The Hilbert manifold of asymptotically flat metric extensions

In [Comm. Anal. Geom., 13(5):845-885, 2005.], Bartnik described the phase space for the Einstein equations, modelled on weighted Sobolev spaces with local regularity $(g,\pi)\in H^2\times H^1$. In particular, it was established that the space of solutions to the contraints form a Hilbert submanifold of this phase space. The motivation for this work was to study the quasilocal mass functional now bearing his name. However, the phase space considered there was over a manifold without boundary. Here we demonstrate that analogous results hold in the case where the manifold has an interior compact boundary, where Dirichlet boundary conditions are imposed on the metric. Then, still following Bartnik's work, we demonstrate the critical points of the mass functional over this space of extensions correspond to stationary solutions. Furthermore, if this solution is sufficiently regular then it is in fact a static black hole solution. In particular, in the vacuum case, critical points only occur at exterior Schwarzschild solutions; that is, critical points of the mass over this space do not exist generically. Finally, we briefly discuss the case when the boundary data is Bartnik's geometric data.


Introduction
It is well-known that the total mass/energy of an isolated body in general relativity is given by the ADM mass, and that the very nature of general relativity precludes the possibility of a local energy density; however, the notion of the mass contained in a given region of finite extent is still an open problem. This question is particularly peculiar,

INTRODUCTION
as it is not that we lack an answer to it, but rather we have many candidates for what this mass should be (See [26] for a detailed review), many of which are incompatible. Bartnik's quasilocal mass [3] is considered by many to give the best answer to this question, if only it were possible to compute in general. The Bartnik mass is described as follows: Given a subset Ω of some (M ,g,π), an initial data set satisfying the Einstein constraints, let PM be the set of asymptotically flat initial data sets satisfying the positive mass theorem, in which Ω isometrically embeds, with no horizons outside of Ω. The Bartnik mass is then taken as the infimum of the ADM mass over PM. It is conjectured that this infimum is indeed realised; however, while some progress has been made (see [1,5,11,13,20] and references therein), this is still an open problem in general. In the case where Ω is bounded by a minimal surface, this is known to be false; in a recent paper of Mantoulidis and Schoen [13], a sequence of extensions to a stable minimal surface is constructed, whose mass converges to the Bartnik mass. In light of black hole uniqueness theorems, the only possible limit for this sequence is a Schwarzschild solution, so if Ω is not contained in a slice of Schwarzschild then the infimum is not realised. However, this conjecture is still wide open when the boundary is not a minimal surface.
There are also interesting results by Corvino [11] and Miao [20], which demonstrate that if a mass-minimising extension exists, then it must be static and satisfy Bartnik's geometric boundary conditions [4]. That is, the metric is Lipshitz across the boundary and the mean curvature on each side of the boundary agree. Bartnik's work on the phase space for the Einstein equations [6] was, in part, motivated by the idea of putting Corvino and Miao's work in a more variational setting. Here we work to this end. For more details pertaining to the space PM and the Bartnik mass, the reader is referred to [3,4]. In this paper, we consider a larger set of extensions to such a bound domain Ω, described by asymptotically flat manifolds with Dirichlet boundary conditions imposed on a compact interior boundary, Σ. The initial data we consider has local regularity (g, π) ∈ H 2 × H 1 , with g prescribed on Σ in the trace sense.
The structure of this article is as follows. In Section 2, we review the mapping properties of the Laplace-Beltrami operator M and show that this is an isomorphism between certain weighted spaces over M with Dirichlet boundary conditions imposed. In Section 3, we apply Bartnik's phase space analysis to the case considered here, where M has an interior boundary and g satisfies Dirichlet boundary conditions. In particular, we prove that the space of asymptotically flat solutions to the constraints, satisfying the Dirichlet boundary conditions, is a Hilbert manifold. Finally, in Section 4, we prove a result intimately related to the static metric extension conjecture and Bartnik's quasilocal mass. We prove that critical points of the mass over the space of extensions to Ω with Dirichlet boundary conditions, correspond to stationary solutions with vanishing stationary Killing vector on Σ. In particular, if g is sufficiently smooth, this implies Σ Stephen McCormick   2 THE LAPLACE-BELTRAMI OPERATOR ON AN ASYMPTOTICALLY FLAT  MANIFOLD WITH INTERIOR BOUNDARY is the bifurcation surface of a bifurcate Killing horizon that is non-rotating, and by a staticity result of Sudarsky and Wald [25], one expects that the extension is therefore static. We also obtain a version of this result related to the geometric boundary data (Corollary 4.6).
2 The Laplace-Beltrami operator on an asymptotically flat manifold with interior boundary The constraint equations form a system of geometric PDEs that do not conform exactly to any of the standard classifications; however, it is well-known that morally they behave as an elliptic system. In fact, a great deal of the research on the constraint equations explicitly relies on this "morally elliptic" structure. For this reason, we first discuss some preliminary results regarding the Laplace-Beltrami operator on an asymptotically flat manifold with interior boundary. The results in this section are to be entirely expected in light of classical results and their counterparts on asympotically flat manifolds without boundary (cf. [2,8,18]), however, it worthwhile to present them here. It is well-known that while the Laplace operator is not Fredholm on R n when considered as a map H 2 → L 2 , it is in fact an isomorphism between certain weighted Sobolev/Lebesgue spaces (cf. [23]). In this section we discuss some properties of the Laplace-Beltrami operator on an asymptotically flat manifold when Dirichlet boundary conditions are imposed.
Throughout, we let M be a smooth, connected manifold with compact boundary Σ. Further assume that there exists a compact set K ⊂ M ∪ Σ such that the complement M ∪ Σ \ K consists of N connected components, each diffeomorphic to R n minus the closed unit ball, B. For concreteness, we denote these connected components by N i , and the associated diffeomorphisms by φ i : N i → R n \ B. Equip M with a smooth background Riemannian metricg, equal to the pullback of the Euclidean metric to each of these ends. Let r be a smooth function on M such that r(x) = |φ i (x)| on each N i , and 1 2 < r < 2 on K. In terms of this background asymptotically flat structure, we define the usual weighted Lebesgue and Sobolev norms, respectively as follows: Stephen McCormick

THE LAPLACE-BELTRAMI OPERATOR ON AN ASYMPTOTICALLY FLAT MANIFOLD WITH INTERIOR BOUNDARY
Norms of sections of bundles are defined in the usual way. Note that our convention follows [2], where δ directly indicates the asymptotic behaviour; that is, u ∈ L p δ behaves as o(r δ ) near infinity. We denote the completion of the smooth compactly supported functions with respect to these norms, by L p and W k,p δ . Note that W k,p δ is a space of functions that vanish on the boundary in the trace sense, along with their first k − 1 derivatives. We use W k,p δ to denote the completion of the smooth functions with bound support, and also use the convention H k δ = W k,2 δ and H 2 δ = W k,2 δ . It is well-known that weighted versions of the usual Sobolev-type inequalities hold for these norms; see, for example, Theorem 1.2 of [2]. While these inequalities are generally considered on manifolds without boundary, it is obvious that the proofs remain valid when a boundary is present. One can easily check this, as the proof in [2] relies only on splitting the norms into integrals over annular regions, rescaling the integrals to integrals over an annulus of fixed radius, then applying the usual local inequalities. The reader is referred to [8,18] for more results pertaining to these weighted spaces.
In terms of these weighted Sobolev spaces, we make precise the notion of asymptotically flat manifolds considered here.
Definition 2.1. An asymptotically flat manifold with N ends and interior boundary, is a manifold M , satisfying the properties described above, equipped with a Riemannian metric g satisfying (g −g) ∈ W 2,k 5/2−n , for some k > n/2.
Note that the condition k > n/2 ensures that the metric is Hölder continuous, by the Sobolev-Morrey embedding.
By comparison to the Laplacian on a bounded domain, it is expected that boundary conditions must be enforced if we hope for ∆ g , the Laplace-Beltrami operator associated with g, to be an isomorphism. We impose Dirichlet boundary conditions here, however Neumann boundary conditions could easily be used instead (cf. [14]).
First note the following elementary estimate, which follows immediately from Proposition 1.6 of [2].
Proof. First note that ∆ g is asymptotic to the background Laplacian in the sense of [2] (Definition 1.5). Further note that the proof of Theorem 1.10 of [2] remains valid on an asymptotically flat manifold with boundary, so we have the scale-broken estimate, which does indeed suffice to prove Fredholmness. From which, we prove (2.4) using a standard argument. Assume, to the contrary, that there exists a sequence u i such that u i 2,2,δ = 1 and ∆ g u i → 0. Passing to a subsequence, u i converges weakly in H 2 δ and by the weighted Rellich compactness theorem it converges strongly in L 2 0 . Now (2.3) implies u i is Cauchy and therefore converges in H 2 δ . By continuity, we have ∆ g u = 0, and therefore we have a non-trivial element of ker(∆ g ). However, it can be seen directly from the maximum principle that From Lemma 2.3, we establish the following.
Proposition 2.4. For any δ ∈ (2 − n, 0), the map ∆ g : Proof. We simply must prove that ∆ g is surjective, which is achieved by proving the range is closed and ∆ * g has trivial kernel. It is a fairly standard argument to demonstrate that ∆ g has closed range, which is as follows. Take a sequence u i ∈ H 2 δ ∩ H 1 δ (M ) such that φ i = ∆ g u i is Cauchy; that is, any Cauchy sequence in the range. By (2.4), u i is convergent to some u, and by continuity, φ i → ∆ g u. It follows that ∆ g has closed range.
It remains to prove that ∆ * g has trivial kernel. Note that the adjoint here does not equal the formal L 2 adjoint, but rather we interpret the equation ∆ * g v = 0 in the weak sense: , and from this standard elliptic regularity theory implies v ∈ H 2 loc . In particular, for any Ω ⊂⊂ M , we have , and therefore ∆v = 0 on M . It then follows that Since v ∈ L 2 −5/2 is H 2 loc and vanishes on ∂M , v vanishes everywhere, again by the maximum principle.

The phase space
In this section we adapt Bartnik's phase space construction to an asymptotically flat manifold with an interior boundary. In particular, we show that the set of asymptotically flat initial data, with g fixed on the boundary, is a Hilbert submanifold of the phase space. For simplicity, we restrict ourselves to the physically relevant case, n = 3. Several of the results in the case considered here follow by entirely identical arguments as used by Bartnik, so we simply refer to the appropriate places in Ref. [6] for proofs in these instances. In addition to this, many proofs given here involve only small modifications to those given by Bartnik. The constraint map is given by detg is a volume form, and π is related to the second fundamental form K, by π ij = (K ij − g ij tr g K) √ g. For a given energy-momentum source (s, S i ), the constraint equations are Φ(g, π) = (s, S i ); in particular, the vacuum constraints are simply Φ(g, π) = 0. Now let (M,g) be an asymptotically flat manifold as described in Section 2, where g will serve as a background metric. As we are motivated by considering extensions to a given compact manifold with boundary, Ω, one should considerg near Σ as coming from the metric on Ω, which is to be extended. More concretely, one may choose M such that it can be glued to Ω along Σ, andg would then be a smooth extension of the metric on Ω. However, we avoid further discussion on Ω by simply consideringg to be some given background metric. We define the domain and codomain of Φ in terms of weighted Sobolev spaces: where Λ k is the space of k-forms on M , and S 2 and S 2 are symmetric covariant and contravariant 2-tensors on M respectively. The phase space is the set of prospective initial data, F = G × K. The proofs of Proposition 3.1 and Corollary 3.2 of [6] apply directly in the case considered here, and it therefore follows immediately that Φ : F → N is a smooth map of Hilbert manifolds. It is interesting to note that at the time of publication, Bartnik's phase space concerned initial data that was slightly too rough to apply known results on the wellposedness of the Cauchy problem; however, through the positive resolution of the bounded L 2 curvature conjecture, Klainerman, Rodnianski and Szeftel [12] have recently improved the local existence and uniqueness results to the case considered by Bartnik, and indeed the case considered here.
The key to proving that the level sets of Φ are Hilbert submanifolds, is a standard implicit function theorem style argument. As such, we study the linearisation of Φ, which at at a point (g, π) ∈ F, is given by for (h, p) ∈ T (g,π) F. The formal L 2 adjoint is then computed as 6) where (N, X) ∈ N * = L 2 −5/2 (Λ 0 × T M ) and L is the Lie derivative on M . Note that we use the superscript 'F ' for the formal adjoint, rather than ' * ', which we reserve for the true adjoint.
We first give a coercivity estimate for DΦ F (g,π) . It should be noted that this is simply Bartnik's Proposition 3.3 of [6]; however, particularly since there is a minor modification to the proof at the end, there is no harm in presenting the computation here. Furthermore, there is a minor omission in the argument of Bartnik that relies on a local version of this estimate, which we address in the proof of Proposition 3.3. Note that for simplicity of presentation, we write ξ = (N, X), which may be interpreted as a 4-vector in the spacetime.
Proof. We will need to make use of the difference of connections tensor, which is clearly controlled in W 1,2 −3/2 , for g ∈ G. Rearranging (3.5) gives where S is given by From this, we can then write which gives an estimate for ∇ 2 N : Noting that (g, π) is fixed and ξ = (N, X), the standard weighted Sobolev-type inequalities give
By inserting these estimates back into (3.12), we obtain choosing ǫ small enough and applying the interpolation inequality gives Applying the interpolation inequality again and noting ξ 2:M \E R 0 ≤ C ξ 2,0 completes the proof.  [2]). In particular, we have where C is independent of R. However, we do not have the same control on ξ 2,2,−1/2:A R , as the constant in the Poincaré inequality depends on A R .
Note that the true adjoint of the linearised constraint map, DΦ * (g,π) , is only defined in the weak sense, which is why we make the distinction between DΦ * (g,π) and DΦ F (g,π) . In order to study the kernel of DΦ * (g,π) we must first demonstrate that weak solutions to the equation DΦ * (g,π) [ξ] = 0 are sufficiently regular to consider this as a bona fide differential equation. Proof. We first note that local regularity follows directly from Bartnik's proof of Proposition 3.5 in Ref. [6]. The only possible place in Bartnik's proof where the boundary terms may come in to play are in choosing (h, p) supported in some coordinate neighbourhood. Clearly our boundary conditions do not prevent this, so there is no obstruction to applying Bartnik's proof directly. That is, ξ ∈ H 2 loc . In the following, let B R be an open "ball" of radius R; for R > 2, B R := {x ∈ M : r(x) < R}, and define M ǫR := {x ∈ B R : dist(Σ, x) > ǫ}, for some small ǫ. π) [ξ] on M ; that is, the formal adjoint is indeed the true adjoint when (f 1 , f 2 ) ∈ L 2 −5/2 × H 1 −3/2 , as expected.
It remains to demonstrate that ξ satisfies the boundary conditions and exhibits the correct asymptotics. To this end, we introduce a new smooth cutoff function χ ∈ C ∞ c (M ) such that χ ≡ 1 on B R 0 , for some R 0 > 2 and χ = 0 on B 2R 0 . Define χ R (x) = χ(xR 0 /R), so that χ R has support on B 2R . Clearly χ R ξ ∈ W 2,2 −1/2 , therefore Proposition 3.1 gives noting that χ R ξ → ξ in L 2 0 . From this we can show that χ R ξ is uniformly bounded in W 2,2 −1/2 . Obtaining control of χ R ξ 2,2,−1/2 independent of R is the minor omission in Ref. [6] mentioned above, however this is easily resolved as follows.
Almost identically, we have Inserting the estimates above into (3.17) we arrive at Unfortunately we are unable to ensure ∇ 2 ξ 2,−5/2:A R ∇ 2 (χ R ξ) 2,−5/2 , so we can not absorb the last term into the left-hand side of (3.18). Recalling Remark 3.2, we apply the local version of Proposition 3.1 to obtain Finally we obtain the desired uniform bound: It follows that χ R ξ converges weakly to ξ in H 2 −1/2 . Now, since the formal adjoint agrees with the true adjoint, the boundary terms arising from integration by parts necessarily vanish; explicitly (cf. eq. (4.6)), It follows that ξ vanishes on Σ and therefore, Proof. By the implicit function theorem, we simply must demonstrate that DΦ (g,π) is surjective and the kernel splits. The kernel trivially splits with respect to the Hilbert structure, so we simply must prove that DΦ * (g,π) has trivial kernel and DΦ (g,π) has closed range. It is clear from the above, that elements in the kernel of DΦ * (g,π) indeed satisfy DΦ F (g,π) = 0. Once we have this, note that Bartnik's proof of the triviality of ker(DΦ F (g,π) ) relies only on the structure of the equation and the asymptotics assumed 2 -it is entirely unaffected by the inclusion of an interior boundary. Therefore this proof applies here and we simply must prove that DΦ (g,π) is surjective, which is again adapted from Bartnik's arguments to deal with the boundary. The key to making this argument work is the estimate given earlier by Lemma 2.3.
The idea is to consider a restriction of DΦ (g,π) to variations of a particular form, so that the operator becomes elliptic. Then we simply must show that this restricted operator has closed range and finite dimensional cokernel. We consider and the formal adjoint is given by It follows from the proof of Proposition 3.3, that (z, Z) satisfying F * [z, Z] = 0 are H 2 −1/2 and the boundary terms arrising from integration by parts vanish; that is, (z, Z) ∈ H 2 −1/2 ∩ H 1 −1/2 (M ). From Lemma 2.3, it is straightforward to show using the weighted Hölder, Sobolev and interpolation inequalities (cf. eq. (3.42) of [6]), that we have the scale-broken estimate: It is now a standard argument to demonstrate that F has closed range and finite dimensional cokernel (cf. Ref. [8], Theorem 6.3, and Ref. [2], Theorem 1.10). Let (y, Y ) i be a sequence in ker(F ) satisfying (y, Y ) 2,2,−1/2 ≤ 1; that is, a sequence 2 The proof essentially makes use of the asymptotics to show that any element of the kernel must be supported away from infinity, then shows if an element of the kernel vanishes on a portion of a small ball then it vanishes on the entire ball. By covering M with balls of this (fixed) size, and noting M is connected, the conclusion follows. It is clear a boundary has no impact on this argument.
in the closed unit ball in ker(F ). By the weighted Rellich compactness theorem, passing to a subsequence, (y, Y ) i n converges strongly in L 2 0 , which in turn implies via (3.22) that (y, Y ) i n converges strongly in H 2 −1/2 . That is, the closed unit ball in ker(F ) is compact, and therefore ker(F ) is finite dimensional. It follows that the domain of F can be split as H 2 δ ∩ H 1 δ = ker(F ) ⊕ Z, for some closed orthogonal complementary subspace, Z. Now, for (y, Y ) ∈ Z, we prove By the above argument, passing to a subsequence, we have that (y, Y ) i n converges strongly to (y, Y ) ∈ W . By continuity, F [y, Y ] = 0, while (y, Y ) 2,2,−1/2 = 1, implying the intersection of ker(F ) and W is nontrivial. That is, by contradiction, (3.23) holds. An identical argument to that used in the proof of Proposition 2.4 now shows that F has closed range.
Furthermore, since F F has the same form as F , an estimate of the form of (3.22) also holds for (z, Z) ∈ ker(F * ), which implies that ker(F * ) is finite dimensional. Since the range of F is contained in the range of DΦ, we have surjectivity of DΦ and therefore completes the proof.

Critical points of the ADM mass
In [6] Bartnik discusses a result of Corvino, which states that if there exists an asymptotically flat extensions to a compact manifold with boundary, minimising the ADM energy, then it must be a static metric [11]. Specifically, Bartnik argues that it would be more natural to obtain Corvino's result from the Hamiltonian considerations he uses to prove a similar result for complete manifolds with no boundary. Here we give such an argument, considering the mass rather than the energy, and obtain that critical points of the mass functional only occur if the exterior is stationary. Furthermore, if these stationary solutions are sufficiently regular, they must in fact be static black hole exteriors. It should be noted that our set of extensions is larger than the usual set of admissible extensions in the context of the Bartnik mass. In order to ensure the validity of the positive mass theorem, we would also require conditions on the mean curvature of Σ (see [21]); however it is not clear how to modify the arguments here to include mean curvature boundary conditions. The content of this section has also been discussed in a recent note [17] using stronger boundary conditions than considered here, and indeed stronger than the preferred mean curvature boundary conditions mentioned above; however, an analogous analysis to that in Section 3 was not given.
As in the preceding section, we quote Bartnik's results where the proofs require no modifications to this case. Furthermore, the results established here are again based on adapting Bartnik's arguments to deal with the boundary. The results of Section 3 are precisely what is needed for these arguments to work in the case considered here. The energy-momentum covector P µ = (m 0 , p i ) is defined by It is useful to consider the pairing of the energy-momentum vector with some asymoptotic translation, By writing this as scalar-valued flux integral at infinity, we can make sense of this as an integral over M through the divergence theorem. To extend ξ ∞ to a scalar function and vector field over M , we identify ξ 0 ∞ with a constant function and ξ i ∞ with agparallel vector field in a neighbourhood of infinity; that is, we identify ξ ∞ with somẽ ξ, defined near infinity and satisfying∇ξ ≡ 0. We then choose any smooth bounded supported away from Σ and with ξ ref ≡ξ near infinity to represent ξ ∞ . This allows us to write the energy-momentum as Now it should be noted that P is not well-defined everywhere on F; however, it is well-defined on any constraint manifold C(s, S) with (s, S) ∈ L 1 = L 1 −3 . In Section 4 of [6], it is shown that this definition is equivalent to the usual definition of the ADM energy-momentum and is in fact a smooth map on each C(s, S) with (s, S) ∈ L 1 .
It is well known that the mass must be added to the ADM Hamiltonian in order to generate the correct equations of motion [24]. The formal equations of motion arising from the ADM Hamiltonian are indeed the correct evolution equations, however the boundary terms, coming from the integration by parts, correspond to the linearisation of the energy-momentum; that is, for (g, π) ∈ F, the correct Hamiltonian to generate the equations of motion is given by While the separate terms in (4.5) are not well-defined on F, by combining the terms into a single integrand, the dominant terms in each component cancel exactly (cf. [6]). Henceforth, we consider the Hamiltonian to be this regularised one, with the dominant terms canceled. Note that the boundary conditions imposed on ξ are required to ensure that the surface integrals on Σ, due to integration by parts in obtaining the equations of motion, do indeed vanish. This can be seen by considering the following: The surface integrals at infinity are exactly cancelled by the term 16πξ µ ∞ P µ (cf. [6]). In particular, we have for all (g, π) ∈ F, (h, p) ∈ T (g,π) F and ξ ∈ Ξ, The ability to express the variation of the Hamiltonian in this form is precisely what we mean by the statement that the correct equations of motion are generated. In this form, we can interpret the variation of the Hamiltonian density with respect to each of g and π; that is, δH (ξ) δg = DΦ F 1 (g,π) [ξ]. We then can write Hamilton's equations as where t is interpreted as the flow parameter of (N, X) in the full spacetime; this is exactly the Einstein evolution equations. This also motivates a result of Moncrief [22], equating solutions to DΦ F (g,π) [ξ] = 0 with Killing vectors in the spacetime. For this reason, we say an initial data set (g, π) is stationary if there exists ξ, asymptotic to a constant timelike translation, satisfying DΦ F (g,π) [ξ] = 0. It is evident that the Hamiltonian (4.5) has the form of a Lagrange function, where we seek to find extrema of ξ µ ∞ P µ subject to the constraints being satisfied. As such, we need to make use of the following Lagrange multipliers theorem for Banach manifolds (cf. Theorem 6.3 of [6]).
Theorem 4.1. Suppose K : B 1 → B 2 is a C 1 map between Banach manifolds, such that DK u : T u B 1 → T K(u) B 2 is surjective, with closed kernel and closed complementary subspace for all u ∈ K −1 (0). Let f ∈ C 1 (B 1 ) and fix u ∈ K −1 (0), then the following statements are equivalent: where , refers to the natural dual pairing.
From this, we prove the following.
Conversely, assuming (ii) holds at some (g,π), it follows from (4.7) that for all (h, p) ∈ T (g,π) F. Then by the definition of H (ξ) , we have for all (h, p) ∈ C(s, S); that is, (i) holds.
Physically, E (ξ ∞ ) is interpreted as the total energy viewed by an observer at infinity, whose worldline is generated by ξ ∞ . So Theorem 4.2 may be interpreted as the statement that critical points of the energy measured by ξ ∞ , correspond to solutions with Killing vectors asymptotic to ξ ∞ .
Further define the total mass, m = . Recall that we have not imposed conditions on the boundary mean curvature; that is, we include initial data for which the positive mass theorem fails. Away from m = 0, this is a smooth function on C(s, S) when (s, S) ∈ L 1 . With this in mind, we have the following corollary of Theorem 4.2.
(ii) (g, π) is a stationary initial data set, whose stationary Killing vector is proportional to P at infinity and vanishes on Σ.
It is worth noting that a Killing vector that is asymptotically constant, must in fact be proportional to P at infinity [7].
Provided g is sufficiently smooth, the stationarity conclusion can in fact be replaced with staticity by the following argument. It is well-known that if a Killing vector field vanishes identically on a closed spacelike 2-surface, then that 2-surface is the bifurcation surface of a bifurcate Killing horizon (see, for example [27]). Furthermore, a result of Chruściel and Wald [10] implies the existence of a maximal spacelike hypersurface in the full spacetime containing the bifurcation surface. Then a staticity theorem of Sudarsky and Wald can be applied [25] (cf. Section 7 of [9]), which states, under the assumption of the existence of a maximal spacelike hypersurface, if the stationary Killing vector generates the horizon, then the solution is static. That is, for the vacuum case, critical points of the mass occur exactly when the solution is the region exterior to a Schwarzschild black hole. It follows that for generic choices ofg on Σ, there are no smooth critical points of the mass functional.
Remark 4.4. The same analysis can be performed with π ≡ 0, considering only the Hamiltonian constraint. In this case, the mass and energy are interchangeable, and we only have the lapse as the Lagrange multiplier. The conclusion from the above analysis is then that critical points of the mass correspond to static solutions, as the Killing vector is necessarily hypersurface orthogonal (cf. Theorem 8 of [11]).

Geometric boundary data
The asymptotic value of the stationary Killing vector field predicted by Theorem 4.2, comes from our choice of ξ ref , which above we chose to be supported away from Σ. However, if we allow ξ ref to be nonzero on Σ then the energy-momentum can no longer be expressed as integrals over M , and expression (4.7) no longer holds. To deal with this, we leave ξ ref unchanged in the definition of P and we introduce ξ Σ = (ξ 0 Σ , 0, 0, 0) with support near Σ and ξ 0 Σ constant on Σ. We then modify the Hamiltonian to allow for ξ ∈Ξ := {ξ : ξ − ξ ref − ξ Σ ∈ H 2 −1/2 ∩ H 1 −1/2 (M )} H (ξ) (g, π) = 16(πξ µ ∞ P µ − ξ 0 Σ m BY (g; Σ)) − M ξ µ Φ µ (g, π), (4.11) where m BY is the Brown-York mass. This is given by where h is the mean curvature of Σ in M and h 0 is the mean curvature of Σ isometrically embedded in R 3 , both computed with respect to the unit normal pointing towards infinty. Note that we could simply replace the term −16πm BY with twice the mean curvature of Σ in M , as the addition of a constant does not change the equations of motion; however, this Hamiltonian is more intuitive as it gives a sensible measure of the energy of the system. In coordinates adapted to Σ, the linearisation of m BY (g; Σ) is given by (cf. (ii) (g, π) is a stationary initial data set, whose stationary Killing vector is proportional to P µ at infinity and (−m 0 , 0, 0, 0) on Σ, with the same constant of proportionality. By choosing different conditions on ξ, both at infinity and on Σ, we will obtain different conditions for solutions to be stationary; essentially, these ideas can be used to find the appropriate condition for the existence of a Killing vector with prescribed boundary conditions. In [15], we use similar ideas to prove that the first law of black hole mechanics gives a condition for stationarity, when the boundary conditions on the Killing vector are inspired by bifurcate Killing horizons. Here we can include the quasilocal generalised angular momentum used in [15] to obtain a similar result, sans the area/surface gravity term (as the metric is fixed on Σ here). One can also infer that E (ξ ref ) has no critical points when ξ ∞ = 0 from the fact that DΦ F has trivial kernel in L 2 −1/2 . That is, one immediately has the expected, or perhaps even obvious, result that the Brown-York mass (equivalently, the mean curvature of Σ) has no critical points.