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A note on mass-minimising extensions Stephen McCormick∗ School of Science and Technology University of New England Armidale, 2351 5 Australia 1 0 2 November 10, 2015 v o Accepted for publication in General Relativity and Gravitation N The final publication is available at Springer via http://dx.doi.org/10.1007/s10714-015-1993-2 9 ] c q Abstract - r g A conjecture related to the Bartnik quasilocal mass, is that the infimum of [ the ADM energy, over an appropriate space of extensions to a compact 3-manifold 5 with boundary, is realised by a static metric. It was shown by Corvino [Comm. v Math. Phys. 214(1), (2000)] that if the infimum is indeed achieved, then it is 5 4 achieved by a static metric; however, the more difficult question of whether or not 0 the infimum is achieved, is still an open problem. Bartnik [Comm. Anal. Geom. 5 0 13(5), (2005)] then proved that critical points of the ADM mass, over the space . of solutions to the Einstein constraints on an asymptotically flat manifold without 1 0 boundary, correspond to stationary solutions. In that article, he stated that it 5 should be possible to use a similar construction to provide a more natural proof of 1 : Corvino’s result. v In the first part of this note, we discuss the required modifications to Bartnik’s i X argument to adapt it to include a boundary. Assuming that certain results con- r cerning a Hilbert manifold structure for the space of solutions carry over to the a case considered here, we then demonstrate how Bartnik’s proof can be modified to consider the simpler case of scalar-flat extensions and obtain Corvino’s result. In the second part of this note, we consider a space of extensions in a fixed conformalclass. Sufficientconditionsaregiventoensurethattheinfimumisrealised within this class. ∗ [email protected] 1 Stephen McCormick 1 INTRODUCTION 1 Introduction 1 The Bartnik mass is often cited as the gold standard definition of a quasilocal mass , if only it were possible to compute for a generic domain. The mass of a domain Ω in some initial data 3-manifold is as taken to be the infimum of the ADM mass over a space of admissible extensions to Ω, satisfying the Einstein constraints. In Ref. [2], where Bartnik first defined the quasilocal mass now bearing his name, it is conjectured that this infimum is realised by a static extension to Ω. In 2000, Corvino proved part of this conjecture (Theorem 8 of Ref. [9]); he proved that if a minimal ADM energy extension exists then it must be static. Note that we differentiate between the energy and the mass – the latter being the absolute value of the energy-momentum four-vector, while the former refers to the component that is orthogonal to the Cauchy surface. It was then shown by Miao [16] that this static extension must also satisfy Bartnik’s geometric boundary conditions; that is, the metric and boundary mean curvature agree on either side of ∂Ω. Later, Bartnik suggested that a variational proof of Corvino’s result, based on extending his work on the phase space [3] to manifolds with boundary, would be more natural. In the first part of this note, we discuss how Bartnik’s analysis may be modified to the case where the data is fixed on the boundary to provide an alternate proof of Corvino’s result. The extensions considered here fix the first derivative of the metric on the boundary, which is a stronger condition than the usual Bartnik data. In the context of the Bartnik conjecture, and in light of Miao’s result, one would like to consider extensions that fix the mean curvature of the boundary while, rather than fixing the first derivative of the metric. It would also be interesting if one can obtain Miao’s result within this framework. However, it is not obvious how to develop the appropriate variational principle in this case; this is to be the subject of future work. In Section 2 we discuss the Hilbert manifold of extensions to be considered, which is essentially Bartnik’s phase space with boundary conditions imposed. In Section 3, we introduce energy, momentum and mass definitions, and demonstrate how Corvino’s result on static extensions can be obtained. Finally, in Section 4, we consider a space of extensions in a prescribed conformal class. We give sufficient conditions to ensure that the infimum is realised within the fixed conformal class. However, as above, the boundary conditions considered here are not appropriate to be of direct significance to the Bartnik mass. It would be interesting to find a larger class of initial data for which a similar argument is possible, and impose Bartnik’s geometric boundary conditions. 1We borrow the phrase “gold standard” from a quote by Hubert Bray in a Duke University press release. 2 Stephen McCormick 2 THE PHASE SPACE 2 The phase space Let be a smooth asymptotically flat 3-manifold with smooth boundary, Σ. We also M assume that has only a single asymptotic end; that is, there exists a compact set M K Σ such that K is diffeomorphic to R3 minus the closed unit ball, φ : K ⊃ M\ M\ → R3 B (1). On K we define˚g to be the pullback of the Euclidean metric via φ, and 0 \ M\ let r be the Euclidean radial coordinate function composed with φ. On K,˚g is extended to be smooth, bounded and positive definite, while r is smooth and bounded between 1 2 and 2. Throughout, “ ” will indicate quantities defined with respect to the background ◦ metric˚g. In order to include the asymptotics and prescribe the data on the boundary, we define the weighted Sobolev spaces, which are equipped with the following norms: 1/p p −δp−3 u r dµ , p < , u =  M| | o ∞ (2.1) k kp,δ  e(cid:16)sRs sup (r−δ u), (cid:17) p = , M | | ∞ k u = ˚ju . (2.2) k kk,p,δ k∇ kp,δ−j Xj=0 p k,p The spaces L and W are defined as the completion of smooth, compactly sup- δ δ ported functions on Σ with respect to these norms. Spaces of sections of bundles M\ k,2 k are defined as usual and we use the standard notation W = H . We also make use of δ δ k,p k the spaces W and H , defined as the completion of smooth functions with bounded δ δ support on . That is, the overline indicates spaces of functions that vanish on the M boundary, in the trace sense. Initial data for the Einstein equations is given by a Riemannian metric g and a contravariant symmetric 2-tensor density π, on a 3-manifold . Motivated by the M Bartnik mass, we are interested in the space of asymptotically flat extensions to a region Ω in a given initial data set ( ˜,g˜,π˜). In the context considered here, an extension to Ω M is an asymptotically flat manifold with boundary Σ that may be identified with ∂Ω M via a diffeomorphism, such that the initial data agrees across the boundary. Let ˚π be some fixed symmetric 2-tensor density that is supported near Σ. We then consider a choice of ˚g, which we are free to specify near Σ, and ˚π as providing our boundary conditions; explicitly, we define the spaces 2 2 3 1 := g S :g > 0,g ˚g H , := π S Λ :π ˚π H , 2 −1/2 −3/2 G { ∈ − ∈ } K { ∈ ⊗ − ∈ } 3 2 where Λ is the space of 3-forms on , and S and S are symmetric covariant and 2 M contravariant tensors on respectively. Initial data (g,π) := on is to be M ∈F G×K M 3 Stephen McCormick 2 THE PHASE SPACE thought of as an extension of some Ω, where˚g and˚π are given by extending gãnd π˜ into ; that is, Ω can be glued to along the boundaries and data on Ω can be extended M M into . However, while this is the motivation for fixing the data on the boundary, we M do not make reference to g˜ and π˜, as ˚g and ˚π may be freely specified. Note that the space imposes both the asymptotics and boundary conditions. F 2 3 ∗ 3 The constraint map, Φ : L (Λ T Λ ), is given by −5/2 F → N ⊂ × M⊗ 1 ij k 2 −1/2 Φ (g,π) = R(g)√g (π π (π ) )g , (2.3) 0 − ij − 2 k k Φ (g,π) = 2 π . (2.4) i k i ∇ The constraint equations are then given by Φ(g,π) = (16πρ,16πj ), where ρ and j are i i the source energy and momentum densities respectively. Bartnik’s work on the phase space relies on the use of weighted Sobolev-type inequalities, most of which remain valid on an asymptotically flat manifold with boundary (see Theorem 1.2 of Ref. [1]), although some care should be taken with the use of the weighted Poincaré inequality. As such, it is straightforward to verify that the majority of Bartnik’s proof, showing the level sets of Φ are Hilbert manifolds (cf. Conjecture 2.1, below), is valid in the case where has a boundary and the initial data has the M boundary conditions imposed by . The place that Bartnik’s proof, when applied to F this case, breaks down is in proving thelinearised constraint map is surjective. In fact, if 2 3 ∗ 3 = L (Λ T Λ ) then this is almost certainly false. Nevertheless, we expect −5/2 N × M⊗ the following conjecture to be true and intend on pursuing a proof of this as part of future work. Conjecture 2.1 (cf. Theorem 3.12 of Ref. [3]). For some , Φ is a smooth map of N Hilbert manifolds and DΦ(g,π) is surjective at each point (g,π) K. It then follows ∈ G× from the implicit function theorem that the level sets of Φ are Hilbert submanifolds of . F That is, the space of possible extensions to a given domain Ω is a Hilbert manifold; we refer to this as the constraint submanifold, and use the notation (ρ,j) = C −1 Φ (16πρ,16πj). It should be noted that such a result is required for the arguments outlined in Section 3. 4 Stephen McCormick 3 STATIC METRIC EXTENSIONS 3 Static metric extensions The total ADM energy and momentum are respectively given by 16πE := ˚gik(˚ g ˚ g )dSj, (3.1) k ij j ik I ∇ −∇ ∞ j 16πp := 2 π dS . (3.2) i ij I ∞ Note that we omit reference to ˚g in writing dS, as the definitions are independent of the asymptotically flat metric used to define the area measure (cf. Lemma 3.3 below, particularly its application in the proof of Proposition 3.2). Note that we have also made a slight abuse of notation here, as π is in fact a density. Often the quantity E is 2 2 called the mass, however we reserve the term mass for the quantity, m = E p ; q −| | we assume the dominant energy condition here to ensure this is real. We are now in a position to discuss critical points of the mass/energy over the space ofextensions,andinparticularshowhowBartnik’sworkiseasilyadaptedtogiveanother proof of Corvino’s result on static metric extensions. Previously, the author considered evolution exterior to a 2-surface [14]; however, the data was not fixed on the boundary so the conclusion is somewhat different. In the context of the static metric extension conjecture, it is more interesting to consider fixed boundary data. It should be emphasised here that this section is to be understood as a discussion, or commentary, on the implications of Bartnik’s earlier work, rather than a new independent work. It is our hope that Conjecture 2.1 is established in the near future, and furthermore that the boundaryconditions can be replaced with Bartnik’s geometric boundary conditions. A variational argument such as this, with boundary conditions directly relevant to the Bartnik mass, would be of significant interest. Proposition 3.1 (cf. Corollary 6.2 of Ref. [3]). Fix (g,π) (ρ,j), where (ρ,j) ∈ C ∈ 1 L . Assume that Conjecture 2.1 holds, and further assume that a weak solution, λ to ∗ 2 2 3 1 DΦ(g,π) [λ] = f for f L (S Λ ) H (S ), is indeed a strong solution; that −5/2 −3/2 2 ∈ ⊗ × ∗ is, if λ satisfies ∈ N λ DΦ(g,π)[h,p] = f (h,p) Z · Z · M M ∗ for all (h,p) T , then DΦ(g,π) [λ] = f. Then if Dm(g,π)[h,p] = 0 for all (g,π) ∈ F (h,p) T (ρ,j), (g,π) is a stationary initial data set. (g,π) ∈ C The assumption that weak solutions imply strong solutions is a result obtained in proving the analogue of Conjecture 1 in [3]; it is expected to be true here too. Note 5 Stephen McCormick 3 STATIC METRIC EXTENSIONS that Conjecture 2.1 and the additional condition regarding weak solutions are precisely the requirements for the arguments given in Sections 5 and 6 of [3] to hold. In fact, Proposition 3.1 follows directly from these arguments if one replaces the function spaces used there with ours, which include boundary conditions. The proof of Proposition 3.2 is essentially the same argument; as such, we only present this and refer the reader to [3] for the proof of Proposition 3.1. Note that Proposition 3.1 above differs from Corvino’s static extension result, which in our framework is essentially Proposition 3.2 below. Let R(g) = R(g)√g, and note that Conjecture 2.1 implies DR(g) : T T g g 0 G → N 3 is surjective, where is the projection of onto the first (Λ ) factor. In the following, 0 N N a static initial data metric is to be taken as a metric g such that there exists a function ∗ N,asymptotic toaconstant, satisfyingDR(g) [N] = 0. Awell-known resultofMoncrief [18] (see also Ref. [5]) implies that the evolution of such an initial data metric is static. In fact, explicit calculation shows N ∗ i j ij ij DR(g) [N] = ( N ∆ N NRic + Rg )√g, ∇ ∇ − g − 2 which reduces to the well-known static vacuum equations (cf. Ref. [9]) when R = 0. It is worth noting that such an N is unique up to scaling, provided the initial data is not flat [17]. Stationary initial data in the context of Proposition 3.1 is to be understood similarly. Proposition 3.2. Fix g ˆ(ρ) = g : R(g) = 16πρ , where ρ L1. Assume that ∈ C { ∈ G } ∈ ∗ Conjecture 2.1 holds, and further assume that a weak solution, λ to DR(g) [λ ] = f 0 0 2 2 3 ∗ for f L (S Λ ), is indeed a strong solution; that is, if λ satisfies −5/2 0 0 ∈ ⊗ ∈ N λ DR(g)[h] = f h 0 Z Z · M M for all h T , then DR(g)∗[λ ] = f. Then if for all h T ˆ(ρ), we have DE(g)[h] = 0, g 0 g ∈ G ∈ C it follows that g is a static initial data metric. Proof. Fix some constant N R, then for N satisfying (N N ) H2 ( ), ∞ ∞ −1/2 ∈ − ∈ M consider the following Lagrange function for the Hamiltonian constraint: L(g;N) = N E(g) NR(g)√g. (3.3) ∞ −Z M Note that this is essentially the Regge-Teitelboim Hamiltonian with the momentum set to zero. While this Lagrange function is well-defined on ˆ(ρ), it is not the case for a C generic g . However, by writing the energy as the volume integral of a divergence ∈ G over (cf. Proposition 4.5 of [3]) the terms combine and the dominant terms cancel M 6 Stephen McCormick 3 STATIC METRIC EXTENSIONS out. In particular, the regularised Lagrange function, L (g;N) = (N N)R(g) reg ∞ Z − M + N (˚gik˚gjl(˚ ˚ g ˚ ˚ g ) ˚g R(g)), (3.4) ∞ k l ij i k jl Z ∇ ∇ −∇ ∇ − M p is well defined on all of , and equal to L(g;N) where the latter is defined. The first G 1 integral clearly converges since ρ L and the second is bound by noting that the ∈ dominant term in R(g) is ˚gik˚gjl(˚ ˚ g ˚ ˚ g ), when expressed in terms of the k l ij i k jl ∇ ∇ − ∇ ∇ background connection (see Proposition 4.2 of [3], and the explicit expression for R(g) 2,2 can be found in Appendix A of [15]). Now we show that if (N N ) W then we − ∞ ∈ −1/2 have ∗ DL (g;N)[h] = h DR(g) [N], (3.5) reg −Z · M ∗ where DR(g) is the formal adjoint of DR(g). This is easily seen by direct calculation, making use of the following Lemma. Lemma 3.3 (Lemma 4.4 of Ref. [3]). Let S be the Euclidean sphere of radius R, E R R be the exterior region to S – the connected component of S containing infinity – R R M\ 1 and A be the annular region between S and S . Suppose u H (E ), then for R R 2R ∈ −3/2 R0 every R R , 0 ≥ 1/2 udS cR u , (3.6) 1,2,−3/2:A I | | ≤ k k R S R where c is independent of R. Note first h DR(g)∗[N] NDR(g)[h] = i(N(˚ hk jh )+h ˚jN hk˚ N)√g, i k ij ij k i · − ∇ ∇ −∇ ∇ − ∇ andthentakingtheintegral ofthisdivergence over resultsinseveralboundaryterms; M those on ∂M vanish due to the boundary conditions. The boundary terms at infinity of the form h˚N are o(r−2), and controlled by ∇ h˚N O(R1/2)sup h N = o(1) k ∇ kL1(SR) ≤ S | |k k2,2,−1/2 R therefore the surface integrals at infinity also vanish. Now, by rewriting i(N(˚ hk jh )) = i((N N )(˚ hk jh )+N (˚ hk jh )) i k ij ∞ i k ij ∞ i k ij ∇ ∇ −∇ ∇ − ∇ −∇ ∇ −∇ 7 Stephen McCormick 3 STATIC METRIC EXTENSIONS we see that the integral of the first term again vanishes, since (N N )˚h O(R1/2)sup N N h = o(1). k − ∞ ∇ kL1(SR) ≤ S | − ∞|k k2,2,−1/2 R We are therefore left with (h DR(g)∗[N] NDR(g)[h]) = (N i(˚ hk jh ))√g. ∞ i k ij Z · − Z ∇ ∇ −∇ M M By making similar use of ˚ and √g √˚g, we establish (3.5), which is valid for all ∇−∇ − h T . g ∈ G We now employ the following theorem of Lagrange multipliers for Banach manifolds (see Theorem 6.3 of [3]). Theorem 3.4. Suppose K :B B is a C1 map between Banach manifolds, such that 1 2 → DK(u) : T B T B is surjective, with closed kernel and closed complementary u 1 K(u) 2 → −1 1 −1 subspace for all u K (0). Let f C (B ) and fix u K (0), then the following 1 ∈ ∈ ∈ statements are equivalent: (i) For all v kerDK , we have u ∈ Df(u)[v] = 0. (3.7) ∗ (ii) There is λ (T B ) such that for all v T B , K(u) 2 u 1 ∈ ∈ Df(u)[v] = λ,DK(u)[v] , (3.8) h i where , refers to the natural dual pairing. h i Let K(g) = R(g) 16πρ, so that ˆ(ρ) = K−1(0) and T ˆ(ρ) = ker(DK(g)), and let g − C C f(g) = L (g;N). reg Thenif g is acritical pointof theADM energy over thespaceof extensions satisfying R(g) = 16πρ, we have DE(g)[h] = 0 for all h ker(DK(g)); that is, (i) in the above ∈ ∗ theorem is satisfied. It follows that there exists λ (T ) , such that K(g) 0 ∈ N ∗ h DR(g) [N] = Df(g)[h] = λDR(g)[h] −Z · Z M M ∗ ∗ forallh T . Thatis,λisaweaksolutiontoDR(g) [λ] = F,whereF = DR(g) [N] g ∈ G − ∈ 2 2 3 L (S Λ ). By assumption, this is a strong solution and we therefore have −5/2 ⊗ ∗ DR(g) [λ+N] = 0; that is, g is a static initial data set, with static potential (λ+N) N at infinity. ∞ → 8 Stephen McCormick 4 ENERGY MINIMISERS IN A FIXED CONFORMAL CLASS It is clear that Theorem 3.4 should imply a converse statement, however little can be said about this without explicitly knowing . N Remark 3.5. If g C3 then an argument of Fischer-Marsden [11] (cf. Ref. [9], ∈ Proposition 2.3) says staticity implies R(g) = 0. In particular, the above result then implies for ρ = 0, any critical points of the mass (subject to the hypotheses holding) 6 3 3 2 should be rougher than C . If the condition g C can be weakened to g H , then loc ∈ ∈ one concludes that there are no critical points of the mass for ρ= 0. 6 4 Energy minimisers in a fixed conformal class A standard approach to simplify the constraint equations is to look for solutions within afixedconformalclass(seeRef. [4]andreferences therein); inthiscase, theHamiltonian constraint becomes elliptic. Here we make use of this simplification by considering the space of extensions to Ω within a given conformal class. Specifically, we consider a fixed 4φ 2 metric g˜ and consider extensions of the form g(φ) = e g˜, with φ H . For −1/2 ∈ G ∈ simplicity, we assume that is diffeomorphic to R3 B (1); that is, we consider the 0 M \ mostnaturalextensions toΩ. Thisaffordsustheuseoftheweighted Poincaré inequality (see, for example, Lemma 3.10 of Ref. [3]). It should be emphasised that the boundary conditions imposed by the condition 2 φ H are too strong to be of direct significance to the Bartnik mass. While −1/2 ∈ motivated by the Bartnik mass, the results in this section are simply of mathematical interest; itisthehopethatsimilarideascanbeusedtoprovetheexistenceofaminimiser in a much larger class of extensions, and therefore gain insight into the computability of the Bartnik mass. One natural candidate for a larger class of extensions would be to consider Brill initial data, using a variation of Dain’s mass functional (see, for example, [10]). 4φ The scalar curvature of g = e g˜ is given by the well-known formula, R(g) = e−4φ(R˜ 8 ˜φ2 8∆˜φ), − |∇ | − where indicates quantities defined with respect to g˜. This allows us to write the ∼ conformal constraint map, Φˆ :H2 ( ) L2 (Λ3 T∗ Λ3), as −1/2 −5/2 M ×K → × M⊗ 1 Φˆ (φ,π) = e2φ (R˜ 8 ˜φ2 8∆˜φ) g˜ g˜ g˜ (πijπkl πikπjl)g˜−1/2 , (4.1) 0 (cid:20) − |∇ | − − ik jl − 2 (cid:21) p Φˆ (φ,π) = 2e4φ g˜ ˜ πkp+4g˜ πkp˜ φ 2g˜ πjp˜ φ . (4.2) i ip k ip k jp i ∇ ∇ − ∇ (cid:16) (cid:17) From this point on, we will raise and lower indices, and consider the weighted Sobolev 9 Stephen McCormick 4 ENERGY MINIMISERS IN A FIXED CONFORMAL CLASS norms, with respect to g˜ rather than g or ˚g. Note that the domain of Φˆ enforces the boundary conditions on (g,π); in particular, the conformal metric g˜ must itself be an extension of Ω, although it need not necessarily satisfy the constraints. Proposition 4.1. For any φ H2 , we have g = e4φg˜ . −1/2 ∈ ∈ G 4φ Proof. Itis clear thate g˜is positive-definite, andusingthestandard weighted Sobolev- type inequalities we have, e4φg˜ g˜ c g˜ ( e4φ 1 + e4φ˜φ 2,2,−1/2 ∞,0 2,−1/2 2,−3/2 k − k ≤ k k k − k k ∇ k + e4φ˜2φ ) 2,−5/2 k ∇ k c g˜ ( e4φ 1 + e4φ ˜φ ). ∞,0 2,−1/2 ∞,0 1,2,−3/2 ≤ k k k − k k k k∇ k 4φ Note that φ is continuous by the Morrey embedding and e 1 < 5φ near infinity, | − | | | 4φ so e 1 < . 2,−1/2 k − k ∞ Proposition 4.2. Assume (φ,π) H2 ( ) satisfies Φˆ (φ,π) = 16πρ, where −1/2 0 ∈ M × K 1 3 ρ L (Λ ( )) is the source energy density. The ADM energy can then be expressed −3 ∈ M as, 1 16πE = 16πE˜ + (8 ˜φ2 R˜) g˜+(πijπ (πk)2)/ g˜+16πe−2φρ , (4.3) Z |∇ | − ij − 2 k M(cid:0) p p (cid:1) where E˜ is the ADM energy of g˜. Proof. First we write E in terms of φ and g˜, 16πE = ˚gike4φ 4˚ (φ)g˜ +˚ g˜ 4˚ (φ)g˜ ˚ g˜ dSj k ij k ij j ik j ik I∞ (cid:16) ∇ ∇ − ∇ −∇ (cid:17) = ˚gik 4˚ (φ)g˜ +˚ g˜ 4˚ (φ)g˜ ˚ g˜ dSj k ij k ij j ik j ik I∞ (cid:16) ∇ ∇ − ∇ −∇ (cid:17) + ˚gik(e4φ 1) 4˚ (φ)g˜ +˚ g˜ 4˚ (φ)g˜ ˚ g˜ dSj. (4.4) k ij k ij j ik j ik I∞ − (cid:16) ∇ ∇ − ∇ −∇ (cid:17) Lemma 3.3 can now be used to control the second integrand in Eq. (4.4), ˚gik(e4φ 1) 4˚ (φ)g˜ +˚ g˜ 4˚ (φ)g˜ ˚ g˜ dSj k ij k ij j ik j ik (cid:12)IS − (cid:16) ∇ ∇ − ∇ −∇ (cid:17) (cid:12) (cid:12) R (cid:12) (cid:12) c e4φ 1 ( g˜ ˚φ + ˚g˜ ) (cid:12) ∞:S ∞:S 1:S 1:S ≤ k − k R k k Rk∇ k R k∇ k R O(R1/2) e4φ 1 ( g˜ ˚φ + ˚g˜ ). ∞:S ∞:S 1,2,−3/2 1,2,−3/2 ≤ k − k R k k Rk∇ k k∇ k 4 Now making use of the continuity and asymptotics of e φ and g˜, the right-hand-side simply becomes o(1) and therefore vanishes as R tends to infinity. Eq. (4.4) now 10