2. One has

(2.1)  Met_{S}^{m,α}(N) ≃ Met^{m,α}(M) × C_{+}^{m,α}(M),

where C_{+}^{m,α}(M) is the space of positive C^{m,α} functions on M. The space 𝔼_{S} = 𝔼_{S}(N) of static Einstein (Ricci-flat) metrics on N is equivalent to the space of pairs g_{N} = (g_{M}, u) ∈ Met(M)×C_{+}(M) satisfying (1.1) or (1.3).(The smoothness indices will be occasionally dropped when unimportant).

Recall that a complete metric g ∈ Met^{m,α}(E) on an end E is asymptotically flat if E is diffeomorphic to ℝ^{3} \ B, where B is a 3-ball, and there exists a diffeomorphism F : ℝ^{3} \ B → E such that, in the chart F,

(2.2)  g_{ij} = δ_{ij} + O(r^{−1}), ∂_{k}g_{ij} = O(r^{−2}), ∂_{k}∂_{ℓ}g_{ij} = O(r^{−3}),

in the standard Euclidean coordinates on ℝ^{3}. The static vacuum equations (1.1) are invariant under multiplication of the potential u by constants. Throughout the paper, we assume that u is normalized so that u → 1 at infinity, and that u is asymptotically constant in the sense that

(2.3)  u = 1 + O(r^{−1}), ∂_{k}u = O(r^{−2}), ∂_{k}∂_{ℓ}u = O(r^{−3}).

Thus the 4-metric g_{N} is asymptotic to the product S^{1} × ℝ^{3} at infinity.

It is proved in [A2] that ends of static vacuum solutions (M, g, u) are either asymptotically flat or parabolic, where parabolic is understood in the sense of potential theory; equivalently, parabolic ends have small volume growth in that the area of geodesic spheres grows slower than r^{1+ε}, for any ε > 0. Moreover, asymptotically flat ends are strongly asymptotically flat in that the metric and potential have asymptotic expansions of the form

(2.4)  g_{ij} = (1 + \frac{2m}{r})δ_{ij} + ⋯, u = 1 − \frac{2m}{r} + ⋯,

where the mass m may apriori be any value m ∈ ℝ. These two behaviors, asymptotically flat and parabolic, are radically different and there is no curve of asymptotic structures for static vacuum solutions which joins them. The finer behavior of asymptotically flat ends is described by the mass parameter m and higher multipole moments, cf. [BS].

Let g̃ be a fixed asymptotically flat (background) metric in 𝔼_{S}; henceforth 𝔼_{S} will denote the space of asymptotically flat static vacuum Einstein solutions. The static Einstein equations are not elliptic, due to their invariance under diffeomorphisms, and for several reasons one needs to choose an elliptic gauge. To begin, we consider the Bianchi gauge, and define

(2.5)  Φg̃ : Met_{S}^{m,α}(N) → S^{m−2,α}(N),
Φg̃(g) = Ric_{g} + δ_{g}^{∗}βg̃(g),

where βg̃ is the Bianchi operator, βg̃(g) = δg̃(g) + \frac{1}{2}dtrg̃(g). Also, (δ^{∗}X)(A, B) = \frac{1}{2}(⟨∇_{A}X, B⟩ + ⟨∇_{B}X, A⟩) and δX = −tr(δ^{∗}X) is the divergence. The operator Φg̃ is a C^{∞} smooth map into the space S^{m−2,α}(N) of static symmetric bilinear forms on N.

Using standard formulas for the linearization of the Ricci and scalar curvatures, cf. [Be] for instance, the linearization of Φ at g = g̃ ∈ 𝔼_{S} is given by

(2.6)  L(h) = 2(DΦ_{g})(h) = D^{∗}Dh − 2R(h).

Clearly, L is elliptic and formally self-adjoint. In §3 we will discuss boundary value problems for Φ and L.

Next, the asymptotic behavior in the asymptotically flat end E requires the introduction of suitable weighted function spaces. We will use the standard weighted Hölder spaces, although one could equally well use weighted Sobolev spaces. Thus, define Met_{δ}^{m,α}(N) ⊂ Met_{S}^{m,α}(N) to be the subspace of metrics which decay to Euclidean data at a rate r^{−δ} at infinity; more precisely, the component functions g_{ij} and u of g_{N} should satisfy

g_{ij} − δ_{ij} ∈ C_{δ}^{m,α}(ℝ^{3} \ B), u − 1 ∈ C_{δ}^{m,α}(ℝ^{3} \ B).

Here C_{δ}^{m} consists of functions v such that

||v||_{C_{δ}^{m}} = ∑_{k=0}^{m} sup r^{k+δ}|∇^{k}v| < ∞,

while C_{δ}^{m,α} consists of functions such that

||v||_{C_{δ}^{m,α}} = ||v||_{C_{δ}^{m}} + sup_{x,y}[min(r(x), r(y))^{m+α+δ} \frac{|∇^{m}v(x) − ∇^{m}v(y)|}{|x − y|^{α}}] < ∞,

cf. [B1], [LP]. Throughout the following, we assume the decay rate δ is fixed, and chosen to satisfy

(2.7)  \frac{1}{2} < δ < 1.

It is well-known, cf. [B1], [LP], that the Laplacian on functions, and Laplace-type operators on tensors, as in (2.6), are Fredholm when acting on these weighted Hölder spaces.

The map Φ in (2.5) clearly induces a smooth map

(2.8)  Φ : Met_{δ}^{m,α}(N) → S_{δ}^{m−2,α}(N).

Observe that g is Einstein if Φg̃(g) = 0 and βg̃(g) = 0, so that g is in Bianchi gauge with respect to g̃. (Although Φg̃ is defined for all g ∈ Met_{δ}^{m,α}(N), we will only consider it acting on g near g̃).

Given g̃ ∈ 𝔼_{S}, let Met_{C}^{m,α}(N) ⊂ Met_{δ}^{m,α}(N) be the space of C^{m,α} smooth static AF Riemannian metrics on N, satisfying the Bianchi gauge constraint

(2.9)  βg̃(g) = 0 at ∂N.

As above,

Φ : Met_{C}^{m,α}(N) → S_{δ}^{m−2,α}(N).

Similarly let Z_{C}^{m,α} be the space of metrics g ∈ Met_{C}^{m,α}(N) satisfying Φg̃(g) = 0, and let

(2.10)  𝔼_{C} ⊂ Z_{C}

be the subset of static Einstein metrics g = g_{N}, Ric_{g} = 0 in Z_{C}. The following result justifies the use of the operator Φ to study 𝔼_{C}.

Proposition 2.1. Any static metric g = g_{N} ∈ Z_{C} sufficiently close to g̃ is necessarily Einstein, g ∈ 𝔼_{C}. Moreover, this also holds infinitesimally in the following sense. Let κ be an infinitesimal deformation of g ∈ Z_{C}, i.e. κ ∈ KerDΦ. If βg̃(g) = 0, (for example g̃ = g), then

(2.11)  βg̃(κ) = 0,

and κ is an infinitesimal Einstein deformation, i.e. the variation of g in the direction κ preserves (1.3) to 1^{st} order.

Proof: Since g ∈ Z_{C}, one has Φ(g) = 0, i.e.

Ric_{g} + δ_{g}^{∗}βg̃(g) = 0.

Applying the Bianchi operator β_{g} and using the Bianchi identity β_{g}(Ric_{g}) = 0 gives

(2.12)  β_{g}(δ_{g}^{∗}(βg̃(g))) = 0.

Setting V = βg̃(g), a standard Weitzenbock formula shows that 2β_{g}δ_{g}^{∗}(V) = D^{∗}DV − Ric(V). Also, since g, g̃ ∈ Met_{δ}^{m,α}(N), V ∈ χ_{1+δ}^{m−1,α}(M), where χ_{1+δ}^{m−1,α}(M) is the space of vector fields whose components are in C_{1+δ}^{m−1,α}(M). When acting on vector fields V with V = 0 on ∂M, as in (2.9), the operator D^{∗}D is positive, with trivial kernel. Namely, if W ∈ C_{1+δ}^{m−1,α} is in the kernel of D^{∗}D, then integrating by parts gives

∫_{B(r)} |DW |^{2} + ∫_{S(r)}⟨W, ∇_{N}W ⟩ = 0,

where B(r) = {x ∈ M : dist(x, ∂M) ≤ r} and N is the outward unit normal. (Since W = 0 on ∂M, there is no boundary term at ∂M). Letting r → ∞, the boundary integral tends to 0 and so DW = 0, which in turn implies W = 0.

Since D^{∗}D is self-adjoint and Fredholm, it has a smallest positive eigenvalue bounded away from 0. For g sufficiently close to g̃, |Ric(V)| = O(ε) pointwise and Ric(V) ∈ C_{3+δ}^{m−3,α}(M), so that we may assume that 2β_{g}δ_{g}^{∗}(V) is a positive operator on V. Hence, again since V = 0 on ∂M, the only solution of (2.12) is V = 0, which implies g ∈ 𝔼_{C}.

To prove the second statement, let g_{t} = g + tκ. Applying the Bianchi operator β_{g_{t}} to Φ(g_{t}) gives

(2.13)  β_{g_{t}}Φ(g_{t}) = β_{g_{t}}δ_{g_{t}}^{∗}(βg̃(g_{t})).

Taking the derivative with respect to t at t = 0, one has (β_{g_{t}}Φ(g_{t}))′ = β′Φ + βΦ′. Both terms here vanish since g ∈ Z_{C} and κ is formally tangent to Z_{C}. Hence the variation of the right hand side of (2.13) vanishes. Since βg̃(g) = 0, this gives β_{g}δ_{g}^{∗}(βg̃(κ)) = 0. The equation (2.11) then follows exactly as following (2.12), with V = βg̃(κ).

Let D_{1}^{m+1,α} denote the space of C_{δ}^{m+1,α} static diffeomorphisms of N which equal the identity on ∂N. Thus such diffeomorphisms decay to the identity at the rate r^{−δ} and are independent of the t or θ-variable in (1.2). The group D_{1} acts freely and continuously on Met(N) and 𝔼_{S} by pullback and one has the following local slice theorem for this action; we refer to [A3] for the proof.

Lemma 2.2. Given any g̃ ∈ 𝔼_{S}^{m,α} and g ∈ Met_{δ}^{m,α}(N) near g̃, there exists a unique diffeomorphism ϕ ∈ D_{1}^{m+1,α} close to the identity, such that

(2.14)  βg̃(ϕ^{∗}g) = 0.

In particular, ϕ^{∗}g ∈ Met_{C}^{m,α}(N).

Lemma 2.2 implies that if g ∈ 𝔼_{S}^{m,α} is a static Einstein metric near g̃, then g is isometric, by a diffeomorphism in D_{1}^{m+1,α}, to an Einstein metric in 𝔼_{C}^{m,α}, so that 𝔼_{C}^{m,α} is a slice for 𝔼_{S}^{m,α} under the action of D_{1}^{m+1,α}.

To prove that the moduli space E is a smooth Banach manifold, (cf. Theorem 3.6), it is important to have a gauge with choice of boundary data in which the Einstein equations form a self-adjoint elliptic boundary value problem. This is not the case for the operator Φ and we are not aware of geometrically natural self-adjoint boundary conditions for Φ. For this reason, we will also consider another natural gauge, namely the divergence-free gauge.

To do this, instead of Φ, consider

(2.15)  Φ̂(g) = Φ̂g̃(g) = Ric_{g} − \frac{s}{2}g + δ_{g}^{∗}δg̃g,

where s is the scalar curvature of g = g_{N}. The linearization of Φ̂ at g = g̃ ∈ 𝔼_{S} is given by

(2.16)  L̂(h) = 2(DΦ̂g̃)_{g}(h) = D^{∗}Dh − 2R(h) − (D^{2}trh + δδh g) + Δtrh g.

In analogy to (2.9), define

(2.17)  Met_{D}^{m,α}(M) = {g ∈ Met_{δ}^{m,α}(N) : δg̃g = 0 at ∂N}.

Similarly, let Z_{D}^{m,α} = Φ̂^{−1}(0) ∩ Met_{D}^{m,α}(M) and 𝔼_{D} ⊂ Z_{D} be the space of Einstein metrics in divergence-free gauge with respect to g̃ ∈ 𝔼_{S}.

It is easy to see that Proposition 2.1 and Lemma 2.2 hold in this divergence-free gauge in place of the previous Bianchi gauge, with essentially the same proof. Thus 𝔼_{D} = Z_{D} and for g ∈ 𝔼_{D},

(2.18)  δg̃g = 0 on N.

Moreover, the diffeomorphism group D_{1} transforms one gauge choice uniquely to the other. For instance, suppose βg̃(g) = 0. Then we claim there is a unique ϕ ∈ D_{1}^{m+1,α} such that

(2.19)  δg̃(ϕ^{∗}g) = 0.

At the linearized level, with g = g̃, this amounts to finding a vector field V with V = 0 on ∂N such that if βg̃h = 0 then δg̃(h + δ^{∗}V) = 0. This equation is equivalent to the equation δδ^{∗}V = \frac{1}{2}dtrh, which is uniquely solvable for V with V = 0 on ∂N. The local result in (2.19) then follows from the inverse function theorem.

3. The Moduli Space

In this section, we study boundary value problems for the elliptic operators Φ and Φ̂, and use this to prove that the moduli space E_{S} of static vacuum solutions is a smooth Banach manifold for which the boundary map Π_{B} is Fredholm, of Fredholm index 0, cf. Theorem 3.6.

We begin with the Bianchi-gauged Einstein operator Φ in (2.5), i.e.

Φg̃(g) = Ric_{g} + δ_{g}^{∗}βg̃(g).

Let A denote the 2^{nd} fundamental form of ∂M in M, A(X, Y) = ⟨∇_{X}N, Y ⟩, where N is the unit inward normal into M, X, Y tangent to ∂M. Similarly, let H_{M} = trA denote the mean curvature of ∂M in M. Throughout the paper W^{T} will denote the restriction or the orthogonal projection of a tensor W to T (∂N) or T (∂M).

Proposition 3.1. Near any given background solution g̃ ∈ 𝔼_{S}^{m,α}, the operator Φ = Φg̃ in (2.5) with boundary conditions:

(3.1)  βg̃(g) = 0, g|_{T (∂M)} = γ_{M}, H_{M} = h at ∂N,

is an elliptic boundary value problem of Fredholm index 0.

Here the induced metric γ_{M} is in Met^{m,α}(∂M) while the mean curvature H_{M} of ∂M in (M, g_{M}) is in C^{m−1,α}(∂M). Note that the potential u does not enter this boundary data and so is formally undetermined at ∂M. Also the static property implies that βg̃(g) vanishes in the vertical direction, βg̃(g)(∂_{θ}) = 0.

Proof: It suffices to prove that the leading order part of the linearized operators forms an elliptic system. Recall from (2.6) that the linearization of Φ at g̃ = g is given by

L(h) = 2(DΦ_{g})(h) = D^{∗}Dh − 2R(h).

The leading order symbol of L = 2DΦ at ξ′ is

(3.2)  σ(L) = −|ξ′|^{2}I,

where I is the Q × Q identity matrix, with Q = (n(n + 1)/2) + 1; Q is the sum of the dimension of the space of symmetric bilinear forms on ℝ^{n}, together with the extra vertical S^{1} direction. Here n = 3 but we give the proof for general dimensions. For static metrics, all components of the metric are locally functions on ℝ^{n}, and all derivatives in the vertical S^{1} direction are trivial. In the following, the subscript 0 represents the direction normal to ∂M in M, (or ∂N in N), subscript 1 denotes the vertical direction, tangent to S^{1}, while indices 2 through n represent the directions tangent to ∂M. Note that one has h_{1α} = 0, for all α ≠ 1. The positive roots of (3.2) are i|ξ|, where ξ′ = (ξ_{0}, ξ), with multiplicity Q at ξ ∈ T^{∗}(ℝ^{n}).

Writing ξ′ = (z, ξ_{i}), i = 2, …, n, (as above ξ_{1} = 0), the symbols of the leading order terms in the boundary operators are given by:

−2izh_{0k} − 2i ∑_{j≥2} ξ_{j}h_{jk} + iξ_{k}trh = 0, k ≥ 2,

−2izh_{00} − 2i ∑_{k≥2} ξ_{k}h_{0k} + iztrh = 0,

h^{T} = (γ′)^{T} , and (H_{M})′_{h} = ω.

This gives n+ \frac{n(n−1)}{2} +1 = Q boundary equations, as required. Ellipticity requires that the operator defined by the boundary symbols above has trivial kernel when z is set to the root i|ξ|. Carrying this out then gives the system

(3.3)  2|ξ|h_{0k} − 2i ∑_{j≥2} ξ_{j}h_{jk} + iξ_{k}trh = 0, k ≥ 2,

(3.4)  2|ξ|h_{00} − 2i ∑_{k≥2} ξ_{k}h_{0k} − |ξ|trh = 0,

(3.5)  h_{11} = ϕ, h^{T} = 0, (H_{M})′_{h} = 0,

where ϕ is an undetermined function.

Multiplying (3.3) by iξ_{k} and summing gives, via (3.5),

2|ξ|i ∑_{k≥2} ξ_{k}h_{0k} = |ξ|^{2}trh.

Substituting (3.4) on the term on the left above then gives

2|ξ|^{2}h_{00} − 2|ξ|^{2}trh = 0.

Since trh = h_{00} + ϕ, it follows that ϕ = 0.

Next, to compute H′_{M}, we first observe that in general

(3.6)  2A′_{h} = ∇_{N}h + 2A ∘ h − 2δ^{∗}(h(N)^{T}) − δ^{∗}(h_{00}N).

This follows by differentiating the defining formula 2A = L_{N}g, and using the identities 2N′_{h} = −2h(N)^{T} − h_{00}N, L_{N}h = ∇_{N}h + 2A ∘ h. Since H_{M} = tr_{M}A, H′_{h} = tr_{M}A′_{h} − tr_{M}A ∘ h and so

(3.7)  2(H_{M})′_{h} = tr_{M}(∇_{N}h − 2δ^{∗}(h(N)^{T}) − δ^{∗}(h_{00}N)).

Hence the symbol of 2(H_{M})′_{h} is given by ∑_{k≥2}(izh_{kk} − 2iξ_{k}h_{0k}). Setting this to 0 at the root z = i|ξ| gives

(3.8)  ∑_{k≥2}(|ξ|h_{kk} + 2iξ_{k}h_{0k}) = 0.

Via (3.5), this gives −2i ∑_{k≥2} ξ_{k}h_{0k} = 0, and substituting this in (3.4) and using the fact that ϕ = 0 gives

2|ξ|h_{00} − |ξ|h_{00} = 0,

so that h_{00} = 0. It follows from (3.3) that h_{0k} = 0 and hence h = 0. This proves ellipticity of the boundary value problem (3.1) and the Fredholm property follows from the fact that the Laplace-type operator L is Fredholm on Met_{δ}^{m,α}, cf. [LP].

Finally, it is straightforward to verify that the boundary data (3.1) may be continuously deformed through elliptic boundary data to elliptic boundary data for which L is self-adjoint and so of index 0. This is proved in [A3] in a slightly different setting and the proof carries over here with only minor change, and so we refer to [A3] for further details. The homotopy invariance of the index then completes the proof.

As noted in §2, we are not aware of a geometrically natural self-adjoint elliptic boundary value problem for Φ. In particular, the boundary conditions (3.1) are not self-adjoint. This property is important for the proof of Theorem 3.6, and for this reason, we turn to the operator Φ̂ in (2.15) with linearization at g̃ = g given by L̂ in (2.16).

Regarding boundary conditions for L̂, for h ∈ S_{δ}^{m−2,α}(N), let h^{T} = h|_{∂N} and [h^{T}]_{0} be the projection of h^{T} onto the space of forms trace-free with respect to γ = γ_{N}. Similarly, H′_{h} denotes here the linearization of the mean curvature H = H_{N} of ∂N ⊂ N.

We then have:

Lemma 3.2. The operator L̂ with boundary conditions

(3.9)  δh = 0, [h^{T}]_{0} = 0, H′_{h} = 0,

is a self-adjoint elliptic operator. Moreover, under the first two conditions δh = 0 and [h^{T}]_{0} = 0, the operator L̂ is self-adjoint exactly for the boundary condition H′_{h} = 0.

Proof: It is a rather long (and uninteresting) calculation to prove that the operator L̂ with boundary data (3.9) forms an elliptic system; this has been verified by computer computation using Maple. More conceptually, instead we will make use of Proposition 3.1 to simplify the proof. First, recall, [ADN], [T], that ellipticity of a boundary value problem is equivalent to the existence of a uniform estimate

(3.10)  ||h||_{C^{m,α}} ≤ C(||L̂(h)||_{C^{m−2,α}} + ||B_{j}(h)||_{C^{m−j,α}} + ||h||_{C^{0}}),

where B_{j} is the part of the boundary operator of order j, together with such an estimate for the adjoint operator. As seen below, the boundary value problem is self-adjoint, so it suffices to establish (3.10).

First, it is simple to prove (3.10) for L in place of L̂ via a slight modification of the proof of Proposition 3.1. Namely, for the boundary condition [h^{T}]_{0} = 0, we have h^{T} = ϕγ on ∂N, (in place of (3.5)). Note also that (3.3)-(3.4) hold, but without the trh terms. The analog of (3.3) then gives

|ξ|h_{0k} = iξ_{k}ϕ,

and hence, via the analog of (3.4),

|ξ|^{2}h_{00} = −|ξ|^{2}ϕ,

so that h_{00} + ϕ = 0. Next, via the condition H′_{h} = 0, the analog of (3.8) becomes

∑_{k≥1}(|ξ|h_{kk} + 2iξ_{k}h_{0k}) = 0,

which gives

n|ξ|ϕ = −2i ∑ ξ_{k}h_{0k} = 2|ξ|ϕ.

Since n ≥ 3, this implies ϕ = 0, and so h_{00} = 0, hence h_{0k} = 0. It follows that h = 0, which proves ellipticity of L with the boundary conditions (3.9). Thus, (3.10) holds with L in place of L̂.

Next, one has

(3.11)  L̂ = L − (D^{2}trh − Δtrh g) − δδh g.

Thus to prove (3.10), it suffices to prove

(3.12)  ||δh||_{C^{m−1,α}} ≤ C(||L̂(h)||_{C^{m−2,α}} + ||B_{j}(h)||_{C^{m−j,α}} + ||h||_{C^{0}}),
(3.13)  ||D^{2}trh||_{C^{m−2,α}} ≤ C(||L̂(h)||_{C^{m−2,α}} + ||B_{j}(h)||_{C^{m−j,α}} + ||h||_{C^{0}}).

From (2.15)-(2.16) and the Bianchi identity, (as in (2.13)), one has δL̂(h) = 2δδ^{∗}(δh) and the operator δδ^{∗} is elliptic with respect to Dirichlet boundary conditions. Since the boundary data δh in (3.9) is included in the boundary operators B_{j}, this proves (3.12).

Using this and taking the trace of (3.11) shows that

||D^{2}trh||_{C^{m−2,α}} ≤ C(||L̂(h)||_{C^{m−2,α}} + ||B_{j}(h)||_{C^{m−j,α}} + ||NN(trh)||_{C^{m−2,α}} + ||h||_{C^{0}}),

so that it suffices to prove that the boundary conditions B cover NN(trh). For this, a simple computation using (3.7), (cf. also (3.19) below), gives

(3.14)  N(trh) = 2H′_{h} − δ((h(N))^{T}) − (δh)(N) + O(h),

where O(h) is of order 0 in h. Using the standard interpolation ||h||_{C^{m−1,α}} ≤ ε||h||_{C^{m,α}} + ε^{−1}||h||_{C^{0}} shows that it suffices here and below only to consider terms with the leading number of derivatives of h.

Now the Gauss equations at ∂N are |A|^{2} − H^{2} + s_{γ_{N}} = s_{g_{N}} − 2Ric(N, N) and hence,

(|A|^{2} − H^{2} + s_{γ_{N}})′_{h} = −2L̂(h)(N, N) + 2δ^{∗}δ(h)(N, N) + O(h).

One has s′_{γ_{N}} (h^{T}) = −Δtrh^{T} + δδ(h^{T}) + O(h^{T}) and A′_{h}, H′_{h} only involve first order derivatives in h. Writing then h^{T} = B_{0}(h) + \frac{1}{n}tr_{∂N}hγ_{N}, it follows that tr_{∂N}h at ∂N is controlled by L̂(h), B_{j}(h), in that

||tr_{∂N}h||_{C^{m,α}} ≤ C(||h||_{C^{m−1,α}} + ||L̂(h)||_{C^{m−2,α}} + ||B_{j}(h)||_{C^{m−j,α}}),

and hence

||h^{T} ||_{C^{m,α}} ≤ C(||h||_{C^{m−1,α}} + ||L̂(h)||_{C^{m−2,α}} + ||B_{j}(h)||_{C^{m−j,α}}),

i.e. h^{T} is controlled at ∂N by L̂(h) and B_{j}(h). Next, at ∂N, one has −(δh)(T) = ∇_{N}h(N, T) + ∇_{e_{i}}h(e_{i}, T), which then gives control as above on (∇_{N}h)(N, T), and so control on ∇_{N}(h(N)^{T}). In turn, this gives then control on δ_{∂N}(∇_{N}(h(N)^{T})), which modulo lower order (curvature) terms, equals N(δ(h(N)^{T})). The N-derivative of (3.14) also holds and shows that control of N(δ(h(N)^{T})) implies control of NN(trh), so that (3.13) holds, provided N(H′_{h}) is controlled. But the Riccati equation gives N(H) = −|A|^{2} − Ric(N, N); taking the linearization of this in the direction h shows that N(H′_{h}) is indeed controlled by L̂(h) and the boundary conditions B_{j}. This completes the proof of ellipticity.

Next, we prove the operator L̂ with boundary conditions (3.9) is self-adjoint. To begin, integrating the terms in the expression (2.16) for L̂ by parts over N gives

∫_{N}⟨D^{∗}D(h), k⟩ + ∫_{∂N}⟨∇_{N}h, k⟩ = ∫_{N}⟨D^{∗}D(k), h⟩ + ∫_{∂N}⟨∇_{N}k, h⟩,

∫_{N} δδhtrk + ∫_{∂N}(δh)(N)trk = ∫_{N}⟨h, D^{2}(trk)⟩ − ∫_{∂N} h(N, dtrk),

and

∫_{N}(Δtrh)trk − ∫_{∂N} N(trh)trk = ∫_{N}(Δtrk)trh − ∫_{∂N} N(trk)trh.

Here the boundary terms on S(r) all tend to 0 as r → ∞, since the components of h and k are in C_{δ}^{m,α} and δ > \frac{1}{2}. It follows that

(3.15)  ∫_{N}⟨L̂(h), k⟩ + ∫_{∂N}⟨B(h), k⟩ = ∫_{N}⟨L̂(k), h⟩ + ∫_{∂N}⟨B(k), h⟩,

where

(3.16)  ⟨B(k), h⟩ = ⟨∇_{N}k, h⟩ + h(N, dtrk) − (δk)(N)trh − trhN(trk).

Setting Z(k, h) = ⟨B(k), h⟩ − ⟨B(h), k⟩, we thus need to show that

(3.17)  ∫_{∂N} Z(h, k) = 0,

when h, k satisfy the boundary conditions (3.9).

Thus suppose h and k both satisfy (3.9). A simple calculation shows that (δk)(T) = 0 is equivalent to

(3.18)  (∇_{N}k)(N)^{T} = δ_{∂N}(k^{T}) − α(k(N)),

where α(k(N)) = [A(k(N)) + Hk(N)^{T}], (all taken on ∂N), while (δk)(N) = 0 is equivalent to

(3.19)  N(k_{00}) = δ_{∂N}(k(N)^{T}) + ⟨A, k⟩ − k_{00}H.

The same equations hold for h, and one also has

(3.20)  h^{T} = ϕ_{h}γ, and k^{T} = ϕ_{k}γ.

We thus need to calculate

B(k, h) = ⟨∇_{N}k, h⟩ + h(N, dtrk) − trhN(trk),

and skew-symmetrize. To begin, write ⟨∇_{N}k, h⟩ = ⟨(∇_{N}k)(N), h(N)⟩ + ⟨(∇_{N}k)(e_{i}), h(e_{i})⟩, so that ⟨∇_{N}k, h⟩ = N(k_{00})h_{00} + ϕ_{h}[N(trk) − N(k_{00})] + 2⟨(∇_{N}k)(N), h(N)^{T} ⟩, where we have used the relation tr_{γ}∇_{N}k = tr_{N}∇_{N}k − N(k_{00}). Thus, B(k, h) equals

(3.21) N(k_{00})h_{00}+ϕ_{h}[N(trk)−N(k_{00})]+2⟨(∇_{N}k)(N), h(N)^{T} ⟩−N(trk)[trh−h_{00}]+⟨h(N)^{T} , dtrk⟩.

By (3.18) and (3.20),

2⟨(∇_{N}k)(N), h(N)^{T} ⟩ = −2⟨dϕ_{k}, h(N)^{T} ⟩ − 2α(k, h) = −2ϕ_{k}δ_{∂N}(h(N)^{T}) − 2α(k, h) + ω_{1}

where ω_{1} is a divergence term and α(k, h) = ⟨α(k(N)), h(N)^{T} ⟩. Similarly, by (3.19) and (3.20),

N(k_{00}) = δ_{∂N}(k(N)^{T}) + (ϕ_{k} − k_{00})H,

where here and in the following δ = δ_{∂N}. Note also that ⟨h(N)^{T} , dtrk⟩ = trkδ_{∂N}(h(N)^{T}) + ω_{2}, where ω_{2} is another divergence term. Since (3.17) involves integration over ∂N, in the following we ignore the divergence terms. Substituting these computations in (3.21) gives

δ(k(N)^{T})[h_{00} − ϕ_{h}] + δ(h(N)^{T})[trk − 2ϕ_{k}] − (n − 1)ϕ_{h}N(trk) + H(ϕ_{k} − k_{00})(h_{00} − ϕ_{h}) − 2α(h, k).

When skew-symmetrizing, the last two terms H(ϕ_{k} − k_{00})(h_{00} − ϕ_{h}) − 2α(h, k) cancel, while the first three terms combine to give

−(n − 1)[ϕ_{h}δ(k(N)^{T}) − ϕ_{k}δ(h(N)^{T})] − (n − 1)[N(trk)ϕ_{h} − N(trh)ϕ_{k}],

or equivalently, (after dividing by n − 1),

(3.22)  −ϕ_{h}[N(trk) + δ(k(N)^{T})] + ϕ_{k}[N(trh) + δ(h(N)^{T})].

On the other hand, by (3.6) or (3.7),

2(H′)_{k} = tr[∇_{N}k − 2δ^{∗}(k(N)^{T}) − δ^{∗}(k_{00}N)]
= N(trk) + 2δ(k(N)^{T}) − k_{00}H − N(k_{00}).

Substituting (3.19) gives

2(H′)_{k} = N(trk) + δ(k(N)^{T}) − Hϕ_{k},

so (3.22) becomes

−ϕ_{h}[2(H′)_{k} + Hϕ_{k}] + ϕ_{k}[2(H′)_{h} + Hϕ_{h}] = −2ϕ_{h}(H′)_{k} + 2ϕ_{k}(H′)_{h}.

This vanishes exactly when H′_{k} and H′_{h} vanish. This completes the proof.

The main step in the proof of the manifold theorem, (Theorem 3.6), is the following result.

Theorem 3.3. Suppose π_{1}(M, ∂M) = 0 and m ≥ 3. Then at any g̃ ∈ 𝔼_{S}, the map Φ̂ is a submersion, i.e. the derivative

(3.23)  (DΦ̂)g̃ : Tg̃Met_{D}^{m,α}(N) → T_{Φ̂(g̃)}S_{δ}^{m−2,α}(N)

is surjective and its kernel splits in Tg̃Met_{D}^{m,α}(N).

Proof: The operator L̂ = 2DΦ̂g̃ is elliptic in the interior, and the boundary data in Lemma 3.2 give a self-adjoint elliptic boundary value problem. Let S_{B}^{m,α}(N) be the space of C^{m,α} symmetric bilinear forms on N satisfying the boundary condition B(h) = 0 from Lemma 3.2, i.e.

B(h) = {δh, [h^{T}]_{0}, (H′)_{h}} = (0, 0, 0).

Clearly, S_{B}^{m,α}(N) ⊂ S_{D}^{m,α}(N), where S_{D}^{m,α}(N) = Tg̃(Met_{D}^{m,α}(N)). Throughout the following, we set g̃ = g. The operator L̂, mapping

S_{B}^{m,α}(N) → S_{δ}^{m−2,α}(N),
L̂(h) = f, B(h) = 0 at ∂N,

is then Fredholm, of Fredholm index 0. On S_{B}^{m,α}(N), the image Im(L̂) is a closed subspace of the range S^{m−2,α}(N), of finite codimension, and with codimension equal to dimension of the kernel K.

If K = 0, then L̂ maps S_{B}^{m,α}(N) onto S_{δ}^{m−2,α}(N), which proves the result. Thus suppose K ≠ 0. Then as in (3.15), by the self-adjointness, one has for any h ∈ S_{B}^{m,α}(N) and k ∈ K,

∫_{N}⟨L̂(h), k⟩ = ∫_{N}⟨h, L̂(k)⟩ = 0,

since the boundary terms vanish and L̂(k) = 0. Thus Im(L̂|_{S_{B}^{m,α}(N)}) = K^{⊥}, (where K^{⊥} is taken with respect to the L^{2} inner product). To prove surjectivity on S_{D}^{m,α}(N), it thus suffices to prove that for any k ∈ K, there exists h ∈ S_{D}^{m,α}(N) such that

(3.24)  ∫_{N}⟨L̂(h), k⟩ ≠ 0.

Suppose then (3.24) does not hold, so that

(3.25)  ∫_{N}⟨L̂(h), k⟩ = 0,

for all h ∈ S_{D}^{m,α}(N), i.e. for which δh = 0 on ∂N. Integrating by parts, it follows that

(3.26)  ∫_{N}⟨h, L̂(k)⟩ + ∫_{∂N} Z(h, k) = 0,

for Z(h, k) as following (3.16). As before, the boundary terms at infinity vanish, since δ > \frac{1}{2}. Choosing h ∈ S_{D}^{m,α}(N) arbitrary of compact support in N, it follows from (3.26) that

(3.27)  L̂(k) = 0,

i.e. k is formally tangent to Ẑ = Φ̂^{−1}(0). Of course this is already known, since k ∈ K. Moreover, one also has

(3.28)  δk = 0 on N.

To see this, let h = δ^{∗}V, with V any vector field vanishing on ∂N. Since g is Einstein and so (Ric − \frac{s}{2}g)′_{δ^{∗}V} = 0, it follows from (3.5) and (3.6) that L̂(h) = δ^{∗}Y, where Y = 2δδ^{∗}V. As in Lemma 2.2, the operator δδ^{∗} is surjective, (in fact an isomorphism), on vector fields vanishing at ∂N, so that Y may be arbitrarily prescribed. Moreover, h ∈ S_{D}^{m,α}(N) if and only if Y = 0 at ∂N. Then (3.25) gives

0 = ∫_{N}⟨L̂(δ^{∗}V), k⟩ = ∫_{N}⟨δ^{∗}Y, k⟩ = ∫_{N}⟨Y, δk⟩ + ∫_{∂N} k(Y, N) = ∫_{N}⟨Y, δk⟩,

since Y = 0 on ∂N. Here we have used again the fact that the boundary term at infinity vanishes, since |k| = O(r^{−δ}) and |Y | = O(r^{−1−δ}). Since Y is otherwise arbitrary, this gives (3.28).

Returning now to (3.26), (3.27) gives

(3.29)  ∫_{∂N} Z(h, k) = 0,

for all h with δh = 0 on ∂N. Next, we choose certain test forms h ∈ S_{D}(N) in (3.29). First, choose h such that h = 0 on ∂N. Then ∇_{N}h is freely specifiable, subject to the divergence constraint δh = 0; all computations here and below are at ∂N. Since h = 0, this constraint gives (∇_{N}h)(N) = 0, which is equivalent to the tangential and normal constraints:

(3.30)  (∇_{N}h)(N, T) = 0,
(3.31)  N(h_{00}) = 0,

for any T tangent to ∂N. Choosing h and ∇_{N}h satisfying h = 0 and (3.30)-(3.31) at ∂N, the terms (∇_{N}h)(T_{1}, T_{2}) are freely specifiable on ∂N, where T_{1}, T_{2} are any vectors tangent to ∂N. Substituting such h in (3.29) and using (3.28), it follows that

(3.32)  ∫_{∂N}⟨∇_{N}h, k⟩ + (k_{00} − trk)N(trh) = 0.

Now choose ∇_{N}h = fg^{T} , where g^{T} = g|_{T (∂N)}. This choice satisfies the constraints (3.30)-(3.31). The integrand in (3.32) then becomes ftr^{T} k − N(trh)tr^{T} k. Since N(trh) = ⟨∇_{N}h, g⟩ = nf, and since f is arbitrary, it follows that tr^{T} k = 0. In turn, since the tangential part of ∇_{N}h is arbitrary, (3.32) implies

(3.33)  k^{T} = 0, on ∂N.

Lemma 3.4. At ∂N, one has

(3.34)  (A′_{k})^{T} = 0,

i.e. (∇_{N}k)^{T} = 2[δ^{∗}(k(N)^{T})]^{T} + k_{00}A, since k^{T} = 0, cf. (3.6).

Proof: The proof is a straightforward, but rather long computation. To begin, as preceding (3.21) and using (3.33), one has ⟨∇_{N}h, k⟩ = 2⟨(∇_{N}h)(N), k(N)^{T} ⟩ + N(h_{00})k_{00}. By (3.18), (∇_{N}h)(N)^{T} = δ_{∂N}(h^{T}) − α(h(N)), so that

(3.35)  ∫_{∂N}⟨∇_{N}h, k⟩ = ∫_{∂N} 2⟨δ_{∂N}(h^{T}), k(N)^{T} ⟩ + N(h_{00})k_{00} − 2α(h, k)

= ∫_{∂N} 2⟨h^{T} , (δ_{∂N})^{∗}(k(N)^{T})⟩ + N(h_{00})k_{00} − 2α(h, k).

Further, for Z tangent to ∂N, one has (δ_{∂N})^{∗}(k(N)^{T})(Z, Z) = ⟨∇_{Z}^{T}k(N)^{T} , Z⟩ = ⟨∇_{Z}k(N)^{T} , Z⟩ = δ^{∗}(k(N)^{T})(Z, Z), where now δ^{∗} is taken with respect to the ambient metric g_{N}, (not the boundary metric γ_{N}). So this gives

(3.36)  ∫_{∂N}⟨∇_{N}h, k⟩ = ∫_{∂N}⟨h^{T} , 2δ^{∗}(k(N)^{T})⟩ + N(h_{00})k_{00} − 2α(h, k).

On the other hand, one computes ⟨∇_{N}k, h⟩ = ⟨(∇_{N}k)^{T} , h^{T} ⟩+⟨∇_{N}k(N), h(N)⟩ = ⟨(∇_{N}k)^{T} , h^{T} ⟩− ⟨α(k(N)), h(N)^{T} ⟩+N(k_{00})h_{00}, again by (3.18) and (3.33). Taking the difference of this with (3.36) and noting that α is symmetric, gives

(3.37)  ∫_{∂N}⟨h^{T} , (∇_{N}k)^{T} − 2δ^{∗}(k(N)^{T})⟩ + N(k_{00})h_{00} − N(h_{00})k_{00} = E,

where via (3.16)-(3.17), E is given by

E = ∫_{∂N}[k(N, dtrh) − h(N, dtrk)] − [N(trh)trk − trhN(trk)].

Computing this term-by-term gives: k_{00}N(trh) + ⟨k(N)^{T} , d^{T} trh⟩ − h_{00}N(trk) − ⟨h(N)^{T} , d^{T} trk⟩ − N(trh)trk + trhN(trk). Since trk = k_{00}, the first and second-to-last terms cancel. Integrating over ∂N and using the divergence theorem shows that

(3.38)  E = ∫_{∂N} trhδ_{∂N}(k(N)^{T}) − k_{00}δ^{T} (h(N)^{T}) − h_{00}N(trk) + trhN(trk).

Next we claim that

(3.39)  δ_{∂N}(h(N)^{T}) = N(h_{00}) + Hh_{00} − ⟨A, h⟩,

and similarly for k. This follows from the following computation: δ_{∂N}(h(N)^{T}) = δ_{∂N}(h(N)) − δ^{T} (h_{00}N) = δ_{∂N}(h(N))+Hh_{00}, while δ_{∂N}(h(N)) = δ(h(N))+N(h_{00}). Since δ(h(N)) = (δh)(N)− ⟨A, h⟩, this gives the claim. Substituting (3.39) into (3.38), and using ⟨A, k⟩ = 0 implies that

E = ∫_{∂N} trh(N(k_{00}) + Hk_{00}) − k_{00}(N(h_{00}) + Hh_{00} − ⟨A, h⟩) − h_{00}N(trk) + trhN(trk),

and rearranging terms gives

(3.40) E = ∫_{∂N}⟨A, h⟩k_{00} + N(trk)[trh − h_{00}] + trhN(k_{00}) − k_{00}N(h_{00}) + H(trhk_{00} − trkh_{00}).

Now substitute (3.40) into (3.37): the k_{00}N(h_{00}) term cancels to give

(3.41)  ∫_{∂N}⟨h^{T} , (∇_{N}k)^{T} − 2δ^{∗}(k(N))^{T} ⟩ − ⟨A, h⟩k_{00} =

− ∫_{∂N} N(k_{00})h_{00} − N(trk)[trh − h_{00}] − trhN(k_{00}) − H(trhk_{00} − trkh_{00}).

The integrand on the right combines to: −N(k_{00})(h_{00}−trh)−N(trk)[h_{00}−trh]−Htrk(h_{00}−trh) = −[N(k_{00}) + N(trk) + Htrk](h_{00} − trh). Since h_{00} − trh = −tr^{T} h = −⟨h^{T} , g^{T} ⟩, and since h^{T} may be chosen arbitrarily, (the constraint δh = 0 imposes no constraint on h^{T}), it follows that

(3.42)  (∇_{N}k)^{T} = 2[δ^{∗}(k(N)^{T})]^{T} + k_{00}A + [N(k_{00}) + N(trk) + Htrk]g^{T} .

To complete the proof of (3.34), we thus need to show that

(3.43)  N(k_{00}) + N(trk) + Htrk = 0.

To obtain (3.43), take the g^{T} -trace of (3.42). One has ⟨∇_{N}k, g^{T} ⟩ = N(trk) − N(k_{00}), while ⟨δ^{∗}(k(N)^{T}), g^{T} ⟩ = ⟨∇_{e_{i}}k(N)^{T} , e_{i}⟩ = ⟨∇_{e_{i}}k(N), e_{i}⟩−k_{00}H = ⟨(∇_{e_{i}}k)(N), e_{i}⟩−k(∇_{e_{i}}N, e_{i})−k_{00}H = −N(k_{00}) − k_{00}H, the last equality using (3.33) and (3.28). This gives

N(trk) − N(k_{00}) = −2N(k_{00}) − 2k_{00}H + k_{00}H − n[N(k_{00}) + N(trk) + Htrk],

which implies (3.43). This completes the proof of the Lemma.

To complete the proof of Theorem 3.3, (3.33) and (3.34) show that

k^{T} = (A′_{k})^{T} = 0,

at ∂N. One also has L̂(k) = δk = 0 on N, so that k is an infinitesimal Einstein deformation on N. By the local unique continuation result of [AH], together with the global hypothesis π_{1}(M, ∂M) = 0, it follows that k = 0. This shows that L̂ is surjective. The fact that its kernel splits is standard, cf. [A3]. This completes the proof.

Via the implicit function theorem, one obtains:

Corollary 3.5. Suppose π_{1}(M, ∂M) = 0 and m ≥ 3. Then the local spaces 𝔼_{D}^{m,α} are infinite dimensional C^{∞} Banach manifolds, with

(3.44)  Tg̃𝔼_{D} = Ker(DΦ̂g̃).

Proof: This is an immediate consequence of Theorem 3.3, the fact from Proposition 2.1 that 𝔼_{D} = Z_{D}, (cf. (2.18)), and the implicit function theorem, (or regular value theorem), in Banach spaces.

This leads to the main result of this section.

Theorem 3.6. Suppose π_{1}(M, ∂M) = 0 and m ≥ 3. Then the moduli space E_{S} = E_{S}^{m,α} is a C^{∞} smooth infinite dimensional Banach manifold for which the boundary map

(3.45)  Π_{B} : E_{S} → Met^{m,α}(∂M) × C^{m−1,α}(∂M),

is a C^{∞} smooth Fredholm map, of Fredholm index 0.

Proof: Recall from §1 that the moduli space E_{S} of static vacuum Einstein metrics is defined to be the quotient 𝔼_{S}^{m,α}/D_{1}^{m+1,α}. The local spaces 𝔼_{D} are smooth Banach manifolds and depend smoothly on the background metric g̃, since the divergence-free gauge condition (2.18) varies smoothly with g̃. As noted preceding Lemma 2.2, the action of D_{1} on 𝔼 is free and the local spaces 𝔼_{D} are smooth local slices for the action of D_{1} on 𝔼_{S}. Hence the global space 𝔼_{S} is a smooth Banach manifold, as is the quotient E_{S}. The local slices 𝔼_{D} represent local coordinate patches for E_{S}.

Proposition 3.1 implies that the boundary map Π_{B} : 𝔼_{S}^{m,α} → Met^{m,α}(∂M) × C^{m−1,α}(∂M) is smooth and Fredholm, of Fredholm index 0. Moreover, Π_{B} is invariant under the action of D_{1}^{m+1,α}(M) on E_{S}^{m,α} and so it descends to a smooth Fredholm map as in (3.45), still of index 0.

The self-adjointness of the boundary conditions (γ, H) for the operator Φ̂ suggests they may arise naturally from a variational principle (Lagrangian) on a space of static (non-vacuum) metrics. This is in fact the case.

To describe this, let S(h) = −Δtrh + δδ(h) − ⟨Ric, h⟩ be the linearization of the scalar curvature s, with adjoint S^{∗} given by

S^{∗}u = D^{2}u − Δu g − uRic.

It is well-known that the static vacuum equations are given by S^{∗}u = 0 and s = 0 on M.

Proposition 3.7. For (M, g, u) as above, the Bartnik boundary conditions (γ, H) give a well-defined variational problem for the Lagrangian

(3.46)  L(g, u) = ∫_{M} usdV_{g} − 16πm_{ADM} : Met_{δ}^{m,α}(M) × C_{δ}^{m,α}(M) → ℝ,

where m_{ADM} is the ADM mass of (M, g, u). The gradient ∇L of L at (g, u) is given by

(3.47)  ∇L = (S^{∗}u + \frac{1}{2}usg, s, uA − N(u)γ, 2u),

in the following sense: if (h, u′) is a variation of (g, u) inducing the variation (h^{T} , H′_{h}) of the boundary data, then

(3.48)  dL(h, u′, h^{T} , H′_{h}) = ∫_{M}[⟨S^{∗}u + \frac{1}{2}usg, h⟩ + su′] + ∫_{∂M}[⟨uA − N(u)γ, h^{T} ⟩ + 2uH′_{h}].

In particular, the static vacuum equations are critical points for L with data (γ, H) fixed.

Proof: Suppose that D is compact domain in M, with N the outward unit normal from D. Varying (g, u) in the direction (h, u′) then gives

(3.49)  DL_{0}(h, u′) = ∫_{D}(us′ + u′s + us(dV)′) = ∫_{D}⟨u, s(h)⟩ + \frac{1}{2}us⟨g, h⟩ + su′,

where L_{0} = ∫_{D} usdV_{g}. A straightforward integration by parts gives

(3.50)  ∫_{D}⟨S(h), u⟩ = ∫_{D}⟨h, S^{∗}u⟩ + ∫_{∂D} −uN(trh) − (δh)(N)u − ⟨h(N), du⟩ + trhN(u).

The equations (3.49) and (3.50) imply immediately the bulk Euler-Lagrange equations - the first two terms in (3.47). If the bulk term (over D) vanishes, then since u′ is arbitrary s = 0, so this gives

S^{∗}u = 0,

with s = 0, which are the static vacuum equations.

For the boundary terms, from (3.7) one has

2H′_{h} = N(trh) + 2δ(h(N)^{T}) − h_{00}H − N(h_{00}).

Also by a simple calculation

(δh)(N) = δ(h(N)^{T}) + ⟨A, h⟩ − h_{00}H − N(h_{00}),

so that,

2H′_{h} − (δh)(N) = N(trh) + δ(h(N)^{T}) − ⟨A, h⟩.

This gives

∫_{∂D} u(−N(trh) − (δh)(N)) = ∫_{∂D} −2uH′_{h} + ⟨du, h(N)^{T} ⟩ − u⟨A, h⟩.

It follows that the boundary term in (3.50) is given by

(3.51) ∫_{∂D} −2uH′_{h} − u⟨A, h⟩ − N(u)h_{00} + N(u)trh = ∫_{∂D} −2uH′_{h} − u⟨A, h^{T} ⟩ + N(u)⟨h^{T} , γ⟩.

Now let the outer boundary of D equal S(r) and consider the limit r → ∞. Then u → 1 and ∫_{S(r)} N(u)⟨h^{T} , γ⟩ → 0. It follows that

lim_{r→∞} ∫_{S(r)} −2uH′_{h} − u⟨A, h^{T} ⟩ = lim_{r→∞} ∫_{S(r)}(−N(trh) − (δh)(N)) = 16π(m_{ADM})′,

where the second equality follows from standard formulas for the ADM mass and its variation, cf. [RT], [B3]. Here the variation (m_{ADM})′ is taken in the direction h. The formula (3.50)-(3.51) is also valid at the inner boundary ∂M, with respect to the outward normal. Changing to the inner normal changes the sign of each term, and (3.47) and (3.48) then follow immediately.

On-shell, i.e. on the space of solutions E, the Lagrangian

L = −16πm_{ADM} : E → ℝ

is a smooth function whose derivative is given by the boundary term in (3.48), a result basically due to Bartnik [B3]. After writing this work, we learned that a special case of Proposition 3.7 has been noted, without proof, in a paper of Miao, cf. [M2].

4. Curvature estimates and Properness of Π^{o}.

In contrast to the previous section, where most of the computations were done on the 4-manifold (N, g_{N}), in this section we focus mostly on the 3-dimensional data (M, g, u). Let inj_{∂M} denote the injectivity radius of the normal exponential map from ∂M in M and let R denote the (full) curvature tensor of g. The main result of this section is the following collection of apriori estimates.

Theorem 4.1. For (M, g, u) ∈ E^{o} = (E^{m,α})^{o} with m ≥ 2, one has a global pointwise estimate

(4.1)  |R| ≤ Λ,

on M, where Λ depends only on bounds for the boundary data (γ, H). Moreover, at ∂M, one has the bounds

(4.2)  |A| ≤ Λ, inj_{∂M} ≥ Λ^{−1}.

The estimates (4.1), (4.2) also hold for higher derivatives of R and A, up to order m − 2, m − 1 respectively.

Proof: For points in the interior of M, of bounded distance away from ∂M, this follows directly from the apriori interior estimates in [A1] which state

(4.3)  |R|(x) ≤ \frac{λ}{t^{2}(x)}, |d log u|(x) ≤ \frac{λ}{t(x)},

where t(x) = dist(x, ∂M), where λ is an absolute constant. Similar (scale-invariant) estimates hold for all higher derivatives of R and log u. So one only needs to consider the behavior near ∂M. At ∂M, the Gauss and Gauss-Codazzi (constraint) equations are given by:

(4.4)  |A|^{2} − H^{2} + s_{γ} = −2R_{NN},
(4.5)  δ(A − Hγ) = −u^{−1}D^{2}u(N, ·).

Also, −R_{NN} = −Ric(N, N) = −u^{−1}NN(u) = u^{−1}(Δ_{∂M}u + HN(u)), so that

(4.6)  u(|A|^{2} − H^{2} + s_{γ}) = 2(Δ_{∂M}u + HN(u)).

From (4.4), a bound on |R| at ∂M gives immediately a bound on |A| at ∂M, given control of (γ, H). Similarly, a bound on |R| on M gives a lower bound on the distance d_{con} to the conjugacy locus of the normal exponential map exp_{∂M}.

Now, again under a bound on |R|, the outer-minimizing property (1.8) implies a lower bound on the distance δ_{∂M} to the cut locus of exp_{∂M}. To see this, suppose that δ_{∂M} << 1 but Λ in (4.1) is bounded, Λ ∼ 1. Then since d_{con} is bounded below, there is a geodesic γ of length 2δ_{∂M} in M meeting ∂M orthogonally at points p_{1}, p_{2}. Let T be the boundary of the tubular neighborhood of γ of radius r. Then T intersects ∂M in the boundary of two discs D_{1}, D_{2} of radius approximately r, (for r small). If δ_{∂M} << r, then areaT < area(D_{1}∪D_{2}). Further, T ⊂ M is homologous to D_{1}∪D_{2} in M. Removing then D_{1} ∪ D_{2} from ∂M and attaching T shows that ∂M is not outer-minimizing in M, giving a contradiction.

Thus it suffices to obtain a curvature bound at or arbitrarily near ∂M. The higher derivative estimates may then be obtained by standard elliptic regularity methods. The proof of (4.1) is by a blow-up argument. If the curvature bound in (4.1) is false, then there is a sequence (M, g_{i}, u_{i}, x_{i}) ∈ E^{o} with bounded Bartnik boundary data such that

|R_{g_{i}}|(x_{i}) → ∞.

Without loss of generality, assume that the curvature of g_{i} is maximal at x_{i}. We then rescale the metrics g_{i} to g′_{i} so that |R| is bounded, and equals 1 at x_{i},

(4.7)  |R_{g′_{i}}|(x_{i}) = 1, |R_{g′_{i}}|(y_{i}) ≤ 1,

for any y_{i} ∈ (M, g′_{i}). One may also need to rescale the potential u. For reasons that will be clearer below, choose points y_{i} ∈ M such that dist_{g′_{i}}(y_{i}, ∂M) = 1 and dist_{g′_{i}}(y_{i}, x_{i}) = 1, and rescale u_{i} so

(4.8)  u_{i}(y_{i}) = 1.

The sequence (M, g′_{i}, u_{i}) has uniformly bounded curvature and uniform control of the boundary geometry, (boundary metric, 2^{nd} fundamental form and normal exponential map). By (4.8) and the Harnack inequality, the potential u_{i} is also uniformly bounded in compact sets. It follows from the convergence theorem in [AT] for manifolds-with-boundary that a subsequence converges weakly, (i.e. in C^{1,α}), to a C^{1,α} static limit (X, g, u, x) with boundary (∂X, γ, u). By the normalization in (4.7), the limit (X, g) is complete, (without singularities). Since ∂M is outer-minimizing in (M, g_{i}), the C^{0} convergence to the limit implies that ∂X is weakly outer-minimizing in X: if D is any compact smooth domain in ∂X and D′ is a surface in X with ∂D′ = ∂D, then

(4.9)  areaD′ ≥ areaD.

One has ∂X = ℝ^{2}, the boundary metric γ is flat, H = 0, so ∂X is a minimal surface in X. One has u > 0 in the interior of X, (by the maximum principle), but may have u = 0 somewhere or everywhere on ∂X. The bound (4.7) and the static equations imply that u_{i} is uniformly bounded up to ∂X, within bounded distance to x_{i} and the limit potential u extends at least C^{1,α} up to ∂X.

We will prove below that the convergence to the limit is smooth, so that in particular

(4.10)  |R|(x) = 1,

where x = lim x_{i} and R = R_{X}.

On the blow-up limit (X, g), (4.6) holds and becomes

\frac{1}{2}u|A|^{2} = Δ_{∂M}u,

on ∂X. This equation holds weakly on ∂X with u ∈ C^{1,α}(∂X); elliptic regularity then implies it holds strongly, and u ∈ C^{3,α}(∂X). Since u is harmonic, u is thus C^{3,α} up to ∂X. Also, by the Riccati equation N(H) = −|A|^{2} − R_{NN} = −|A|^{2} + \frac{1}{2}(|A|^{2} − H^{2} + s_{γ}), so that

(4.11)  N(H) = −\frac{1}{2}(|A|^{2} + H^{2} − s_{γ}).

This holds pointwise on the blow-up sequence (M, g′_{i}, u_{i}) and since s_{γ} → 0 and H → 0 for g′_{i}, it follows that N(H) is defined pointwise on the limit ∂X and on ∂X,

(4.12)  N(H) ≤ 0,

with equality on any domain only when A = 0.

Since ∂X is minimal, (4.12) and the outer-minimizing property (4.9) imply that

(4.13)  N(H) = 0,

on ∂X. In more detail, (4.9) and the fact that H = 0 on ∂X implies the 2^{nd} order stability of ∂X, in that the 2^{nd} variation of the area of ∂X is non-negative. Thus, for all f of compact support on ∂X, one has

(4.14)  ∫_{∂X}(|df|^{2} + f^{2}N(H)) ≥ 0.

Choose f = f_{R,S}(r) such that f = 1 on D(R) ⊂ ∂X = ℝ^{2} and, for r ≥ R, f = (log r−log S)/(log R− log S), for S >> R >> 1. One may choose R and S sufficiently large such that ∫_{∂X} |df|^{2} < ε, for any given ε > 0. This together with (4.12) implies (4.13).

It follows that A = 0 and hence by the Liouville theorem on ℝ^{2}, u = const on ∂X. Using the divergence constraint (4.5), we also now have 0 = δ(A − Hγ) = −u^{−1}D^{2}(N, ·), and so 0 = D^{2}(N, ·) = dN(u) − A(du) = dN(u), so that N(u) = const.

Thus, the full Cauchy data (γ, u, A, N(u)) for the static vacuum equations is fixed and trivial: γ is the flat metric, A = 0 and u, N(u) are constant. Observe that this data is realized by the family of flat metrics on (ℝ^{3})^{+} with either u = const or u equal to an affine function on (ℝ^{3})^{+}.

Suppose first

(4.15)  u = const > 0 on ∂X.

The static vacuum equations (1.1) are then non-degenerate up to ∂X. The unique continuation property for Einstein metrics with boundary, cf. [AH], implies that the Cauchy data uniquely determine the solution locally. Alternately, since the static vacuum equations are non-degenerate up to ∂X and since the boundary data (∂X, γ, H) are real-analytic, elliptic regularity implies that the solution (M, g, u) is real-analytic up to ∂M. Such solutions are uniquely determined (locally) by their Cauchy data. Hence, the limit (X, g, u) is flat in this case.

Moreover, the convergence to the limit is smooth everywhere. This again follows from nondegeneracy and ellipticity. Briefly, the potential equation Δu = 0 gives a boost on the regularity of u, (given background regularity on g). One substitutes this into the main static vacuum equation uRic = D^{2}u, giving thus a boost to the regularity of Ric, inducing then a boost to the regularity of g. This in turn further boosts the regularity of u via the potential equation. Bootstrapping gives C^{m,α} convergence, up to the boundary, in regions where u > 0, (given that u is C^{m,α} at the boundary). In sum, one has a contradiction to (4.10).

Thus, suppose instead

(4.16)  u = 0 on ∂X.

This situation is more complicated. It is also more difficult to prove smooth convergence in this situation (one may have x in (4.10) at ∂X). Moreover, there are in fact non-flat static vacuum solutions with flat Cauchy data as above with u = 0 on ∂X, (so-called toroidal black holes, cf. [P] and [T]). Thus the unique continuation results used above are false in this degenerate situation where the boundary becomes characteristic.

We observe first that the solution (X, g) is still real-analytic up to ∂X in this case. This follows since the 4-metric g^{4} = u^{2}dθ^{2} + g_{X} is Einstein (Ric = 0) and is C^{1,α} up to the horizon or vanishing locus ∂X = {u = 0}. Elliptic regularity for the Einstein equations then implies that g^{4} is realanalytic, and hence so are u, g_{X} up to ∂X.

To see this in more detail, let U be a chart neighborhood of ∂X in X, so that U is diffeomorphic to a half-ball in ℝ^{3} with boundary a disc D^{2} ⊂ ℝ^{2}. Over each p ∈ U, one has a circle of length 2πu(p), with u → 0 as p → ∂X. From the work above, u is C^{1,α} up to ∂X with N(u) = const at ∂X. Note that N(u) ≠ 0 at ∂X. For if N(u) = 0 at ∂X, since also u = 0 at ∂X and u is harmonic (Δu = 0), the unique continuation property for harmonic functions implies that u = 0 in X, giving a contradiction. By rescaling u if necessary, one may thus assume that N(u) = 1 at ∂X. This implies that the 4-metric g^{4}, defined on B^{4} \ D^{2} extends to a C^{1,α} metric on the 4-ball B^{4}. (The coordinate θ is an angular variable in ℝ^{2} in polar coordinates, shrinking down to the origin on approach to ∂X). It is well-known, cf. [Be], that any C^{1,α} weak solution to the Einstein equations is real-analytic (in harmonic or geodesic normal coordinates), which gives the claim above.

To prove the limit is in fact flat, and that one has strong convergence, we need to use the outerminimizing property again. Thus, first note that (4.11) holds everywhere on the limit (X, g) near ∂X, not just at ∂X; here A is the 2^{nd} fundamental form of the level sets S(t) of t = dist(∂X, ·), etc. We have already established N(H) = 0 at ∂X, via the outer-minimizing property and the corresponding stability of the 2^{nd} variation operator (4.14). Taking then the derivative of (4.11) in the normal direction gives,

(4.17)  NN(H) = −⟨A′, A⟩ − ⟨A^{2}, A⟩ − H′H + \frac{1}{2}s′_{γ},

where A′ = ∇_{N}A. At ∂X, the first three terms vanish while (s′_{γ})_{k} = −Δ(trk)+δδk −⟨Ric_{γ}, k⟩ = 0, since k = 2A = 0. Thus

(4.18)  NN(H) = 0,

at ∂X and it follows that the 3^{rd} variation of the area of ∂X in the unit normal direction vanishes.

Now choose f = f_{R,S} as following (4.14) with R, S large. Let S_{tf} = exp_{∂X}(tf(x)), where exp_{∂X} is the normal exponential map of ∂X into X. Letting v(t) = areaS_{tf}, one has

(4.19)  v(t) = v(0) + \frac{1}{2}v′′(0)t^{2} + \frac{1}{6}v′′′(0)t^{3} + \frac{1}{24}v′′′′(0)t^{4} + O(t^{5}).

The expansion (4.19) is valid for all t sufficiently small, |t| ≤ δ_{0}, with δ_{0} independent of R, S, since the area and its derivatives are integrals of local expressions, and the local geometry of X is uniformly bounded in a tubular neighborhood of radius 1 about ∂X. By the 2^{nd} variational formula (4.14) and (4.13), for any given ε > 0, one has

v′′(0) ≤ ε,

for R, S sufficiently large. For the same reasons via (4.18),

v′′′(0) ≤ ε.

It follows then from the outer-minimizing property (4.9) and (4.19) that for R, S sufficiently large, one must have

(4.20)  v′′′′(0) ≥ −ε,

again for any ε = ε(R, S) > 0. Using the vanishing of the lower order terms, one computes that (4.20) gives

(4.21)  ∫_{∂X} f^{4}NNN(H) − 6f^{2}⟨df · df, A′⟩ ≥ −ε.

On the other hand, taking the normal derivative of (4.17) gives

(4.22)  NNN(H) = −⟨A′, A′⟩ − (H′)^{2} + \frac{1}{2}s′′_{γ}.

We have H′ = N(H) = 0 and (s′_{γ})_{2A} = 2ΔH + 2δδA − ⟨Ric_{γ}, 2A⟩. For s′′_{γ}, one has (ΔH)′ = Δ′H + ΔH′ = 0 and ⟨Ric_{γ}, A⟩′ = ⟨(Ric_{γ})′, A⟩ + ⟨(Ric_{γ}), A′⟩ = 0. So at ∂X, \frac{1}{2}s′′_{γ} = δδA′. It follows then from (4.21)-(4.22) that for f = f_{R,S} as above

(4.23)  ∫_{∂X} −f^{4}|A′|^{2} + f^{4}δδA′ − 6f^{2}⟨df · df, A′⟩ ≥ −ε.

Integrating the second term by parts gives ∫ ⟨D^{2}f^{4}, A′⟩ = ∫ ⟨4f^{3}D^{2}f + 12f^{2}⟨df · df, A′⟩. Using the Cauchy-Schwarz and Young inequalities, (4.23) then implies, for any μ small,

∫_{∂X} f^{4}|A′|^{2} ≤ μ ∫_{∂X} f^{4}|A′|^{2} + Cμ^{−1} ∫_{∂X} f^{2}|D^{2}f|^{2} + Cμ^{−1} ∫_{∂X} |df|^{4} + ε.

Choosing μ small, the first term on the right may be absorbed into the left, while simple computation shows that the last two terms become arbitrarily small for R and S sufficiently large. It follows that

(4.24)  A′ = 0

and so NNN(H) = 0 at ∂X. The Riccati equation

(4.25)  A′ = ∇_{N}A = −A^{2} − R_{N},

where R_{N}(V, W) = ⟨R_{g}(V, N)N, W ⟩, thus gives R_{N} = 0 at ∂X, and so via the Gauss and Gauss-Codazzi equations R_{g} = 0 at ∂X. Thus the full ambient curvature vanishes at ∂X.

One can now continue inductively in the same way to see that A and R vanish to infinite order at ∂X. A simpler method proceeds as follows. The Riccati equation (4.25) holds along the level sets S(t) of t = dist(∂X, ·). Since s_{g} = 0 and dimX = 3, R_{N} = − ∗ Ric, i.e. R_{N}(v, v) = −Ric(w, w), where (N, v, w) are an orthonormal basis. Via the static vacuum equations, this gives ∇_{N}A = −A^{2} + u^{−1} ∗ (D^{2}u). Rescale u if necessary so that N(u) = 1 at ∂X and set v = u − t. Since A = D^{2}t, one then obtains

(4.26)  ∇_{N}A = −A^{2} + u^{−1}(∗A) + u^{−1}(∗D^{2}v).

This is a system of ODE’s for A, singular at ∂X = ℝ^{2}, but with indicial root 1. From the work above, we have v = O(t^{2}) and A = O(t^{2}). Writing A = t^{2}B and substituting in (4.26) shows that A = O(t^{3}). Also, by the computation following (4.22), s_{γ} = O(t^{3}) on S(t), and hence using the scalar constraint (4.4) and the relation R_{NN} = u^{−1}NN(u), this in turn implies v = O(t^{3}), and so on. It follows that (g, u) agree with a flat solution to infinite order at ∂X. Since the solution (X, g, u) is analytic up to ∂X, it follows that (X, g, u) is flat, as claimed.

Next we claim that one has strong convergence to the limit, so that (4.7) is preserved in the limit, i.e. (4.10) holds, contradicting the fact that the limit is flat. Note first that if x in (4.10) is in the interior of X, then strong (C^{∞}) convergence is immediate, by the interior estimates (4.3), i.e. their higher derivative analogs. Thus we may assume that x ∈ ∂X.

From the work above, we know that |R| is uniformly bounded everywhere on (M, g′_{i}) and |R| → 0 everywhere away from ∂M → ∂X, so |R| jumps quickly from 1 to 0 near x_{i}. The main point is to prove that

(4.27)  |R|(x) → 0,

for all x ∈ ∂M → ∂X; it is then easy to prove that |R| → 0 on (M, g′_{i}), cf. (4.38) below. To prove (4.27), note that the estimates (4.11)-(4.24) above at ∂X also hold on the blow-up sequence at ∂M, (since H → 0 and s_{γ} → 0 on ∂M). It follows then by these arguments that for any R < ∞ and D(R) the R-ball about any base point y_{i} ∈ (∂M, γ′_{i}) converging to y ∈ ∂X,

(4.28)  ∫_{D(R)} |A|^{2} + |∇_{N}A|^{2} → 0.

To proceed further, consider the divergence constraint δA = −dH − Ric(N, ·) on (∂M, γ′_{i}), as in (4.5). The equations δA = χ_{1}, dtrA = χ_{2} form an elliptic first order system in 2-dimensions, and so one has elliptic estimates. Since H → 0 ∈ C^{m−1,α} on (∂M, γ′_{i}), dH → 0 in C^{m−2,α}. Also, by assumption (4.7), Ric(N, ·) is bounded in L^{∞}. It follows then from elliptic regularity that A is bounded in L^{1,p}. By the scalar constraint (4.4), R_{NN} is then also bounded in L^{1,p} and since trR_{N} = R_{NN}, it follows from (4.28) and (4.25) that

(4.29)  A → 0 and R_{NN} → 0 in C_{loc}^{α}(∂M).

Next the Einstein equation Ric_{g_{N}} = 0 on (N, g_{N}) implies that δ_{N}R_{g_{N}} = 0. Hence

(4.30)  0 = (δ_{g_{N}} R)(N, N, ·) = δ_{g_{N}} (R(·, N)N, ·) + 2R(e_{α}, ∇_{e_{α}}N)N = δ_{g_{N}} (R(·, N)N, ·),

since one may choose a basis in which ∇_{e_{α}}N = A(e_{α}) = λ_{α}e_{α}. Let V be the unit vertical vector, and note that ∇_{V} V = −dν, where ν = log u. Then ∇_{V} (R(V, N)N, ·) = R(dν, N)N, while ∇_{N}(R(N, N)N, ·) = 0. Hence, for R_{N} as in (4.25), these computations on ∂M give

δR_{N} = −R_{N}(dν),

where the divergence δ and R_{N} are taken on (∂M, γ′_{i}). Since R is bounded in L^{∞} and trR_{N} = R_{NN} is bounded in L^{1,p}, elliptic regularity gives

(4.31)  ||R_{N}||_{L^{1,p}} ≤ C||dν||_{L^{p}},

again on compact domains in ∂M converging to a compact domain in ∂X.

To control dν in (4.31), recall that by (4.7) the ambient curvature R is bounded, and so R restricted to ∂M is also bounded. Via the static vacuum equations (1.1), this implies that

(4.32)  N(ν)A + u^{−1}D^{2}u

is bounded on (∂M, γ′_{i}), where D^{2}u is the Hessian of u|_{∂M} : ∂M → ℝ^{+}. Now we claim that each term in (4.32) is bounded, i.e. there exists K such that

(4.33)  |u^{−1}D^{2}u| ≤ K, |N(ν)A| ≤ K,

pointwise, on domains converging to a bounded domain in ∂X. To prove (4.33), suppose instead that |u^{−1}D^{2}u| → ∞ at some sequence of base points y_{i} → y ∈ ∂X. Without loss of generality, we may assume the points y_{i} realize the maximum of |u^{−1}D^{2}u| on D_{y_{i}}(10) ⊂ (∂M, γ′_{i}), (possibly up to a factor of 2). As before, one may then rescale the metrics g′_{i} further to g′′_{i} so that |u^{−1}D^{2}u|(y_{i}) = 1 and hence the full curvature R → 0 in this scale. Also, renormalize u if necessary so that u(y_{i}) = 1. Note that |du|(y_{i}) must be bounded. For if |du|(y_{i}) were too large, it follows, (e.g. by a still further rescaling), that u would be close to an affine function on ℝ^{2} and hence u would assume negative values in bounded distance to y_{i}. Since u > 0 everywhere, this is impossible. Thus, by integration along paths, u is bounded in L^{1,∞} in the scale g′′_{i}, within bounded distance to y_{i}.

Now working in the scale and normalization above, from divergence constraint (4.5), one has −uδ(A − Hγ) = dN(u) − A(du). Since A is bounded in L^{1,p} and u and du are bounded in L^{∞}, it follows that dN(u) is bounded in L^{p}. Thus, N(u) = c + ϕ, where ϕ is bounded in L^{1,p}. Here c is a constant which may, and in fact does, go to ±∞, in the u-normalization u(y_{i}) = 1 above.

The trace equation (4.6) in this scale and normalization gives

(4.34)  Δu + H(c + ϕ) = uf,

where f is bounded in L^{1,p}. Also, Hc is bounded, since N(u)A is bounded via (4.32), and so Hc → c′, for some constant c′ on ∂X. Since ϕ is also bounded in L^{1,p}, it follows that u is bounded in L^{3,p}, and hence (in a subsequence), u converges in C^{2,α} to its limit on ∂X = ℝ^{2}. Hence D^{2}u converges in C^{α} to its limit on ℝ^{2}. On the limit, since H → 0, the trace equation (4.34) becomes

(4.35)  Δu + c′ = 0.

If c′ = 0 then since u > 0, u = const and hence D^{2}u = 0, giving a contradiction. If c′ ≠ 0, then again since u > 0, one must have c′ < 0 and u is a quadratic polynomial on ℝ^{2}. Since u is harmonic on the blow-up sequence and c′ < 0 implies N(u) → −∞, this is also easily seen to be inconsistent with the requirement u > 0 everywhere. This proves (4.33) holds.

Returning to (4.31), we may work in the normalization above that u(y_{i}) = 1 and then (4.33) implies that dν = u^{−1}du is bounded in L^{∞} in bounded domains about y_{i}. Hence by (4.31), R_{N} is bounded in L^{1,p} and so bounded in C^{α}. Since R_{N} → 0 in L^{2} locally on ∂M, one has

(4.36)  R_{N} → 0 in C_{loc}^{α}(∂M).

This is the main part of the estimate (4.27).

Next, one needs the same result for the Ric(N, T) term. To do this, take the normal derivative of the scalar constraint (4.6), to obtain

(4.37) ΔN(u) + Δ′u + N(H)N(u) + HNN(u) = \frac{1}{2}N(u)[|A|^{2} − H^{2} + s_{γ}] + \frac{1}{2}uN[|A|^{2} − H^{2} + s_{γ}].

Since γ′ = 2A, a standard formula for the variation of the Laplacian, (cf. [Be]), gives \frac{1}{2}Δ′u = −⟨D^{2}u, A⟩+⟨du, β(A)⟩ which is bounded in L^{∞}. Also the terms N(H), H and NN(u) = −\frac{u}{2}[|A|^{2}− H^{2} + s_{γ}] all go to 0 in L^{∞}. For the right side of (4.37), the coefficient of the first term goes to 0 in L^{∞} while via (4.36) above, uN[|A|^{2} − H^{2} + s_{γ}] is bounded in L^{p} provided N(s_{γ}) is, and this in turn follows from an L^{p} bound on δδA.

To obtain this, return to (4.30) but with (N, X) in place of (N, N), with X tangent to the boundary. Arguing as above, it follows that δ(R(·, N)X, ·) is bounded in L^{p}. Via the Gauss-Codazzi equations dA(X, Y, Z) = ⟨R(N, X)Y, Z⟩, it follows that δdA is bounded in L^{p}. Similarly, in the normalization above, δA = D^{2}u(N, ·) = ∇_{N}du modulo lower order terms. Taking the exterior derivative d of this, one easily obtains that dδA is bounded in L^{p}. This shows that ΔA is bounded in L^{p}, which of course gives the same for δδA by elliptic regularity.

Finally, as in the proof of (4.33), N(u) = c + ϕ with ϕ bounded in L^{1,p}. Thus it follows from the divergence constraint as preceding (4.34) and elliptic regularity that dN(u) is bounded in L^{1,p} and so converges in C^{α} on ∂M. Since

−uRic(N, ·) = dN(u) − A(du),

it now follows that Ric(N, ·) converges to its limit, necessarily 0, in C^{α} on ∂M. Lastly, δA = −dH − Ric(N, ·) → 0 in C^{α} and hence by elliptic regularity, A → 0 in C^{1,α} so that again via the Gauss-Codazzi equations dA(X, Y, Z) = ⟨R(N, X)Y, Z⟩ → 0 in C^{α}. Combining the computations above now proves the estimate (4.27).

To complete the proof, we have |R| → 0 on ∂M → ∂X. On the 4-manifold N = M ×_{u} S^{1}, the Einstein equations give the inequality

(4.38)  Δ_{N}|R_{g_{N}} | + c|R_{g_{N}} |^{2} ≥ 0,

where c is a fixed numerical constant, Δ_{N} is the 4-Laplacian and R_{g_{N}} is the curvature tensor on N. One has |R_{g_{N}} | = |R|, (up to a constant). We have proved above that R → 0 in L^{2} locally on (N, (g_{N})_{i}) and R → 0 pointwise at ∂N. It follows from the deGiorgi-Nash-Moser estimates for domains with boundary, (cf. [GT, Thm. 8.25]), that R → 0 pointwise on N and hence on M, contradicting (4.7). This completes the proof.

Next, we show that the potential function u is also controlled by the boundary data of Π_{B}.

Corollary 4.2. For (M, g, u) ∈ E^{o}, there is a constant U_{0} depending only on the boundary data (γ, H) ∈ Π^{o}(E^{o}) such that

(4.39)  u ≤ U_{0},

on M. Moreover, if H ≥ H_{0} > 0 on ∂M, then there exists U_{0} as above depending in addition only on H_{0}, such that on M,

(4.40)  u ≥ U_{0}^{−1}.

Proof: Let S(s) = {x ∈ (M, g) : dist(x, ∂M) = s} be the geodesic ’sphere’ about ∂M. Choose a fixed base point x_{0} ∈ S(1) and suppose one has the bound

(4.41)  c_{0}^{−1} ≤ u(x_{0}) ≤ c_{0}.

By Theorem 4.1, the geometry of the annular region A(\frac{1}{2}, 2) about S(1) is uniformly controlled by the boundary data (γ, H) and so by integration of the static vacuum equations uRic = D^{2}u along paths in A(\frac{1}{2}, 2), one has

(4.42)  C_{0}^{−1} ≤ u ≤ C_{0},

in A(\frac{3}{4}, \frac{3}{2}), where C_{0} depends only on c_{0} and (γ, H).

To remove the dependence of C_{0} in (4.42) on c_{0} in (4.41), we need better control on the large-scale behavior of u. To do this, it is proved in [An2, Lemma 3.6], that there is a constant K, depending only on C_{0}, such that for all s ≥ 1,

(4.43)  sup_{S_{c}(s)} |du| ≤ K(v_{c}(s))^{−1},

where v_{c}(s) = areaS_{c}(s) and S_{c}(s) is any component of the geodesic sphere S(s). We point out that (4.43) holds for general static vacuum solutions, not only those in E^{o} for instance. The estimate (4.43) is proved by studying the behavior of the harmonic potential log u on the Ricci-flat 4-manifold (N, g_{N}, log u) and then reducing to (M, g_{M}, u).

Consider now the conformally equivalent metric

(4.44)  g̃ = u^{2}g.

It is well-known that the static vacuum Einstein equations (1.1) are equivalent to the equations R̃ic = 2(dν)^{2} ≥ 0, Δg̃ν = 0, ν = log u. The metric g̃ thus has non-negative Ricci curvature with harmonic potential ν. These are exactly the properties used to prove (4.43), and a brief examination of its proof shows that (4.43) also holds with respect to g̃, i.e.

(4.45)  sup_{S̃_{c}(s)} |dν|g̃ ≤ K(ṽ_{c}(s))^{−1},

again with K = K(C_{0}). Since (M, g) is asymptotically flat and u → const at infinity, the area growth of geodesic spheres ṽ(s) in (M, g̃) satisfies ṽ(s)/s^{2} → ω_{2}, where ω_{2} = areaS^{2}(1). It follows then from the volume comparison theorem for Ricci curvature, (cf. [Pe]), that

(4.46)  ṽ(s) ≥ ω_{2}s^{2},

for all s ≥ 1. As above, by integration of (4.45) along a geodesic ray starting from a suitable base point x_{1} ∈ S(1) out to infinity, one sees that (4.39) holds then globally on M \ B(1), with U_{0} again depending only on c_{0} in (4.41). Using the static vacuum equations, the same integration along paths gives such a bound within B(1). Thus, we see that (4.39) follows from (4.41).

To prove (4.41), suppose one has a static vacuum solution (M, g, u) with

(4.47)  u(x_{0}) = ε.

Renormalize u to ū = u/u(x_{0}), so that ū(x_{0}) = 1 and ū → ε^{−1} at infinity. Then (4.41) holds, and hence so does (4.45)-(4.46). Again by integration along geodesics starting at x_{0} and diverging to infinity, it follows that

u ≤ U_{1},

where U_{1} depends only on the boundary data of Π_{B}. This proves the lower bound in (4.41). To prove the upper bound, suppose instead

(4.48)  u(x_{0}) = ε^{−1}.

Then again we renormalize u to ū as above, so that now u → ε at infinity. This does not directly give a lower bound on ε via (4.45)-(4.46) as above. However, one may proceed as follows. First, it is well-known that static vacuum solutions come in “dual” pairs, in that if (M, g, ū) is a static vacuum solution, then so is (M, ĝ, û) with ĝ = ū^{4}g, û = ū^{−1}, cf. [A2] for instance. Then (4.45)-(4.46) hold for (M, ĝ, û) which as before by integration gives an upper bound û ≤ U_{1} at infinity. Since near infinity, û ≃ ε^{−1}, this again gives a bound on ε^{−1}. This completes the proof of (4.39).

To prove (4.40), note that (4.41) has been proved above, and hence by the maximum principle and normalization u → 1 at infinity, (4.40) holds in the exterior region M \ B(1). Thus, one only needs to consider the behavior near ∂M. For this, suppose (M, g, u) is a static vacuum solution, C^{2,α} up to ∂M with u ≥ 0 on M̄ = M ∪ ∂M. If H ≥ H_{0} > 0 on ∂M, we claim that necessarily

u > 0 on ∂M.

For if u = 0 at some point z ∈ ∂M, then by (4.6), Δ_{∂M}u + HN(u) = 0 at z. Since 0 = u(z) is a global minimum for u, one has Δ_{∂M}u ≥ 0 and by the Hopf maxmimum principle, N(u) > 0 at z. This gives a contradiction if H > 0. The same arguments prove the existence of a lower bound (4.40) by a contradiction argument, taking a sequence and passing to a limit, using Theorem 4.1.

The previous results now lead quite easily to the following main result of this section.

Corollary 4.3. The boundary map

Π^{o} : E^{o} → Met^{m,α}(∂M) × C_{+}^{m−1,α}(∂M),

is almost proper.

Proof: Let (M, g_{i}, u_{i}) be a sequence of static vacuum solutions in E^{o}, with Π^{o}(g_{i}, u_{i}) = (γ_{i}, H_{i}). Supposing (γ_{i}, H_{i}) → (γ, H) in Met^{m,α}(∂M) × C_{+}^{m−1,α}(∂M), we need to prove that the sequence (g_{i}, u_{i}) has a subsequence converging in C^{m,α}(M), modulo diffeomorphisms, to a limit (M, g, u) ∈ E.

The curvature bound (4.1) and control of the intrinsic and extrinsic geometries of the boundary metrics first implies the metrics g_{i} cannot collapse within bounded distance to ∂M, i.e. there is a fixed constant i_{0} > 0 such that the injectivity radius of (M, g_{i}) satisfies

inj_{g_{i}}(x) ≥ i_{0},

for dist_{g_{i}}(x, ∂M) ≤ K. By the compactness theorem in, for instance [AT], it follows that a subsequence of (M, g_{i}) converges in C^{m,α}, (and C^{∞} in the interior), uniformly on bounded domains containing ∂M, to a limit (M′, g). One has ∂M′ = ∂M and g is a complete Riemannian metric on M′, C^{m,α} up to ∂M and C^{∞} in the interior.

By Corollary 4.2, the potential functions u_{i} also converge in C^{m,α}, (in a subsequence) to a limit potential function u on M′, and the pair (g, u) gives a solution of the static vacuum Einstein equations. Since u = lim u_{i}, it follows that

(4.49)  u ≤ U_{0},

on M′, for U_{0} as in (4.39). Clearly the boundary metric and mean curvature of (M′, g, u) are given by the limit values (γ, H). To prove that (M′, g, u) ∈ E, one then needs to prove that (M′, g, u) is asymptotically flat and M′ is diffeomorphic to M. Note that since the convergence above is only uniform on compact sets, apriori there need not be any relation between the asymptotic structure of (M′, g, u) and (M, g_{i}, u_{i}) for any given i.

The equation (4.43) holds on each (M, g_{i}, u_{i}) and by Corollary 4.2, the constant K is uniform, independent of i. Moreover, as in the proof of Corollary 4.2, there is a geodesic ray σ = σ_{i} starting at any fixed base point in S(1) and diverging to infinity, such that on the component S_{c}(s) of S(s) containing σ, one has

sup_{S_{c}(s)} |du_{i}| ≤ Ks^{−2},

where K is independent of i. Since u_{i} is harmonic, by elliptic regularity, (and scaling), a similar estimate holds for higher derivatives of u_{i}, and via the static vacuum equations, it follows that

sup_{S_{c}(s)} |R_{g_{i}}| ≤ Cs^{−3},

with again C independent of i. This means that the metrics (M, g_{i}, u_{i}) become asymptotically flat at infinity uniformly, at a rate independent of i. For R sufficiently large, R ≥ R_{0} independent of i, the geodesic spheres S(R) and annuli A(R, 2R) are close to Euclidean spheres and annuli, (when scaled by R^{−1}), and hence the geometry is close to that of Euclidean space; there can be no branching or joining of different components of S(R) for R ≥ R_{0}. This implies that the limit (M′, g, u) has a single asymptotically flat end, and M′ is diffeomorphic to M.

Remark 4.4. The results above also show that the boundary map Π_{B} is almost proper not only on E^{o} but also its closure \overline{E}^{o}. In other words, if (M, g_{i}, u_{i}) is a sequence of static vacuum solutions in E^{o} with boundary data (γ_{i}, H_{i}) and (γ_{i}, H_{i}) → (γ, H) in Met^{m,α}(∂M) × C^{m−1,α}(∂M), then (M, g_{i}, u_{i}) converges in C^{m,α} (in a subsequence) to a limit (M, g, u) in \overline{E}^{o}, with H ≥ 0.

To see this, note that the constant Λ in Theorem 4.1 does not depend on a positive lower bound on H, and so (4.1)-(4.2) hold for the sequence (M, g_{i}, u_{i}) above. Of course we are using here the fact that ∂M is outer-minimizing on the sequence (M, g_{i}, u_{i}). Similarly in Corollary 4.2, the upper bound U_{0} on u_{i} does not depend on a lower bound for H. One may then use the argument concerning (4.47) to show that u_{i} cannot go to 0 on M away from ∂M. The proof of Corollary 4.3 also does not require a bound on H away from 0.

It is also worth pointing out a brief examination of the proof shows that the results of this section only require that ∂M is outer-minimizing in a neighborhood of arbitrarily small but fixed size (depending on (M, g, u)) about ∂M.

5. Degree of Π^{P}.

By [Sm], a smooth proper Fredholm map F : B_{1} → B_{2} of Fredholm index 0 between connected Banach manifolds B_{1}, B_{2} has a well-defined degree (mod 2). Namely, if y is a regular value of F, then F^{−1}(y) is a finite set of points, and deg_{ℤ_{2}}F is just the cardinality of F^{−1}(y) (mod 2). In fact if B_{1} and B_{2} are oriented, then F has a well-defined degree in ℤ, cf. [ET], [BFP].

From the discussion in §1, the boundary map Π_{B} : E^{+} → Met(∂M) × C_{+}(∂M), although Fredholm, is not proper. However, by Theorem 1.2, the restricted boundary map Π^{o} : E^{o} → Met(∂M)×C_{+}(∂M) is almost proper. This in fact suffices to obtain a ℤ_{2}-valued degree on natural domains within E^{o}.

Thus, as in the Introduction, let ∂E^{o} be the boundary of E^{o} within the space E_{S} of static vacuum solutions. Hence, (M, g, u) ∈ ∂E^{o} if and only if (M, g, u) satisfies (1.10) but not (1.8). Let Z = Π_{B}(∂E^{o}) ⊂ Met^{m,α}(∂M) × C^{m−1,α}(∂M) be the image of ∂E^{o} under the boundary map Π_{B} and set

E^{P} = (Π^{o})^{−1}([Met^{m,α}(∂M) × C^{m−1,α}(∂M)] \ Z).

Then, by Theorem 1.2 and construction, the induced boundary map

(5.1)  Π^{P} : E^{P} → [Met^{m,α}(∂M) × C^{m−1,α}(∂M)] \ Z

is proper. In particular E^{P} has only finitely many components E^{P_{i}} and on each component the ℤ_{2}-valued Smale degree is well-defined.

Next, we use Proposition 3.7 to prove that the boundary map Π^{P} on each component E^{P_{i}} of E^{P} has a well-defined and computable degree in ℤ.

Given a static vacuum solution (M, g, u) ∈ E^{+}, as preceding (1.2) let N = ℝ × M with Ricci-flat static 4-metric g = u^{2}dθ^{2} + g. A simple computation shows that the Lagrangian L in (3.46) may be expressed in terms of (N, g) as

(5.2)  L(g) = ∫_{N} s_{g}dV_{g} − 8πm + 2 ∫_{∂N} N(ν)dV_{γ_{N}} ,

where ν = log u and N(ν) is the normal derivative of ν into N. As in (2.16), let L̂ be the linearization of the divergence-gauged Einstein operator Ric_{g} − \frac{s_{g}}{2} g + δ_{g}^{∗}βg̃(g). This gives the (gauged) 2nd

27

variation of the Lagrangian (5.2) on the space of static metrics with boundary conditions (γ, H) fixed. As in Proposition 3.7, the operator L̂ is elliptic, Fredholm and self-adjoint with respect to the boundary conditions h^{T} = H′_{h} = 0. It follows that L̂ has a complete basis of eigenvectors in L^{2}, with discrete eigenvalues diverging to +∞. Let

(5.3)  ind(g, u) ∈ ℤ,

be the L^{2} index of the operator L̂ at (N, g), i.e. the maximal dimension of the subspace of L_{δ}^{2,2}(N, g) on which L̂ is a negative definite bilinear form, with respect to the L^{2} inner product. This equals the dimension of the space on which L̂ is negative definite on a sufficiently large compact domain in M, with zero boundary values on both inner and outer boundaries. The nullity of (M, g, u) is the dimension of the L^{2} kernel K, consisting of forms (h, u′) = h satisfying

L̂(h) = 0, with h^{T} = 0, H′_{h} = 0 at ∂M.

Such indices are well-studied in the context of geodesics and minimal surfaces, the basic idea going back to Morse.

Let Π^{P_{i}} be the restriction of the boundary map Π^{P} to E^{P_{i}} and let T_{i} be the component of [Met^{m,α}(∂M) × C^{m−1,α}(∂M)] \ Z containing ImΠ^{P_{i}}. Thus

Π^{P_{i}} : E^{P_{i}} → T_{i},

is a proper map of connected Banach manifolds.

Proposition 5.1. Let (γ, H) ∈ T_{i} be a regular value of Π^{P_{i}} and define

(5.4)  deg_{ℤ}(Π^{P_{i}}) = ∑_{(g_{k},u_{k})∈(Π^{P_{i}})^{−1}(γ,H)}(−1)^{ind(g_{k},u_{k})}.

Then deg_{ℤ}(Π^{P_{i}}) is well-defined, i.e. independent of the choice of the regular value (γ, H) ∈ T_{i}.

Proof: Similar results have been proved earlier for simpler variational problems, namely for geodesics and minimal surfaces, cf. [Tr], [W1], [W2] for instance and further references therein. Hence we refer to [W1], [W2] for some further details.

First, the general idea is the following. Let (γ_{1}, H_{1}) and (γ_{2}, H_{2}) be regular values of Π^{P_{i}} in T_{i} and let σ̄(t), t ∈ [0, 1] be an oriented curve in T_{i} joining them. By genericity, σ̄ may be assumed to be transverse to Π^{P_{i}}, so that the lift σ = (Π^{P_{i}})^{−1}(σ̄) is a collection of curves in E^{P_{i}}, with boundary in the fibers over (γ_{1}, H_{1}) and (γ_{2}, H_{2}). Define an orientation on σ by declaring that Π^{P_{i}} is orientation preserving at any regular point of σ which has even index, while Π^{P_{i}} is orientation reversing at regular points of σ of odd index. Thus, provided this orientation is well-defined, the map Π^{P_{i}}|_{σ} : σ → σ̄ has a well-defined mapping degree, as a map of 1-manifolds. By construction, this 1-dimensional degree is given by (5.4) at any regular value of Π^{P_{i}}(σ) and hence it follows that (5.4) is well-defined. Thus, it suffices to prove that the orientation constructed above is well-defined.

If DΠ^{P_{i}}(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 DΠ^{P_{i}}(σ′(t_{0})) = 0, so that σ(t_{0}) is a critical point of Π^{P_{i}}; (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. [W1]) one may assume that K = ⟨κ_{0}⟩ is the span of κ_{0}. Also without loss of generality, we assume that σ_{t} is in divergence-free gauge with respect to σ_{0}.

Recall from Theorem 3.3 that the divergence-gauged Einstein operator

Φ̂ : Met_{D}^{m,α}(N) → 𝕊_{δ}^{m−2,α}(N),

is a submersion at any static vacuum solution g_{0} = (g_{0}, u_{0}) ∈ E. Let ℝ^{2} be parametrized by coordinates (t, s), for t as above and s ∼ sκ_{0} parametrizing K. Since Φ̂(g, u) = 0 on the curve σ(t), the implicit function theorem and the self-adjointness of the linearization L̂ imply that there is a map F : ℝ^{2} → Met_{D}^{m,α}(N), such that F (t, 0) = σ(t), F (t, s)−F (t, 0) ∈ K^{⊥} and Φ̂(F (t, s)) ∈ K ≃ ℝ. Define then

ψ(t, s) = π_{K}(Φ̂(F (t, s))),

where π_{K} is the L^{2} projection to K.

The curve σ(t) is thus the 0-set of the smooth function ψ : ℝ^{2} → ℝ. Further ∂_{t}ψ(t, s) ≠ 0 at t = 0, s small, and hence σ(t) is (locally) the boundary of the open set {ψ > 0} in ℝ^{2},

σ(t) = ∂{ψ > 0}.

Now 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.

The discussion above leads to Theorem 1.3. Let E^{P_{0}} be the component of E^{P} containing the standard round flat solution, equal to the exterior of the round ball in ℝ^{3}, with

Π^{P_{0}} : E^{P_{0}} → T_{0}.

Theorem 5.2. The degree of Π^{P_{0}} satisfies

(5.5)  deg_{ℤ}Π^{P_{0}} = 1.

Proof: The proof is based on the black hole uniqueness theorem [I], [R], [BM], that the Schwarzschild metrics

(5.6)  g_{Sch}(m) = (1 − \frac{2m}{r})^{−1}dr^{2} + r^{2}g_{S^{2}(1)}, u = \sqrt{1 − \frac{2m}{r}},

r ≥ 2m, are the unique AF static vacuum metrics with a smooth horizon H = {u = 0}. The induced metric on ∂M is S^{2}(2m) - the round metric γ_{2m} of radius 2m on S^{2}. The mean curvature satisfies H = 0. Of course the Schwarzschild metrics are not in E_{S}, but instead lie at the boundary ∂E_{S}.

Consider any sequence {(g_{i}, u_{i})} ∈ E^{P_{0}} for which Π^{P_{0}}(g_{i}, u_{i}) = (γ_{i}, H_{i}) → (γ_{2m}, 0) smoothly. Clearly, {(g_{i}, u_{i})} is a divergent sequence in E^{o}. By Corollary 4.3 and Remark 4.4, a subsequence of {(g_{i}, u_{i})} converges smoothly to a static vacuum limit (M, g, u). (Of course one may have u = 0 on ∂M). On this limit,

H = 0,

at ∂M, so that ∂M is a minimal surface. From (4.6) one has 2Δ_{∂M}u = u(|A|^{2} + s_{γ}) ≥ 0, and hence it follows from the maximum principle that u = 0 on ∂M. Via the static vacuum equations (1.1), this implies further that A = 0 and N(u) = const at ∂M. The black hole uniqueness theorem then implies that any such limit is the Schwarzschild metric, and so unique up to scaling. Thus one has uniqueness for the boundary data (γ, 0), so that most all boundary metrics γ cannot be realized with H = 0 at ∂M, (the no-hair result).

Given this background, suppose

(5.7)  deg_{ℤ}Π^{P_{0}} = 0.

Then for any regular value (γ, H) of Π^{P_{0}}, the finite set (Π^{P_{0}})^{−1}(γ, H), if non-empty, consists of at least two distinct static vacuum solutions (g^{1}, u^{1}), (g^{2}, u^{2}). The regular values of Π^{P_{0}} are generic (of second category) in the range space, by the Sard-Smale theorem. Choose then a sequence of regular values (γ_{i}, H_{i}) → (γ_{+1}, 0) smoothly. (We set m = 1/2 here). Suppose for now that (Π^{P_{0}})^{−1}(γ_{i}, H_{i}) is non-empty; this will proved to be the case later.

Let (g_{i}^{1}, u_{i}^{1}), (g_{i}^{2}, u_{i}^{2}) be any pair of corresponding distinct sequences in (Π^{P_{0}})^{−1}(γ_{i}, H_{i}). By Corollary 4.3 and Remark 4.4, the sequences (g_{i}^{1}, u_{i}^{1}), (g_{i}^{2}, u_{i}^{2}) have C^{m,α} convergent subsequences to limits (g_{∞}^{1}, u_{∞}^{1}), (g_{∞}^{2}, u_{∞}^{2}) in \overline{E}^{P_{0}} and by the uniqueness above

g_{∞}^{1} = g_{∞}^{2} = g_{Sch}(m),

with m = 1/2, with u_{∞}^{1} = u_{∞}^{2} = u in (5.3).

This implies that near g_{Sch}, the boundary map Π^{P_{0}} is not locally 1-1, and so presumably DΠ_{B} has a non-trivial kernel at g_{Sch}. (Note however that g_{Sch} ∉ E_{S}). We claim this is impossible. To prove the claim, let

g_{Sch} = u^{2}dθ^{2} + g_{Sch},

be the 4-dimensional Schwarzschild metric on ℝ^{2} × S^{2}, and similarly let

g_{i}^{j} = (u_{i}^{j})^{2}dθ^{2} + g_{i}^{j},

be the 4-dimensional static Ricci-flat metrics associated to (g_{i}^{j}, u_{i}^{j}). By Lemma 2.2, without loss of generality we may assume that each g_{i}^{j} is in Bianchi gauge with respect to g_{Sch}, so that, as in (2.10)-(2.11),

β_{g_{Sch}}(g_{i}^{j}) = 0,

for j = 1, 2 and i sufficiently large. By the smoothness of the convergence above, one may write

(5.8)  g_{i}^{j} = g_{Sch} + ε_{i}^{j}κ_{i}^{j} + O((ε_{i}^{j})^{2}),

where L(κ_{i}^{j}) = 0 and L is the linearized Einstein operator (2.6) at g_{Sch}. The data g_{i}^{j}, g_{Sch} and κ_{i}^{j} are all smooth, (up to the boundary). The forms κ_{i}^{j} are only unique up to multiplicative constants, which will be determined by choosing ε_{i}^{j} so that the C^{1,α} norm of g_{i}^{j} − g_{Sch} equals ε_{i}^{j}. Thus the C^{1,α} norm of κ_{i}^{j} is on the order of 1. Note that κ_{i}^{j} decays to 0 at infinity, so it is basically supported within compact regions of M. Let ε_{i} = max(ε_{i}^{1}, ε_{i}^{2}). Then

ε_{i}^{−1}(g_{i}^{2} − g_{i}^{1}) = κ_{i} + O(ε_{i}),

where κ_{i} = ε_{i}^{−1}(ε_{i}^{2}κ_{i}^{2} − ε_{i}^{1}κ_{i}^{1}) → κ, where the convergence is in C^{1,α′}, (in a subsequence). As previously, we need to show that the convergence is strong, so that κ ≠ 0. This follows from a standard linearization and bootstrap argument, as preceding (4.16). In more detail, dropping the index i, we have Δ_{g^{j}}u^{j} = 0, so that

Δ_{g^{2}}(u^{1} − u^{2}) = (Δ_{g^{2}} − Δ_{g^{1}})u^{1}.

In local harmonic coordinates, the right side this equation is on the order of ε in C^{1,α}, and hence by elliptic regularity, u_{1} − u_{2} is on the order of ε in C^{3,α}. Substituting this in the difference of the static equations u^{j}Ric_{g^{j}} = D_{g^{j}}^{2}u^{j} and arguing in the same way shows that the difference g^{1} − g^{2} is then also on the order of ε in C^{3,α}. This proves the strong convergence.

It follows that the limit form

(5.9)  κ = (h, u′),

is a non-zero C^{1,α} weak solution of the linearized static vacuum equations L(κ) = 0 at g_{Sch} and since Π^{P_{0}}(g_{i}^{1}, u_{i}^{1}) = Π^{P_{0}}(g_{i}^{2}, u_{i}^{2}), one has

(5.10)  γ′_{h} = H′_{h} = 0 at ∂M,

where γ′_{h} = h^{T} = h|_{∂M}. As discussed following (4.16), elliptic regularity implies that (h, u′) is smooth and so in particular a strong solution. Below we will use the fact that the data (h, u′) are in fact real-analytic up to ∂M, (again by elliptic regularity).

We claim that

(5.11)  (h, u′) = 0 on M,

which will give a contradiction. This is of course a linearized version of the black hole uniqueness theorem. It is possible that (5.11) can be proved by linearizing one of the existing proofs of black hole uniqueness in [I], [R], [BM]. However, we have not succeeded in doing this and instead (5.11) is proved in a manner similar to the proof of Theorem 5.1.

First, the linearization of (4.6) gives, at ∂M,

u′s_{γ} = 2Δu′.

Since s_{γ} > 0, the maximum principle implies that u′ = 0 at ∂M. Next we claim A′ = 0. To see this, the vacuum equations give uRic = D^{2}u and D^{2}u = N(u)A + (D^{2}u)^{T} when evaluated on tangent vectors to ∂M. Taking then the variation and evaluating tangentially gives (uRic)′ = 0 so 0 = (D^{2}u)′ = (D^{2})′u + D^{2}u′. The first term on the right vanishes when evaluating tangentially and hence so does the second term. This implies 0 = (N(u)A)′ = N(u)A′ + N(u′)A = N(u)A′. Since N(u) = const ≠ 0, it follows that A′ = 0. Similarly, taking the variation of the divergence or vector constraint gives N(u′) = const.

Clearly N(u′) = m′, (up to constants). A simple examination of the proof of black hole uniqueness in [R] applied to an Einstein deformation as in (5.6) and satisfying (5.10), shows easily that N(u′) = 0 at ∂M. (One does not obtain any further information, since the bulk data in the Robinson proof, via divergence identities, are quadratic in the deviation from Schwarzschild).

Thus the variations (γ′, u′, A′, N(u′)) of all the Cauchy data are trivial. As in the proof of Theorem 4.1, we use a bootstrap argument to prove that the data (h, u′) vanish to infinite order at ∂M, in geodesic gauge.

Thus, using geodesic normal coordinates near ∂M, write

g = dt^{2} + g_{t},

where t(x) = dist(x, ∂M). We may assume, (by adding an infinitesimal deformation of the form δ^{∗}V if necessary), that h preserves this gauge, so that h_{0α} = 0, i.e. h(N, ·) = 0, N = ∂_{t}, near ∂M. By the discussion above, we have u′ = N(u′) = 0 at ∂M and similarly h = ∇_{N}h = 0 at ∂M, so that

(5.12)  u′ = O(t^{2}) and h = O(t^{2}).

The variation of the potential equation Δ_{M}u = 0 gives

(5.13)  Δu′ = −Δ′u = ⟨D^{2}u, h⟩ − ⟨β(h), du⟩,

where β is the Bianchi operator, (cf. [Be]). Since β(h) = 0 at ∂M, this gives Δu′ = 0 at ∂M and hence NN(u′) = 0 at ∂M, so that

(5.14)  u′ = O(t^{3}).

Next the linearization of the Riccati equation gives (∇_{N}A)′ = −(A^{2})′ − (R_{N})′ = −(R_{N})′ at ∂M. One computes ∗R_{N} = −(Ric^{T}) = −u^{−1}(D^{2}u)^{T}, so (∗R_{N})′ = u^{−2}u′D^{2}u − u^{−1}(D^{2})′u − u^{−1}D^{2}u′, as a form on T (∂M). It follows from (5.14) that (∗R_{N})′ = 0 at ∂N and hence (∇_{N}A)′ = 0 so that

(5.15)  h = O(t^{3}).

Next taking the normal derivative of (5.13) gives

ΔN(u′) = ⟨∇_{N}D^{2}u, h⟩ + ⟨D^{2}u, ∇_{N}h⟩ + ⟨∇_{N}β(h), du⟩ + ⟨β(h), ∇_{N}du⟩,

which vanishes at ∂M and hence u′ = O(t^{4}). Substituting this in the linearized Riccati equation above and using previous estimates gives h = O(t^{4}), and so on. It follows that (h, u′) vanish to infinite order at ∂M. Since (h, u′) are real-analytic up to ∂M, (in geodesic gauge), this implies that h = u′ = 0 on M, i.e. (5.11) holds.

It remains only to prove that there are regular values (γ, H) of Π^{P_{0}} near (γ_{+1}, 0) whose inverse image under Π^{P_{0}} is non-empty. However, (5.11) proves that KerDΠ_{B} = 0 at the Schwarzschild metric with boundary data (γ_{+1}, 0). By continuity and the black hole uniqueness theorem, it follows that KerDΠ_{B} = 0, for all solutions with boundary data (γ, H) near (γ_{+1}, 0) in ImΠ^{P_{0}}. Thus, all such boundary data are regular values of Π^{P_{0}}. This completes the proof.

Remark 5.3. Although the proof of Theorem 5.2 implies that DΠ_{B} has trivial kernel at the Schwarzschild metric g_{Sch}, one does not expect this to be the case for the cokernel. In fact, one expects that CokerDΠ_{B} is infinite dimensional, in that any boundary variation of the form (k, 0), where k is a variation of the boundary metric, is not tangent to a curve of static metrics with H = 0 at ∂M. This amounts to the linearized version of the no-hair theorem, (which has not been proved as far as we are aware). In particular, we expect DΠ_{B} is not Fredholm at g_{Sch}.

Remark 5.4. The proof of Theorem 5.2 above shows that ImΠ^{P_{0}} and hence ImΠ_{B} contains regular values. In particular, this means that the image ImΠ_{B} has non-empty interior and is in fact an open map near the Schwarzschild metric with boundary the horizon r = 2m. We point out that it has been an open question as to whether Π_{B} has any regular values. For instance, the analysis of Miao in [M1] shows that the exterior of the standard round unit ball B^{3}(1) in flat ℝ^{3} is a critical point of Π_{B}. In fact it is still unknown whether ImΠ_{B} has open interior near such standard boundary data (γ_{+1}, 2).

Remark 5.4 indicates how little has been understood, and how much remains to be understood, regarding the global behavior of the boundary map Π_{B}. The proof of the almost properness of Π^{o} in Theorem 1.2 uses the outer-minimizing property (1.8) in two key, but essentially independent ways. First, it is used to obtain the apriori curvature and related estimates discussed in Section 5, i.e. to prevent blow-up behavior at the boundary. For this, only a small-scale version of (1.8) is necessary, in that one needs stability of the area of the boundary only to 4th order at ∂M, and this only in small discs in ∂M. Second, it is used to keep the boundary ∂M properly embedded in M, i.e. to prevent the passage from embedded to immersed behavior. Again, as mentioned in Remark 4.4, only a local version of (1.8) is needed for this.

For further progress, an important issue is to find some condition on the boundary data (γ, H) which ensures that the two properties above hold. A natural question is whether positivity of the Gauss and mean curvatures is sufficient.

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February, 2011

Dept. of Mathematics, Stony Brook University, Stony Brook, NY 11790

   E-mail address: anderson@math.sunysb.edu
   E-mail address: khuri@math.sunysb.edu