Spacetime Harmonic Functions and Applications to Mass
Hubert Bray, Sven Hirsch, Demetre Kazaras, Marcus Khuri, Yiyue Zhang
SSPACETIME HARMONIC FUNCTIONS AND APPLICATIONS TO MASS
HUBERT BRAY, SVEN HIRSCH, DEMETRE KAZARAS, MARCUS KHURI, AND YIYUE ZHANG
Abstract.
In the pioneering work of Stern [73], level sets of harmonic functions have been shownto be an effective tool in the study of scalar curvature in dimension 3. Generalizations of this idea,utilizing level sets of so called spacetime harmonic functions as well as other elliptic equations, aresimilarly effective in treating geometric inequalities involving the ADM mass. In this paper, wesurvey recent results in this context, focusing on applications of spacetime harmonic functions to theasymptotically flat and asymptotically hyperbolic versions of the spacetime positive mass theorem,and additionally introduce a new concept of total mass valid in both settings which is encoded ininterpolation regions between generic initial data and model geometries. Furthermore, a novel andelementary proof of the positive mass theorem with charge is presented, and the level set approach tothe Penrose inequality given by Huisken and Ilmanen is related to the current developments. Lastly,we discuss several open problems. Introduction
Euclidean space and hyperbolic space possess the following extremality property: any compactlysupported perturbation to these geometries must somewhere decrease their scalar curvature. Thepresent work will survey and explain recent developments related to the 3-dimensional asymptoti-cally flat and hyperbolic positive mass theorems in mathematical general relativity. Geometricallyspeaking, these theorems refine the aforementioned extremality properties of R n and H n into thenonnegativity of a geometric invariant of manifolds which are merely asymptotic to – as opposed toidentically equal to – the model geometries outside of a compact set.The fact that R n admits no compact perturbations increasing its scalar curvature is known as theGeroch conjecture. By identifying faces of a large coordinate cube in R n , one can deduce this factfrom the non-existence of positive scalar curvature (psc) metrics on the torus T n , first observed bySchoen-Yau [65, 67] in dimensions less than 8 and subsequently established by Gromov-Lawson [35] inall dimensions. Though it is far from obvious, an argument due to Lohkamp [52] shows that the morefar-reaching (Riemannian) positive mass theorem also follows from the statement that the connectedsum of a torus and a closed manifold admits no psc metric. See Gromov’s Four Lectures [34, Section3.3] for further discussion of this and other rigidity phenomena related to scalar curvature.A novel approach to the study of scalar has been initiated by Stern [73], in which a central roleis played by the level sets of harmonic functions. Analogies may be drawn between the use ofsuch level sets and the application of stable minimal hypersurfaces instituted by Schoen and Yau[65, 66], while the harmonic functions themselves may be compared with the harmonic spinors foundin the fundamental work of Gromov and Lawson [35, 36]. The level set technique may be expandedby replacing harmonic functions with solutions to other geometrically motivated elliptic equations,leading to new tools with which to investigate initial data sets for the Einstein equations. The mainidea is as follows. M. Khuri acknowledges the support of NSF Grant DMS-1708798, and Simons Foundation Fellowship 681443. a r X i v : . [ m a t h . DG ] F e b BRAY, HIRSCH, KAZARAS, KHURI, AND ZHANG
Consider an orientable compact 3-dimensional Riemannian manifold (
M, g ) with boundary ∂M ,and let u ∈ C ∞ ( M ) be a function that, for simplicity of discussion, has no critical points and admitsthe boundary ∂M as a level set. Then a modified Bochner identity combined with (two traces of) theGauss equations for level sets of u , yields an expression that may be integrated by parts to produce(1.1) (cid:90) M (∆ u ) |∇ u | dV − (cid:90) ∂M H |∇ u | dA = (cid:90) M (cid:18) |∇ u | |∇ u | + ( R − K ) |∇ u | (cid:19) dV, where R is the scalar curvature of g , K is the Gauss curvature of level sets, and H is the meancurvature of the boundary with respect to the outward pointing normal. This formula leads toseveral well-known results by choosing ∆ u appropriately. In particular, the harmonic prescription∆ u = 0 is sufficient to show that the torus T does not admit a metric of positive scalar curvature[16, 73], and can be used to establish the Bray-Brendle-Neves rigidity of S × S [10, 73], as well asthe Riemannian positive mass theorem [12]. Choosing u to satisfy the spacetime harmonic functionequation(1.2) ∆ u = − (Tr g k ) |∇ u | , where k is a symmetric 2-tensor representing extrinsic curvature of a spacelike slice, gives rise to thespacetime version of the positive mass theorem in the asymptotically flat [39] and asymptoticallyhyperbolic settings [11]. Furthermore, if E denotes a divergence free vector field on M , then theassociated drift Laplacian produces an appropriate equation(1.3) ∆ u = (cid:104)E , ∇ u (cid:105) , for the positive mass theorem with charge. Lastly, the equation governing the level set formulationof inverse mean curvature flow(1.4) ∆ u = |∇ u | + (cid:104)∇ u, ∇|∇ u |(cid:105)|∇ u | , together with a weighted version of (1.1) yields the Penrose inequality as proven by Huisken andIlmanen [43].The goal of this paper is to survey these results, with an emphasis on the applications of spacetimeharmonic functions, and to introduce a new concept of mass via interpolation with model geometries.Furthermore, we discuss connections between spacetime harmonic functions and harmonic spinors,and point out relations with versions of the Jang equation. In addition, a novel and elementaryproof of the positive mass theorem with charge is given utilizing (1.3), and several open questionsare discussed. 2. Background
General relativity gives the best description of gravity known to date. From a mathematical pointof view this theory can be described as the study of Lorentzian 4-manifolds ( M , g ), where the metric g arises as a solution to the Einstein field equations(2.1) Ric − Rg = 8 πT, where Ric and R denote Ricci and scalar curvature, and T is the stress-energy tensor which containsall relevant information concerning the matter fields. It is often assumed that T ( v, w ) ≥ v , w , which simply asserts that all observed energy densitiesare nonnegative, and is known as the dominant energy condition . PACETIME HARMONIC FUNCTIONS AND APPLICATIONS 3
It turns out that for many problems in general relativity, including those addressed here, it isnot necessary to consider the entire spacetime, but rather just a spacelike slice. Therefore we willrestrict attention to initial data sets for the Einstein equations (
M, g, k ), consisting of a Riemannian3-manifold M with metric g and a symmetric 2-tensor k representing extrinsic curvature of theembedding into spacetime. As a consequence of this embedding, the tensors g and k must satisfycompatibility conditions, known as the constraint equations , arising from traces of the Gauss-Codazzirelations(2.2) 2 µ := 16 πT ( n, n ) = R + (Tr g k ) − | k | , J := 8 πT ( n, · ) = div g ( k − (Tr g k ) g ) , where R is the scalar curvature of g , n is the unit timelike normal to M , and µ , J are interpreted asthe matter and momentum density of the matter fields. We note that the dominant energy conditionimplies that µ ≥ | J | , which places significant restrictions on the possible geometry and topology ofan initial data set. For instance, in the time symmetric ( k = 0) and maximal (Tr g k = 0) cases thiscondition yields nonnegative scalar curvature.An initial data set will be referred to as asymptotically flat , if outside a compact set M is the disjointunion of a finite number of ends, and for each end there is a diffeomorphism to the complement of aball ψ : M end → R \ B such that(2.3) ψ ∗ g − δ ∈ C l,α − q ( R \ B ) , ψ ∗ k ∈ C l − ,α − q − ( R \ B ) , for some l ≥ α ∈ (0 , q > . See [50] for the definition of weighted H¨older spaces. TheHamiltonian formulation of general relativity given by Arnowitt, Deser, and Misner (ADM) [4] givesrise to the total energy E and linear momentum P of each end. It will be assumed that the energyand momentum densities are integrable µ, J ∈ L ( M ), so that the ADM quantities are well-defined[6, 22] and given by(2.4) E = lim r →∞ π (cid:90) S r (cid:88) i ( g ij,i − g ii,j ) ν j dA, P i = lim r →∞ π (cid:90) S r ( k ij − (Tr g k ) g ij ) ν j dA, where ν is the unit outer normal to the coordinate sphere S r of radius r = | x | and dA denotes its areaelement. The ADM mass is then the Lorentzian length m = (cid:112) E − | P | of the ADM 4-momentum( E, P ). Under the dominant energy condition µ ≥ | J | , the 4-momentum is timelike unless the sliceoriginates from Minkowski space, in which case it vanishes. This is the content of the spacetimepositive mass theorem. Theorem 2.1.
Let ( M, g, k ) be a 3-dimensional, complete, asymptotically flat initial data set forthe Einstein equations. If the dominant energy condition is satisfied, then E ≥ | P | in each end.Moreover, E = | P | in some end only if E = | P | = 0 and the initial data arise from Minkowski space. Initial versions of this result were obtained by Schoen and Yau [67, 68, 69] as well as by Witten [62,76] in the early 1980’s. The non-spacelike nature of the 4-momentum was established by Witten, withthe hypersurface Dirac operator and a generalized Lichnerowicz formula, along with an outline for therigidity statement. The case of equality was investigated further by Ashtekar and Horowitz [5], andYip [78], with a full proof given by Beig and Chru´sciel [8] under the asymptotic assumption µ, | J | = O ( | x | − q − / ). The Schoen and Yau approach treated the time-symmetric case first [67], when k = 0,with stable minimal hypersurfaces, and then reduced the general case to this situation by employingthe Jang equation [68]. This technique proved that E ≥ g k = O ( | x | − q − / ) [30]. Alternatively,Eichmair, Huang, Lee, and Schoen [31] generalized the minimal hypersurface strategy by employing BRAY, HIRSCH, KAZARAS, KHURI, AND ZHANG stable marginally outer trapped surfaces (MOTS) to show | E | ≥ | P | . While the case of equality wasnot treated in [31], Huang and Lee [42] show that the rigidity statement follows from the inequalitybetween E and P . Two other strategies have been used for the Riemannian version of the positivemass theorem, namely a Ricci flow proof by Li [51] and the inverse mean curvature flow proof ofHuisken and Ilmanen [43].In the higher dimensional setting, the MOTS approach [31] extends in a straightforward mannerfor dimensions up to and including 7. The Jang deformation has also been generalized by Eichmair[30] to these dimensions as well. Combining this with the rigidity argument of Huang and Lee [42],yields the desired result for dimensions less than 8 without the spin assumption. For spin manifoldsWitten’s method carries over to all dimensions, with the case of equality given by Chru´sciel andMaerten in [26]. In the case of K¨ahler manifolds, Hein and LeBrun [38] prove the Riemannianpositive mass theorem in all dimensions. A compactification argument by Lohkamp [54], analogousto the Riemannian case [52], reduces the inequality between E and P to the nonexistence of initialdata with a strict dominant energy condition on the connected sum of a torus and a compact manifold.Moreover, in the articles [71] and [53] the higher dimensional Riemannian problem is addressed bySchoen and Yau, and Lohkamp respectively. We recommend the book by Lee [50], for a detaileddiscussion of topics related to the positive mass theorem.Asymptotically hyperbolic manifolds appear naturally in general relativity within two contexts.Namely, as asymptotically totaly geodesic spacelike hypersurfaces in asymptotically anti-de Sitter(AdS) spacetimes, as well as asymptotically hyperboloidal spacelike hypersurfaces in asymptoticallyflat spacetimes. Here we will focus on the second type, the quintessential example of which is thetotally umbilic hyperboloid t = √ r in Minkowski space, whose induced metric b = dr r + r σ isthat of hyperbolic 3-space H , where σ is the round metric on S . An initial data set will be referredto as asymptotically hyperboloidal , if outside a compact set M is the disjoint union of a finite numberof ends, and for each end there is a diffeomorphism to the complement of a ball ψ : M end → H \ B such that(2.5) h := ψ ∗ g − b ∈ C l,α − q ( H \ B ) , p := ψ ∗ ( k − g ) ∈ C l − ,α − q ( H \ B ) , ψ ∗ µ, ψ ∗ J ∈ C l − ,α − − (cid:15) ( H \ B )for some l ≥ α ∈ (0 , q > , and (cid:15) >
0. With these asymptotics, a well-defined notion of totalenergy and linear momentum is possible due to Chru´sciel, Herzlich, Jezierski, and (cid:32)L¸eski [24, 25, 60].Namely, for each function V ∈ C ∞ ( H ) consider the mass functional (2.6) H ( V ) = lim r →∞ π (cid:90) S r [ V (div b h − d Tr b h ) + Tr b ( h + 2 p ) dV − ( h + 2 p ) ( ∇ V, · )] ( ν ) dA b , where again ν is the unit outer normal to the coordinate sphere S r . In order to isolate the 4-momentum, the function V should satisfy the ‘static equation’ Hess b V = V b . A basis of solutionsis obtained by restricting the coordinate functions of Minkowski space to the canonical hyperboloid: √ r , and x i , i = 1 , ,
3. The energy, linear momentum, and mass are then given by(2.7) E = H (cid:16)(cid:112) r (cid:17) , P i = H (cid:0) x i (cid:1) , i = 1 , , , m = (cid:112) E − | P | . In an asymptotically flat spacetime, the difference between the hyperboloidal mass and the ADMmass, is related to the amount of mass lost due to radiation.The study of mass for asymptotically hyperbolic manifolds was initiated by Wang [75]. Theasymptotics utilized in [75] are more restrictive, however they are more concrete in that they identifywhich portion of the metric and extrinsic curvature contribute to the mass. An asymptoticallyhyperboloidal initial data set is said to have
Wang asymptotics if τ = 3, and there are symmetric PACETIME HARMONIC FUNCTIONS AND APPLICATIONS 5 m and p on S such that(2.8) ψ ∗ g = dr r + r (cid:16) σ + m r + O ( r − ) (cid:17) , ψ ∗ ( k − b ) = p r + O ( r − ) . The trace Tr σ ( m + 2 p ) is typically referred to as the mass aspect function and gives rise to the4-momentum in this setting(2.9) E = 116 π (cid:90) S Tr σ ( m + 2 p ) dA σ , P i = 116 π (cid:90) S x i Tr σ ( m + 2 p ) dA σ , i = 1 , , , where x i are Cartesian coordinates of R restricted to the unit sphere S . We point out that theasymptotic conditions for the extrinsic curvature are integral to the definition of the energy andlinear momentum. For instance, there exist asymptotically hyperboloidal slices of the Schwarzschild,as well as asymptotically totally geodesic slices of the AdS Schwarzschild, whose induced metrics arethat of hyperbolic space [21, Remark 1.5]. The hyperboloidal version of the positive mass theoremmay be stated as follows. Theorem 2.2.
Let ( M, g, k ) be a 3-dimensional, complete, asymptotically hyperboloidal initial dataset for the Einstein equations with a single end, and l ≥ . If the dominant energy condition issatisfied, then E ≥ | P | . Moreover, if E = 0 and Wang asymptotics are satisfied, then the initial dataarise from Minkowski space. This theorem as stated is established by Sakovich in [64]. Her method derives from a deformationargument of Schoen and Yau [70], which involves solving the Jang equation so that the solutionadmits hyperboloidal asymptotics. The induced metric on the Jang graph is then asymptoticallyflat, with ADM mass that agrees (up to a positive multiplicative factor) with the hyperbolic mass;further generalizations of this deformation procedure have been studied in [20, 19]. The desired resultthen follows from the positive mass theorem in the asymptotically flat setting. The initial proofsby Wang [75], as well as Chru´sciel and Herzlich [24], relied on spinor techniques and are valid in alldimensions n ≥ R ≥ − n ( n −
1) whichis equivalent to the dominant energy condition when k = g (the umbilic case). There is a substantialliterature in which spinor techniques have been applied to the study of hyperbolic mass, namely[26, 27, 55, 77, 79, 80]. Andersson, Cai, and Galloway [2] were able to remove the spin conditionfor dimensions n ≤ Statement of Results
In this section we present the main results concerning the study of mass via level sets of solutionsto elliptic equations. A novel feature of this approach is that explicit lower bounds for the mass areachieved without the need to assume nonnegative scalar curvature, or more generally the dominantenergy condition. This is in contrast to the spinor technique, which although yields explicit lowerbounds, requires the dominant energy condition in order to establish existence of the appropriateharmonic spinor.
BRAY, HIRSCH, KAZARAS, KHURI, AND ZHANG
Consider the master identity (1.1). As discussed in the introduction, different choices for ∆ u inthis formula lead to different geometric inequalities for the mass. We will first examine the choiceof a harmonic function, ∆ u = 0. Recall that for each asymptotically flat end M end there is a socalled exterior region M ext containing M end , that has minimal boundary, and which is diffeomorphicto R with a finite number of disjoint balls removed [43, Lemma 4.1]. Such a region is desired whenapplying (1.1), in order to avoid possible nonseparating sphere level sets of u that can contributeadversely to the mass inequality. Theorem 3.1.
Let ( M, g ) be a smooth complete asymptotically flat Riemannian 3-manifold havingmass m in a chosen end, with exterior region ( M ext , g ) . If u is a harmonic function on the exteriorregion which is asymptotic to an asymptotically flat coordinate function, and satisfies zero Neumannboundary conditions on ∂M ext , then (3.1) m ≥ π (cid:90) M ext (cid:18) |∇ u | |∇ u | + R |∇ u | (cid:19) dV. Consequently, if the scalar curvature R ≥ then m ≥ . Moreover, m = 0 if and only if ( M, g ) isisometric to ( R , δ ) . In the special case that M is diffeomorphic to R , the integral of (3.1) may betaken over all of M . This theorem, which is established in [12], may be generalized to the case in which (
M, g ) hasa boundary of nonpositive mean curvature, where the mean curvature is computed with respect tothe normal pointing towards the asymptotic end in question. These surfaces, as below, are referredto as ‘trapped’ and are connected with gravitational collapse [74]. When such surfaces are present,and the scalar curvature is nonnegative, the proof of (3.1) produces a strict inequality m >
0. Wemention also that an expression for the mass, related to Theorem 3.1, was obtained by Miao in [59].Furthermore, a version of this result for manifolds with corners is given by Hirsch, Miao, and Tsang[40].The use of Neumann boundary conditions ensures that certain boundary integrals vanish. How-ever, an alternate approach is available in which the harmonic functions need not have boundaryconditions prescribed. This may be accomplished with the Mantoulidis-Schoen neck construction[56], whereby the boundary spheres of the exterior region are capped-off with 3-balls of nonnegativescalar curvature to produce a new manifold that is diffeomorphic to R . The function u may thenbe taken to be harmonic on the new manifold of trivial topology.We now consider the spacetime setting. Let Σ denote a 2-sided closed hypersurface in M withnull expansions given by θ ± = H ± Tr Σ k , in which H is the mean curvature of Σ with respect to υ the unit normal pointing towards a designated end. The surface Σ may be viewed as embeddedwithin spacetime, where the null expansions are then the mean curvatures in the null directions υ ± n ;here n is the future pointing timelike normal to the spacelike hypersurface ( M, g, k ). It follows thatphysically these quantities can be interpreted as measuring the rate of change of area of a shell oflight emanating from the surface in the outward future/past direction, and hence are indicators ofthe strength of the gravitational field. A strong gravitational field is associated with an outer orinner trapped surface, that is when θ + < θ − <
0. Furthermore, Σ is referred to as a marginallyouter or inner trapped surface (MOTS or MITS) if θ + = 0 or θ − = 0.In analogy with the time symmetric setting, it is important to control the topology of regularlevel sets for the relevant function u appearing in (1.1). This was previously achieved with thehelp of an exterior region, obtained from identifying the outermost minimal surface with respect to aparticular end. In the spacetime setting it is not known whether an appropriate exterior region, using PACETIME HARMONIC FUNCTIONS AND APPLICATIONS 7 the outermost MOTS/MITS, always exists. In its place we use the notion of a generalized exteriorregion associated with a designated end. More precisely, as shown in [39, Proposition 2.1], for eachend there exists a new initial data set ( M ext , g ext , k ext ) with the following properties. Namely, ithas a single end that agrees with the original, M ext is orientable with a boundary (possibly empty)comprised of MOTS and MITS, and satisfies the homology condition H ( M ext , ∂M ext ; Z ) = 0.The appropriate choice of equation to use in the spacetime context, when applying the primaryidentity (1.1), is given by(3.2) ∆ u + (Tr g k ) |∇ u | = 0 . Solutions of this equation are called spacetime harmonic functions . The left-hand side of (3.2) arisesas the trace along M of the spacetime Hessian (3.3) ¯ ∇ ij u = ∇ ij u + k ij |∇ u | , which indicates some similarity with the hypersurface Dirac operator introduced by Witten [76].Further discussion of spacetime harmonic functions may be found in Section 6. We will say thata spacetime harmonic function u , on a generalized exterior region M ext , is admissible if it realizesconstant Dirichlet boundary data together with ∂ υ u ≤ ( ≥ )0 on each boundary component satisfying θ + = 0 ( θ − = 0), and there is at least one point on each boundary component where |∇ u | = 0. Theexistence of admissible spacetime harmonic functions that asymptote to a given linear function inthe asymptotically flat end is established in [39, Lemma 5.1]. Theorem 3.2.
Let ( M, g, k ) be a smooth complete asymptotically flat 3-dimensional initial data setfor the Einstein equations, with energy E and linear momentum P in a chosen asymptotic end M end .(i) If the dominant energy condition holds, then a generalized exterior region M ext exists whichis associated with M end , and satisfies the dominant energy condition. Let (cid:104) (cid:126)a, x (cid:105) = a i x i be alinear combination of asymptotically flat coordinates of the associated end, with | (cid:126)a | = 1 . If u is an admissible spacetime harmonic function on M ext , asymptotic to this linear function,then (3.4) E + (cid:104) (cid:126)a, P (cid:105) ≥ π (cid:90) M ext (cid:18) | ¯ ∇ u | |∇ u | + 2( µ − | J | ) |∇ u | (cid:19) dV. Consequently E ≥ | P | . Moreover if E = | P | then E = | P | = 0 , and the data ( M, g, k ) arisefrom an isometric embedding into Minkowski space.(ii) Let ( M ext , g, k ) be a generalized exterior region which does not necessarily satisfy the dominantenergy condition. Then (3.4) still holds.(iii) In the special case that M is diffeomorphic to R , the integral of (3.4) may be taken over M . This theorem is established in [39]. A version also holds where the generalized exterior region hasweakly trapped boundary, instead of an apparent horizon boundary. This means that each boundarycomponent satisfies θ + ≤ θ − ≤
0. In this case, if in addition the dominant energy conditionsis valid, then the conclusion is the strict inequality
E > | P | . Each boundary component thus has anontrivial contribution to the mass, and it would be of interest to more accurately determine thisamount.It turns out that spacetime harmonic functions are instrumental in the study of mass in theasymptotically hyperbolic regime as well. As in the asymptotically flat case, these functions mustgrow linearly in the asymptotic end in order to ‘pluck out’ the mass from the identity (1.1). Inorder to find the appropriate model function to which the spacetime harmonic function shouldapproach, consider the linear function (cid:96) = − t + (cid:104) (cid:126)a, x (cid:105) in Minkowski space. When restricted to the BRAY, HIRSCH, KAZARAS, KHURI, AND ZHANG hyperboloid t = √ r , this function satisfies the spacetime harmonic function equation (3.2), infact its spacetime Hessian vanishes. Moreover, the level sets of (cid:96) intersected with the hyperboloidgive horospheres in hyperbolic space H , and ∇ H (cid:96) is a conformal Killing field on hyperbolic space.These properties suggest the asymptote(3.5) v (cid:126)a = − (cid:112) r + (cid:104) (cid:126)a, x (cid:105) , which is defined in any asymptotically hyperboloidal end. Note that the functions x i may be ex-pressed in the asymptotically hyperboloidal coordinate system by using their polar form. Theorem 3.3.
Let ( M, g, k ) be a smooth complete asymptotically hyperboloidal 3-dimensional initialdata set for the Einstein equations, having one end with energy E and linear momentum P . Fix (cid:126)a ∈ R with | (cid:126)a | = 1 , and assume that the second integral homology group H ( M ; Z ) is trivial. Thenthere exists a spacetime harmonic function u which is asymptotic to v (cid:126)a , and (3.6) E + (cid:104) (cid:126)a, P (cid:105) ≥ π (cid:90) M (cid:18) | ¯ ∇ u | |∇ u | + 2( µ − | J | ) |∇ u | (cid:19) dV. Consequently, if the dominant energy condition holds then E ≥ | P | . Moreover if E = 0 , then the data ( M, g, k ) arise from an isometric embedding into Minkowski space. In the special case that k = g , if E = | P | then E = | P | = 0 and ( M, g ) is isometric to hyperbolic 3-space. This result is established in [11]. The topological assumption H ( M ; Z ) = 0 should not be consid-ered as necessary to obtain mass lower bound (3.6). In the general setting where the homology of M is nontrivial, one should pass to an auxiliary initial data set with trivial homology and an end thatis isometric (as initial data) to the original ( M, g, k ). This type of auxiliary data set is an enhancedversion of the generalized exterior region used for Theorem 3.2. The arguments of Theorem 3.3 canthen be carried out on this secondary space to obtain the desired result in full generality. It shouldbe pointed out that our argument in the case of equality, under the umbilic assumption k = g ,does not require the Huang-Jang-Martin result [41], although their theorem does imply the desiredconclusion. We also mention the recent result of Jang and Miao [45] which provides an expressionfor hyperbolic mass computed via horospheres, in the umbilic case.Consider now a Riemannian 3-manifold ( M, g ) augmented by a smooth vector field E representingan electric field. The triple ( M, g, E ) will be referred to as charged asymptotically flat initial data ,if M has the topology of an exterior region with a single asymptotically flat end M end and a finitenumber of asymptotically cylindrical ends, and E ∈ C ,α − q − ( M end ) with q > , α ∈ (0 , Q = lim r →∞ π (cid:90) S r (cid:104)E , υ (cid:105) dA, where υ is the unit outer normal to coordinate spheres S r in the asymptotically flat end. Typically E will be taken to be divergence free, meaning that there is no charge density, and in this case thetotal charge is finite as it is a homological invariant. The model charged asymptotically flat initialdata are time slices of the Majumdar-Papapetrou spacetime (cid:0) R × (cid:0) R \ ∪ Ii =1 p i (cid:1) , g MP (cid:1) where(3.8) g MP = − φ − dt + φ δ, E MP = ∇ log φ, φ = 1 + I (cid:88) i =1 q i r i , with r i the Euclidean distance to each point p i ∈ R . Each such point represents a degenerate blackhole, in the sense that the horizon is not present within the initial data but rather lies at the bottomof the associated asymptotically cylindrical end. The constants q i > PACETIME HARMONIC FUNCTIONS AND APPLICATIONS 9 of each black hole, and the total charge as well as the total mass agrees with (cid:80) Ii =1 q i . In orderto establish a version of the positive mass theorem with charge, we utilize (1.1) with functions u satisfying the equation(3.9) ∆ u − (cid:104)E , ∇ u (cid:105) = 0 . Solutions to this equation will be referred to as charged harmonic functions , and the associated charged Hessian is given by(3.10) ˆ ∇ ij u = ∇ ij u + E i u j + E j u i − (cid:104)E , ∇ u (cid:105) g ij , where u i denote partial derivatives. Observe that the charged (drift) Laplacian in (3.9) arises froma trace of the charged Hessian. Theorem 3.4.
Let ( M, g, E ) be a smooth complete charged asymptotically flat initial data set, havingdivergence free electric field, mass m , and charge Q . There exists a charged harmonic function u on M which is asymptotic to an asymptotically flat coordinate function in M end , and remains boundedalong asymptotically cylindrical ends, such that (3.11) m − | Q | ≥ π (cid:90) M (cid:32) | ˆ ∇ u | |∇ u | + ( R − | E | ) |∇ u | (cid:33) dV. Consequently, if the charged dominant energy condition is satisfied R ≥ | E | , then m ≥ | Q | . More-over, m = | Q | if and only if ( M, g, E ) is isometric (as charged initial data) to the time slice of aMajumdar-Papapetrou spacetime. Physically, the charged dominant energy condition hypothesis may be interpreted as stating thatthe non-electromagnetic matter fields satisfy the dominant energy condition. The inequality m ≥| Q | , known as the positive mass theorem with charge, was first established by Gibbons, Hawking,Horowitz, and Perry [33] using spinorial techniques (see also [7]). Their result allowed for the inclusionof extrinsic curvature k , but did not allow asymptotically cylindrical ends and thus could not treatthe case of equality. Novel features of Theorem 3.4 include the lower bound (3.11) for the difference m − | Q | , which does not rely on an energy condition, as well as a new approach to the rigiditystatement which does not appear to be fully resolved in all cases [28]. Other related results may befound in [1, Theorem 2.1], [46, Theorem 2], and [47, Theorem 2].Theorem 3.4 will be established in Section 8 below. A discussion of the proofs of Theorems 3.2and 3.3 are given in Sections 6 and 7, respectively. The new concept of mass obtained throughinterpolation with model geometries is also given in Section 7. In Section 4 an outline of the proof ofidentity (1.1) is provided, and it is shown how the inverse mean curvature flow approach [43] to theRiemannian Penrose inequality may be placed within the context of the level set methods presentedhere. Lastly, motivation for the spacetime harmonic function equation as well as connections to otherequations are examined in Section 5, while open questions are proposed in Section 9.4. The Level Set Formula
The purpose of this section is to outline the proof of the main identity (1.1). For simplicity,throughout this section it will be assumed that |∇ u | (cid:54) = 0. This restriction is not necessary, howeverit will allow us to display the essence of the argument without including certain technical details.In order to obtain the general result, |∇ u | should be replaced with (cid:112) |∇ u | + ε in what follows, andthen ε taken to zero after an appropriate application of Sard’s theorem and a Kato inequality. Werefer the reader to [39, Section 3] for details. Primary identity.
First recall Bochner’s identity(4.1) 12 ∆ |∇ u | = |∇ u | + Ric( ∇ u, ∇ u ) + (cid:104)∇ u, ∇ ∆ u (cid:105) , and note that this implies(4.2) ∆ |∇ u | = 1 |∇ u | (cid:0) |∇ u | + Ric( ∇ u, ∇ u ) + (cid:104)∇ u, ∇ ∆ u (cid:105) − |∇|∇ u || (cid:1) . Consider now the t -level set Σ t of u . The unit normal to this surface is given by ν = ∇ u |∇ u | , and thecorresponding second fundamental form and mean curvature respectively take the form(4.3) A ij = ∇ ij u |∇ u | , H = 1 |∇ u | (∆ u − ∇ νν u ) . These formulas imply that(4.4) |∇ u | (cid:0) H − | A | (cid:1) = 2 |∇|∇ u || − |∇ u | + (∆ u ) − u ∇ νν u. Combining this with two traces of the Gauss equations(4.5) Ric( ν, ν ) = 12 ( R − K + H − | A | ) , produces(4.6) ∆ |∇ u | = 12 |∇ u | (cid:16) |∇ u | + ( R − K ) |∇ u | + 2 (cid:104)∇ u, ∇ ∆ u (cid:105) + (∆ u ) − u ∇ νν u (cid:17) . Next observe that(4.7) (cid:104)∇ u, ∇ ∆ u (cid:105)|∇ u | − ∆ u |∇ u | ∇ νν u = div (cid:18) ∆ u ∇ u |∇ u | (cid:19) − (∆ u ) |∇ u | , and therefore(4.8) div (cid:18) ∇|∇ u | − ∆ u ∇ u |∇ u | (cid:19) = 12 |∇ u | (cid:16) |∇ u | + ( R − K ) |∇ u | − (∆ u ) (cid:17) . Integrating this over the manifold M whose boundary is a level set of u , yields the desired result(4.9) (cid:90) M (∆ u ) |∇ u | dV − (cid:90) ∂M H |∇ u | dA = (cid:90) M (cid:18) |∇ u | |∇ u | + ( R − K ) |∇ u | (cid:19) dV, where the mean curvature in the boundary integral is with respect to the unit outer normal. It isinteresting to note the vague similarities between equation (4.8) and [49, Lemma 3.2], in particularwhen u is harmonic.4.2. Relation to Geroch monotonicity.
The Penrose inequality [57] is a conjectural relationbetween the mass and horizon area for initial data satisfying the dominant energy condition, and ismotivated by heuristic physical arguments tying it to the grand cosmic censorship conjecture [63].In the asymptotically flat time symmetric case k = 0, assuming nonnegative scalar curvature, therelation states that(4.10) m ≥ (cid:114) A π where A denotes the area of the outermost minimal surface with respect to a particular end. More-over, equality is achieved only for time slices of the Schwarzschild spacetime. These statements wereestablished by Huisken and Ilmanen [43] for a single black hole, and by Bray [9] for multiple blackholes; Bray and Lee [15] extended this to dimensions n ≤
7. The proof by Huisken and Ilmanen relies
PACETIME HARMONIC FUNCTIONS AND APPLICATIONS 11 on monotonicity of the Hawking mass along inverse mean curvature flow. A level set characterizationof the flow was used to overcome singular behavior. A strategy to generalize these strategies to thenon-time symmetric setting may be found in [13, 14, 37].It is natural to expect some connection between the level set approach for the Penrose inequality,and those that are used to study the positive mass theorem. Indeed, consider a smooth ‘weightfunction’ f = f ( u ) defined on the levels of u , and suppose that the maximum and minimum levelsare min M u = T and max M u = T , which are achieved on the level set boundary of M . Thenmultiplying (4.8) by f and integrating by parts produces − (cid:90) Σ T H |∇ u | f dA + (cid:90) Σ T H |∇ u | f dA = (cid:90) T T f (cid:90) Σ t (cid:20) H + | A | + R + 2 |∇| Σ t ν ( u ) | |∇ u | − H ∆ u |∇ u | (cid:21) dAdt − (cid:90) T T πf χ (Σ t ) dt − (cid:90) T T f (cid:48) (cid:90) Σ t H |∇ u | dAdt, (4.11)where χ (Σ t ) is the Euler characteristic of the t -level set. This expression is obtained with the helpof the coarea formula, the Gauss-Bonnet theorem, and (4.3). Now choose u to solve the level setformulation of inverse mean curvature flow equation, and set the weight function as follows(4.12) ∆ u = |∇ u | + (cid:104)∇ u, ∇|∇ u |(cid:105)|∇ u | , f ( t ) = e ( t − T ) / π (cid:114) | Σ T | π = 116 π (cid:114) | Σ t | π . Inserting this into (4.11), integrating by parts once more, and using H = |∇ u | yields m H (Σ T ) − m H (Σ T )= (cid:90) T T (cid:114) | Σ t | π (cid:20) − χ (Σ t ) + 116 π (cid:90) Σ t (cid:18) |∇ Σ t H | H + R + | A | − H (cid:19) dA (cid:21) dt, (4.13)where(4.14) m H (Σ t ) = (cid:114) | Σ t | π (cid:18) − π (cid:90) Σ t H dA (cid:19) is the Hawking mass. If the level sets remain connected, which is valid in exterior regions [43, Lemma4.2], then χ (Σ t ) ≤ Spacetime Harmonic Functions
Consider an asymptotically flat 4-dimensional spacetime ( M , g ) having initial data set ( M, g, k ).The positive mass theorem in the time-symmetric case, Theorem 3.1, was proven [12] using thelevel sets of asymptotically linear harmonic functions on (
M, g ). We are led to the analogue in thespacetime setting by taking into account intuition from the case of equality. More precisely, when themass vanishes the initial data arise from Minkowski space M , and in the time symmetric case thishypersurface is a constant time slice with the relevant harmonic functions given by a i x i where a i , i = 1 , , x i are the standard cartesian coordinates parameterizing Minkowskispace, with i = 1 , , i = 0 representing the time index. Fornonconstant time slices, the most natural generalization is to utilize linear functions (cid:96)(cid:96)(cid:96) = a x + a i x i of all the coordinates in M . The level set foliation within the initial data ( M, g, k ) is then the intersection of these hyperplanes with the slice. In other words, the relevant function is the restrictionof the spacetime linear combination to M . The issue is then to determine a canonical equationinduced on the slice that is satisfied by this function.To find the appropriate equation, let ∇ and ∇∇∇ be the Levi-Civita connections of the slice andspacetime, respectively. Linear functions of the coordinates in Minkowski space have vanishingspacetime Hessian, and thus when restricted to M they must lie in the kernel of the hypersurfacespacetime Laplacian ∆ = g ij ∇∇∇ ij . This suggests that in a general spacetime, we should considerfunctions u ∈ C ( M ) that satisfy(5.1) 0 = ∆u = g ij ( ∇ ij u − k ij n ( u )) = ∆ u − (Tr g k ) n ( u ) on M, where the unit timelike normal to the slice is denoted by n . Observe that ∆u may be consideredas the divergence of ∇∇∇ u | M , using the ambient connection ∇∇∇ acting on sections of the restrictedbundle T M | M . This is similar to the type of harmonic spinor employed by Witten [76], where thehypersurface Dirac operator is defined by the induced connection from M . The equation (5.1) isnot sufficient, however, because it still depends on the spacetime nature of the function, instead ofbeing solely determined by the restriction u = u | M . In order to rectify this problem, a choice for n ( u ) must be made in terms of intrinsic quantities associated with the data. In this pursuit we areguided by the lower bound for mass that arises from the identity (1.1). It turns out that the desiredchoice is for the spacetime gradient ∇∇∇ u to be null, more precisely n ( u ) = −|∇ u | . Going back tothe intuition from Minkowski space, we see that the level sets of the relevant function u consist ofthe intersection of light cones with the slice. With these considerations, we are then motivated todefine the notion of a spacetime harmonic function to be u ∈ C ( M ) solving the spacetime Laplace equation(5.2) ¯∆ u = ∆ u + (Tr g k ) |∇ u | = 0 , which arises from a trace of the spacetime Hessian (5.3) ¯ ∇ ij u = ∇ ij u + k ij |∇ u | . Association with the Jang equation.
Let us generalize the derivation of the spacetimeLaplacian. Recall that at a certain point a choice was made for the normal derivative n ( u ). Thereare other natural choices that can be made besides dictating that ∇∇∇ u be null. For instance, onepossibility is to take(5.4) n ( u ) = − (cid:112) a + |∇ u | for some constant a ∈ R , assuming that the quantity inside the square root is nonnegative. Thenthe generalized spacetime harmonic function equation becomes(5.5) ∆ u + (Tr g k ) (cid:112) a + |∇ u | = 0 . To gain intuition concerning the level sets of solutions, consider again the example of initial data(
M, g, k ) for Minkowski space. The linear function (cid:96)(cid:96)(cid:96) , by virtue of having a vanishing Hessian, willsatisfy the generalized spacetime harmonic function equation as long as the coefficients a i are chosento satisfy (5.4). Since(5.6) − a + (cid:88) i =1 a i = |∇∇∇ (cid:96)(cid:96)(cid:96) | = − n ( (cid:96)(cid:96)(cid:96) ) + |∇ (cid:96)(cid:96)(cid:96) | , PACETIME HARMONIC FUNCTIONS AND APPLICATIONS 13 we find that (5.4) is satisfied whenever(5.7) a = a − (cid:88) i =1 a i . It follows that the hyperplanes (cid:96)(cid:96)(cid:96) = const are timelike when a >
0, spacelike when a <
0, and nullwhen a = 0. Level sets of the generalized spacetime harmonic function, arising from the restriction of (cid:96)(cid:96)(cid:96) to the slice, are obtained by intersecting M with these hyperplanes. Thus, we find that the foliationwill always be smooth when a ≥
0, as M is a spacelike hypersurface. Based on these observations,it is reasonable to call solutions of (5.5) timelike, spacelike, or null spacetime harmonic functions depending on whether a > a <
0, or a = 0 respectively. Note that such null spacetime harmonicfunctions agree with the original definition of spacetime harmonic functions given at the beginningof the section.As discussed in Section 3, null spacetime harmonic functions play an important role in a proof ofthe spacetime positive mass theorem. It should also be pointed out that timelike spacetime harmonicfunctions, in particular with a = 1, have previously been derived in a different context and appliedto another type of geometric inequality involving mass. Namely, in [17] (see also [18]) the authorsgeneralize the Jang equation, utilized in the Schoen and Yau proof of the spacetime positive masstheorem [68], to be suitable for application to mass-angular momentum inequalities. In short, theJang equation was generalized to include a lapse and shift, so that it can be used to recognizeinitial data from stationary spacetimes, such as Kerr. In this setting the initial data are taken to beaxisymmetric, however as noted in Theorem 2.3 of [17] if the quantity Y ≡ f − (Tr g k ) (cid:112) |∇ f | = 0 . This may be recognized as the generalized spacetime harmonic function equation with a = 1, byreplacing k with − k . Note that the sign associated with k is immaterial as it simply representsthe direction (future or past) of the timelike normal used to compute the extrinsic curvature. Thisequation may be viewed in yet another way. Recall that the classical Jang equation was derivedby taking the trace of the difference of second fundamental forms, namely, the second fundamentalform (cid:0) |∇ f | (cid:1) − / ∇ ij f of the graph t = f ( x ) in the product manifold ( R × M, dt + g ) and thegiven extrinsic curvature k ij . Traditionally the trace was taken with respect to the induced metric¯ g = g + df on the graph, whereas in (5.8) the trace is taken with respect to g .5.2. Association with harmonic spinors.
Here we would like to draw a comparison betweenspacetime harmonic functions and spinorial approaches to mass is general relativity. To this end,consider the following characterization of the spacetime Hessian. Let n denote a normal covectorfield to M of length −
1, and define a map T ∗ M → T ∗ M | M by sending a covector α to the nullcovector I ( α ) := α + | α | n . As above, the Levi-Civita connection on M will be denoted by ∇∇∇ . Thenthe tangential components of ∇∇∇ I ( du ) of a spatial function u coincide with the spacetime Hessian¯ ∇ u .In spinor-based approaches to the mass of 3-dimensional initial data sets [62, 76], one considersthe bundle S of SL ( C ) (Weyl) spinors associated to the restricted bundle T M | M . The spinorbundle S inherits two connections: an intrinsic connection ∇ and one induced by ( M , g ). Usingthe later connection, one constructs the so-called Witten-Dirac operator ¯ /∂ . To analyze the mass of( M, g, k ), spinors solving ¯ /∂ψ = 0 and converging to a constant spinor at spatial infinity are studied.
Clearly, there is an analogy one can make between this approach and the techniques described inthe present paper – the spinor ψ corresponds to the 1-form I ( du ), and the Witten-Dirac harmonicequation corresponds to the spacetime harmonic equation.To make this relationship more concrete, observe that a given Weyl spinor ψ can be ‘squared’ toa null 1-form α defined by(5.9) α ( X ) = Im( X · ψ, ψ )for any vector X ∈ T M , where ( , ) denotes the Hermitian pairing on S and · represents Cliffordmultiplication. A direct calculation reveals(5.10) Tr g ( ∇∇∇ α ) = Im(¯ /∂ψ, ψ ) . In particular, if ¯ /∂ψ = 0 and α = I ( du ) for a spatial function u , then u is spacetime harmonic.Informally, one can interpret a spacetime harmonic function as the integral of the square of a harmonicspinor. It is interesting to note that the level sets of u are obscured from the perspective of ψ , yetthey play a central role in our methods.6. The Spacetime Positive Mass Theorem
Consider the existence problem for asymptotically linear spacetime harmonic functions on asymp-totically flat initial data. Although this is a nonlinear equation, the nonlinearity is relatively mild asit appears only in the first derivatives and is homogeneous of degree 1. For instance, the maximumprinciple still applies to the spacetime Laplace equation. Thus, we expect traditional existence re-sults, mimicking those for linear equations with vanishing zeroth order term, but with a cap on theamount of regularity in general.Let (
M, g, k ) denote a smooth complete asymptotically flat initial data set with boundary ∂M (which may be empty), having a single end; the case of multiple ends may be treated similarly.Given a linear function a i x i defined in the asymptotic end, with (cid:80) i a i = 1, it is convenient to firstconstruct an approximate solution in the asymptotic end that extends to all of M . More precisely,by slightly generalizing [6, Theorem 3.1] we may solve the asymptotically linear Dirichlet problemfor Poisson’s equation(6.1) ∆ v = − Tr g k on M, (6.2) v = 0 on ∂M, v = a i x i + O ( r − q ) as r → ∞ , where q > is as in (2.3), r = | x | , and O indicates in the usual way additional fall-off for eachderivative taken up to order 2. Next let M r be a sequence of exhausting domains as r → ∞ , inwhich each member of the sequence consists of all points inside a large coordinate sphere S r in theasymptotic end. The Dirchlet problem for the spacetime harmonic function equation may be solvedin M r via the Schauder fixed point theorem, with the solution u r prescribed to be v on S r and agiven fixed smooth function c on ∂M . Uniform estimates for the difference u r − v may then beobtained by constructing a barrier function with the appropriate decay. By then passing to a limit,the desired solution u ∈ C ,α ( M ) is found [39, Section 4] for the boundary value problem(6.3) ∆ u + (Tr g k ) |∇ u | = 0 on M, (6.4) u = c on ∂M, u = v + O ( r − q ) as r → ∞ . PACETIME HARMONIC FUNCTIONS AND APPLICATIONS 15
In the next stage of the argument, the solution to (6.3), (6.4) is taken on a generalized exteriorregion M ext , with the inner boundary condition c chosen to be appropriate constants on each com-ponent of the apparent horizon boundary. This solution is then inserted into the primary identity(1.1), or rather a slightly expanded version of it [39, Proposition 3.2], on the exhausting domain M r ⊂ M ext to produce(6.5) (cid:90) ∂ (cid:54) =0 M r ( ∂ υ |∇ u | + k ( ∇ u, υ )) dA ≥ (cid:90) uu (cid:90) Σ t (cid:18) | ¯ ∇ u | |∇ u | + µ + J (cid:16) ∇ u |∇ u | (cid:17) − K (cid:19) dAdt, where ∂ (cid:54) =0 M r is the open subset of ∂M r on which |∇ u | (cid:54) = 0, υ is the unit outer normal, u and u denote the maximum and minimum values of u , and Σ t are t -level sets. The Gauss-Bonnet theoremmay be used to replace the Gauss curvature K with the Euler characteristic of level sets, and thegeodesic curvature of the curves obtained by intersecting Σ t with the outer boundary S r . Note thatthe inner boundary does not contribute a geodesic curvature term, since these boundary componentsare level sets of u . The outer boundary integral of (6.5), together with the Euler characteristic andgeodesic curvature contributions, converges to the 4-momentum expression 8 π ( E + (cid:104) (cid:126)a, P (cid:105) ) as longas the Euler characteristics are not larger than 1. It should be noted that the original computationis carried out for coordinate cubes in the asymptotically flat end, although the corresponding limitagrees with the computation computed on coordinate spheres. This Euler characteristic conditionrequires that there are no spherical level sets, and this follows from the strong maximum principletogether with the topological condition H ( M ext , ∂M ext ; Z ) = 0. More precisely, the constants c prescribed on the boundary are chosen in a particular manner to obtain two properties [39, Lemma5.1]: 1) there exists a point p in each boundary component such that |∇ u ( p ) | = 0, and 2) on eachMOTS (MITS) boundary component ∂ υ u ≤ ( ≥ )0. See Figure 1. ∂M ext (cid:45) u − ( c ) u − (0) u − ( − Figure 1.
Possible level sets of the spacetime harmonic function u near componentsof the boundary ∂M ext .The first property implies, with help from the strong maximum principle, that no regular level setsof u can be homologous to any part of the boundary, the second property guarantees that the innerboundary integral relates to the MOTS and MITS condition(6.6) (cid:90) ∂ (cid:54) =0 M ext ( ∂ υ |∇ u | + k ( ∇ u, υ )) dA = (cid:88) j (cid:90) ∂ j M ext θ ± | υ ( u ) | dA, where the sum is taken over the number of boundary components ∂ j M ext , and the notation θ ± aboveindicates that the integrand contains θ + for a MOTS component and θ − for a MITS component. Itfollows that the boundary integral vanishes, and the desired mass lower bound (3.4) is attained.In the above arguments the topological properties of the generalized exterior region play theimportant role of eliminating undesirable level set topologies. For each end M end of an asymptoticallyflat initial data set satisfying the dominant energy condition, there exists a corresponding generalizedexterior region [39, Proposition 2.1]. To accomplish this there are two primary steps. The first is toidentify suitable (possibly immersed) MOTS and MITS to excise from M in order to obtain a subset M (cid:48) ⊃ M end , whose compactification admits a metric of positive scalar curvature. This is establishedbased on a reorganization of the arguments in [3, Theorem 1.2], and utilizes the dominant energycondition. The goal of the second step is then to reduce the first Betti number of M (cid:48) to zero by aniterative process that involves passing to finite sheeted covers. See Figure 2. ( M, g, k ) ( M ext , g ext , k ext ) Figure 2.
A schematic description of the iterative covering space construction usedto obtain a generalized exterior region. Starting with initial data possessing a non-vanishing first Betti number, pass to a double cover, and then remove an outermostMOTS to reduce the Betti number. This process may require a finite number ofiterations to achieve vanishing first Betti number.7.
The Hyperbolic Positive Mass Theorem
It is natural to expect that the methods utilizing spacetime harmonic functions to establish theasymptotically flat spacetime positive mass theorem, should be applicable in the asymptoticallyhyperboloidal setting as well. Indeed, the model situation in both cases involves a spacelike hyper-surfaces of Minkowski space, one ending at spacelike infinity and the other at null infinity. However,there are several significant technical hurdles that are present in the hyperboloidal framework thatare moot in the asymptotically flat scenario. First, the existence problem for the appropriate asymp-totically linear spacetime harmonic functions is highly nontrivial in the asymptotically hyperboloidalcase. A primary reason for this concerns the nonlinear term (Tr g k ) |∇ u | . Since Tr g k ∼ g k = O ( r − q − ), and therefore plays an important role in determining the asymptotic behavior ofsolutions. Secondly the boundary integrals arising from (6.5) that converge to the mass are extremelydifficult to compute, and require a precise expansion of the solution u . Nonetheless, this complica-tions may be overcome leading to Theorem 3.3, which is established in [11]. Below we discuss inmore detail various aspects of the proof. PACETIME HARMONIC FUNCTIONS AND APPLICATIONS 17
Spacetime harmonic functions in the asymptotically hyperboloidal setting.
In Section5.1 we discussed how null linear functions of the coordinates in Minkowski space, such as (cid:96) = − t + (cid:104) (cid:126)a, x (cid:105) with | (cid:126)a | = 1, restrict to spacelike hypersurfaces to yield canonical spacetime harmonicfunctions. In the asymptotically flat case, where slices are asymptotically totally geodesic, (cid:96) ∼ (cid:104) (cid:126)a, x (cid:105) in the asymptotic end suggesting that this be used as the asymptote for spacetime harmonic functionsin general as in (6.4). In the asymptotically hyperboloidal case(7.1) (cid:96) ∼ − (cid:112) r + (cid:104) (cid:126)a, x (cid:105) = v (cid:126)a , and so it is this function that is used as the asymptote for spacetime harmonic functions in thiscontext. Given a smooth complete asymptotically hyperboloidal 3-dimensional initial set ( M, g, k ),we prove in [11] that there exists u ∈ C ,α ( M ) solving(7.2) ∆ u + (Tr g k ) |∇ u | = 0 on M, u = v (cid:126)a + O ( r − τ | v (cid:126)a | τ ) as r → ∞ , where τ = min (cid:16) , q +12 (cid:17) . As in the previous section, the equation is solved first on finite exhaustingdomains M r , with Dirichlet condition u = v (cid:126)a on the boundary sphere S r , by a fixed point theorem.Then subconvergence to a global solution is obtained as r → ∞ , with the aid of barriers that arequite delicate to construct due to the rather asymmetrical nature of v (cid:126)a . The intricate structure of thebarriers is comparable with the complications faced when solving the Jang equation on asymptoticallyhyperbolic manifolds [64].It turns out that the asymptotics for the solution as expressed in (7.2) are not sufficient to controlthe boundary integrals arising from the identity (1.1). A more careful analysis is needed, and inparticular a precise expansion of several orders is required, which informally is takes the form(7.3) u = v (cid:126)a + | v (cid:126)a | τ r τ (cid:18) A + Br + O ( r − ) (cid:19) where A and B are functions defined on the spherical conformal infinity that are related to one anotherby a certain elliptic PDE. This expression may be compared to the expansion of a harmonic functionsin spherical harmonics. Owing to the nature of the nonlinearities present in the spacetime Laplacian,achieving (7.3) is challenging and technical. Although this expansion can be shown to correctlyidentify the mass from the boundary terms of (6.5), a direct approach is problematic. For thisreason we pursue an alternative approach which utilizes the property that, in several respects, (7.3)improves as the order of asymptotic hyperbolicity q increases. To exploit this, we construct initialdata sets ( ˜ M , ˜ g, ˜ k ) which interpolate between the original initial data set ( M, g, k ), and the modelhyperboloid ( H , b, b ) near infinity. There is an annular interpolation region in ˜ M where geometricalfeatures of ( M, g, k ) such as the dominant energy condition are severely disturbed. Nevertheless, wedemonstrate a sense in which this region remembers the mass of the original initial data set up toan error which shrinks as the interpolation region escapes to infinity.These observations allows us to carry out the following strategy to prove Theorem 3.3. Takingadvantage of the improved expansion for spacetime harmonic functions on ( ˜
M , ˜ g, ˜ k ), it is possible toestablish a version of inequality (3.6) for ( ˜ M , ˜ g, ˜ k ). In this version the boundary integrals converge tozero at infinity, since the asymptotic end is exactly hyperbolic space. We then choose the transitionannular regions in ( ˜ M , ˜ g, ˜ k ) to occur further and further out into the asymptotic end, and show thatthe these bulk transition integrals converge to the mass. Thus, the interpolation regions ‘encode themass’. This will be discussed in more detail within the next subsection. It should be remarked thatthis style of argument seems to be uniquely available to the spacetime harmonic function approach,since it requires the ability to obtain ‘mass formulas’ without the validity of the dominant energy condition as this is violated in the transition to hyperbolic space. Lastly we mention that, as before,the vanishing second homology hypothesis is used to control the Euler characteristic of level sets.7.2. Interpolation and a new concept of mass.
In order to illustrate the concept of interpolationmass , we will give the example of a Schwarzschild metric of mass m > ρ > m/
2, and consider a function m ∈ C ∞ ( R ) satisfying the following properties(7.4) m ( r ) = (cid:40) m r < ρ r > ρ , ≤ m ( r ) ≤ m, ρ | m (cid:48) ( r ) | + ρ | m (cid:48)(cid:48) ( r ) | ≤ C, where C is independent of ρ . This function may be used to define the Riemannian manifold ( R \ B m/ , g ) which is Schwarzschild of mass m for r < ρ , and Euclidean for r > ρ , namely(7.5) g = (cid:18) m ( r )2 r (cid:19) δ. Observe that the scalar curvature takes the form(7.6) R = (cid:40) r ∈ [ m/ , ρ ] ∪ [2 ρ, ∞ ) (cid:0) − r + O ( r − ) (cid:1) m (cid:48)(cid:48) ( r ) r ∈ ( ρ, ρ ) . Let u be a g -harmonic function with zero Neumann data on S m/ , such that u = x + O (1) as r → ∞ ;this may be achieved by a slight modification of [6, Theorem 3.1] and the solution is unique up toaddition of a constant. Then according to Theorem 3.1, and the fact that g is of zero mass, we have(7.7) 0 ≥ (cid:90) R \ B m/ (cid:18) |∇ u | |∇ u | + R |∇ u | (cid:19) dV ≥ (cid:90) B ρ \ B ρ R |∇ u | dV + (cid:90) B ρ \ B m/ (cid:18) |∇ u | |∇ u | + R |∇ u | (cid:19) dV. Treating ρ as a parameter, note that the sequence of metrics g is uniformly asymptotically flat in lightof (7.4), and thus there are uniform estimates such that |∇ u | = 1 + O ( r − ) and | S r | = 4 πr + O ( r ).It follows that(7.8) (cid:90) B ρ \ B ρ R |∇ u | dV = (cid:90) ρρ m (cid:48)(cid:48) ( r ) (cid:18) − r + O ( r − ) (cid:19) (cid:0) πr + O ( r ) (cid:1) dr = − πm + O ( ρ − )as ρ → ∞ . Hence, all the mass is ‘stored’ in the scalar curvature integral over an annulus nearinfinity, and the desired mass formula (3.1) may be obtained by taking the limit in (7.7).To summarize, the mass may be interpreted as measuring the amount of negative scalar curvatureone has to ‘pay’ to deform an asymptotically flat manifold into Euclidean space. In fact, this methodof encoding the mass by interpolation works in the same manner in the non-time symmetric case,where the scalar curvature integral is replaced by a dominant energy condition expression, and it isvalid in the asymptotically hyperboloidal setting as well where it is utilized in the proof of Theorem3.3.7.3. Alternative approach: double interpolation.
In the previous discussion there was a singleinterpolation, either to Euclidean or hyperbolic space, depending on whether the initial data areasymptotically flat or asymptotically hyperboloidal. However, we would like to point out here analternative approach that requires two interpolations. Start with an asymptotically hyperboloidalinitial data set (
M, g, k ). Then perform the first interpolation to achieve the initial data of thehyperboloid in Minkowski space outside a large coordinate sphere, and denote this newly deformeddata by ( M (cid:48) , g (cid:48) , k (cid:48) ). This will of course destroy the dominant energy condition within an annulus A (cid:48) in the asymptotic end, where the mass will be stored. In the second step, interpolate to Euclidean 3-space further out in the asymptotic end to obtain ( M (cid:48)(cid:48) , g (cid:48)(cid:48) , k (cid:48)(cid:48) ). Thus, there are inner and outer annuli PACETIME HARMONIC FUNCTIONS AND APPLICATIONS 19 A (cid:48) and A (cid:48)(cid:48) , in which A (cid:48) encodes the mass but fails the dominant energy condition, while A (cid:48)(cid:48) containsno mass but satisfies the dominant energy condition. Out side of A (cid:48) the data agrees with ( H , b, b ),while outside of A (cid:48)(cid:48) the data agrees with ( R , δ, M (cid:48)(cid:48) , g (cid:48)(cid:48) , k (cid:48)(cid:48) ) is nowasymptotically flat, and hence we may apply the existence result for spacetime harmonic functions[39] to obtain a solution with linear asymptotics. This function may then be used in the mass formula(3.4), following arguments similar to those in Section 7.2, to establish the hyperbolic version of thepositive mass theorem by using the asymptotically flat result. The advantage of this process is that itavoids the complicated computations in the asymptotically hyperbolic end, in which it is shown thatthe boundary integrals from the mass formula converge to the appropriate quantity. On the otherhand, it is more difficult to construct barriers for the spacetime harmonic function on ( M (cid:48)(cid:48) , g (cid:48)(cid:48) , k (cid:48)(cid:48) ),which are independent of the radial parameter determining the location of the two annuli A (cid:48) and A (cid:48)(cid:48) .For this reason, in [11] this strategy is not pursued. Nonetheless, using the interpolation method weare able to prove a special case of the rigidity statement for the hyperbolic positive mass theorem.See for instance [41], where the full result is established. MA (cid:48) H A (cid:48)(cid:48) R Figure 3.
Schematic of the double interpolation.
Proposition 7.1.
Let ( M, g ) be a complete Riemannian 3-manifold with R ≥ − . If, outside acompact set, this manifold is isometric to the compliment of a ball hyperbolic space, then ( M, g ) isglobally isometric to H .Proof. The initial data (
M, g, g ) may be viewed as the end result of a first interpolation, since itagrees with the standard hyperboloid outside a compact set. As described above, there is thenan interpolation ( M (cid:48)(cid:48) , g (cid:48)(cid:48) , k (cid:48)(cid:48) ) between ( M, g, g ) and Euclidean data, which in this case satisfies thedominant energy condition globally. Since this new manifold has zero ADM mass, the spacetimepositive mass theorem, Theorem 3.2, then applies to show that ( M (cid:48)(cid:48) , g (cid:48)(cid:48) , k (cid:48)(cid:48) ) arises from an embeddinginto Minkowski space. Since the interpolated data agrees with the original into its hyperbolic end,there is a portion ( M r , g, g ) lying inside a large coordinate sphere which has an umbilic embeddingin Minkowski space.We claim that the image of the isometric embedding I : ( M r , g ) (cid:44) → M must be a portion of thestandard hyperboloid. To see this consider the ‘center function’ C : M r → M given by(7.9) C ( x ) = I ( x ) − n ( I ( x )) , where n is the unit timelike normal to I ( M r ) and x = ( x , x , x ) are local coordinates. Let T i = ∂ i I ( x ) denote the coordinate vector fields spanning the tangent space along the embedding. Thenthe umbilicity condition yields(7.10) ∂ i C ( x ) = T i − k ji T j = T i − g ji T j = 0 , i = 1 , , . It follows that C ( x ) = C is a constant map, and in fact(7.11) |I ( x ) − C | = − , so that I ( M r ) lies in a Lorentzian sphere, or rather the standard hyperboloid. We may now concludethat ( M, g ) is isometric to H . (cid:3) A Positive Mass Theorem With Charge
In this section we establish Theorem 3.4. Let (
M, g, E ) be a smooth complete charged asymp-totically flat initial data set, having divergence free electric field E , mass m , and total charge Q .In particular, M is diffeomorphic to R \ ∪ Ii =1 p i with a single asymptotically flat end M end and I asymptotically cylindrical ends represented by p i . We say that a subset of M is an asymptoticallycylindrical end if there is a diffeomorphism ψ : M cyl → [1 , ∞ ) × S such that(8.1) | g ∇ l ( ψ ∗ g − g ) | g + | ψ ∗ E − q ∂ s | g = O ( s − ) as s → ∞ for l = 0 , ,
2, where g = ds + σ with σ a metric on S , and q (cid:54) = 0 is a constant that determinesthe charge of the end. This definition is modeled on the asymptotically cylindrical ends present in atime slice of the Majumdar-Papapetrou spacetime.8.1. The basic integral identity.
The first step is to establish the primary integral identity oncompact subdomains Ω ⊂ M . Assume that u satisfies the charged Laplacian equation(8.2) ∆ u − (cid:104)E , ∇ u (cid:105) = 0 on Ω , and recall the charged Hessian(8.3) ˆ ∇ ij u = ∇ ij u + E i u j + E j u i − (cid:104)E , ∇ u (cid:105) g ij which satisfies(8.4) |∇ u | = | ˆ ∇ u | − |E| |∇ u | + (cid:104)E , ∇ u (cid:105) − E i u j ∇ ij u. Inserting equation (8.2) into the main identity (4.8), assuming that |∇ u | (cid:54) = 0, produces(8.5) div (cid:18) ∇|∇ u | − ∆ u ∇ u |∇ u | (cid:19) = 12 |∇ u | (cid:16) | ˆ ∇ u | − E i u j ∇ ij u + |∇ u | ( R − |E| − K ) (cid:17) . Next observe that − |∇ u | E i u j ∇ ij u = − div ( |∇ u |E ) + |∇ u | div E + 1 |∇ u | E i u j ∇ ij u − E i ∂ i |∇ u | = − div ( |∇ u |E ) + |∇ u | div E . (8.6)Combining the last two equations, using the divergence free property of the electric field, and inte-grating over Ω yields(8.7) (cid:90) ∂ Ω (cid:18) ∂ υ |∇ u | − ∆ u ∇ υ u |∇ u | + 2 |∇ u |(cid:104)E , υ (cid:105) (cid:19) dA ≥ (cid:90) Ω (cid:32) | ˆ ∇ u | |∇ u | + ( R − |E| − K ) |∇ u | (cid:33) dV, where υ is the unit outer normal to ∂ Ω. The inequality, rather than equality, arises from theprocedure that passes from the case |∇ u | (cid:54) = 0 to the general setting, see [39, Section 3]. PACETIME HARMONIC FUNCTIONS AND APPLICATIONS 21
Existence of charged harmonic functions and the mass formula.
Consider a 2-parameterfamily of exhausting domains M r,(cid:15) ⊂ M , which are defined to consist of all points lying between alarge coordinate sphere S r in the asymptotically flat end and the collection of coordinate ( s = (cid:15) − cross-section) spheres S (cid:15) in the asymptotically cylindrical ends. In order to better control level settopology, we will cap-off the spheres S (cid:15) with a collection of 3-balls Ω (cid:15) to achieve a new manifold ˜ M r,(cid:15) that is diffeomorphic to a 3-ball. The metric on ˜ M r,(cid:15) is defined to be a smooth extension of g suchthat the curvature of g (cid:15) , the restriction of the extended metric to Ω (cid:15) , remains uniformly bounded. Inaddition, the electric field is extended trivially so that E = 0 on Ω (cid:15) . This unrefined extension of thegeometry and electric field will of course destroy the charged dominant energy condition R ≥ |E| on Ω (cid:15) , however it will be sufficient for our purposes.We first construct a model function v to which the desired charged harmonic function shouldasymptote. Let x be one of the Cartesian coordinates in M end . Since (cid:104)E , ∂ x (cid:105) ∈ C ,α − q − ( M end ), itfollows from [6, Theorem 3.1] that there is a solution of the Dirichlet problem(8.8) ∆ v = (cid:104)E , ∂ x (cid:105) on ˜ M ∞ ,(cid:15) \ ˜ M r ,(cid:15) , (8.9) v = 0 on S r , v = x + O ( r − q ) as r → ∞ , where q is as in (2.3) and r is a large fixed radius. The function v may be extended smoothly sothat it is defined on all of ˜ M ∞ ,(cid:15) , agrees with the solution above for r > r , and vanishes for r < r / w r,(cid:15) − (cid:104)E , ∇ w r,(cid:15) (cid:105) = f on ˜ M r,(cid:15) , w r,(cid:15) = 0 on S r , for r > r where(8.11) f := − ∆ v + (cid:104)E , ∇ v (cid:105) = −(cid:104)E , ∂ x (cid:105) + (cid:104)E , ∂ x + O ( r − q ) (cid:105) = O ( r − q − ) . Note that since E is discontinuous at S (cid:15) , the solution is only guaranteed to be C ,α in a neighborhoodof this surface, although it is smooth everywhere else. By setting u r,(cid:15) = v + w r,(cid:15) , we obtain a solutionof (8.2) on ˜ M r,(cid:15) with u r,(cid:15) = v on S r , and uniform estimates will show that this sequence convergesto the desired charged harmonic function.Uniform C bounds for w r,(cid:15) may be established by constructing a global barrier. More precisely,in light of (8.11) the appropriate uniform sub and super solutions may be found as in [39, Section4.2] with the decay rate r − q in M end . It follows from elliptic theory that there are uniform W ,p estimates on compact subsets, with control of higher order derivatives away from S (cid:15) , and thus w r,(cid:15) subconverges to a solution w (cid:15) . We then have u (cid:15) = v + w (cid:15) ∈ C ,α , and smooth away from S (cid:15) , weaklysatisfying(8.12) ∆ u (cid:15) − (cid:104)E , ∇ u (cid:15) (cid:105) = 0 on ˜ M ∞ ,(cid:15) , u (cid:15) = v + O ( r − q ) as r → ∞ . The barriers imply that u (cid:15) remains uniformly bounded independent of (cid:15) along each capped-off cylin-drical end. Thus, employing a separation of variables argument along each such end shows that forlarge s we have(8.13) u (cid:15) ∼ c + ∞ (cid:88) j =1 c j e (cid:32) q − √ q λj (cid:33) s χ j , for some constants c j where λ j > χ j are the eigenvalues and eigenfunctions of the Laplacian on( S , σ ). It follows that u (cid:15) is nearly constant on S (cid:15) , and since u (cid:15) is harmonic on Ω (cid:15) we find that(8.14) sup Ω (cid:15) |∇ u (cid:15) | → (cid:15) → . We are now in a position to establish the mass formula. In particular, since ˜ M ∞ ,(cid:15) ∼ = R has trivialtopology, no component of a regular level set can be compact as otherwise it would bound a compactdomain leading to a contradiction via the maximum principle. In analogy with [39], we may thenapply (8.7) separately on both components of ˜ M ∞ ,(cid:15) \ S (cid:15) , and add the resulting inequalities together,to obtain 8 π ( m − | Q | ) ≥ (cid:90) ˜ M ∞ ,(cid:15) \ Ω (cid:15) (cid:32) | ˆ ∇ u (cid:15) | |∇ u (cid:15) | + ( R − |E| ) |∇ u (cid:15) | (cid:33) dV + 12 (cid:90) Ω (cid:15) R (cid:15) |∇ u (cid:15) | dV (cid:15) + (cid:90) S (cid:15) |∇ u (cid:15) |(cid:104)E , υ (cid:105) dA, (8.15)where υ is the unit normal to S (cid:15) pointing to the asymptotically flat end, and R (cid:15) , dV (cid:15) are the scalarcurvature and volume form of g (cid:15) . It should be noted that the boundary term ∂ υ |∇ u (cid:15) | , at S (cid:15) , mightappear to cause difficulty due to the fact that u (cid:15) is only C ,α at across this surface. However, preciselythis scenario has been treated in [40], and is shown not to contribute adversely to the inequality.The estimate (8.14), together with the control imposed on the geometry of Ω (cid:15) , guarantees that thetwo integrals on the second line of (8.15) converge to zero. Finally, the uniform estimates for u (cid:15) oncompact subsets shows that this sequence of functions subconverges to a solution u of the chargedLaplacian (8.2) on M , which is bounded along the asymptotically cylindrical ends and satisfies theasymptotics of (8.12) in M end . Moreover, (8.15) and Fatou’s lemma imply that(8.16) m − | Q | ≥ π (cid:90) M (cid:32) | ˆ ∇ u | |∇ u | + ( R − |E| ) |∇ u | (cid:33) dV, which gives the desired inequality of Theorem 3.4.8.3. The case of equality.
When the charged dominant energy condition R ≥ |E| holds, (8.16)implies that m ≥ | Q | . Here we consider the rigidity phenomena when m = | Q | . This immediatelyimplies that(8.17) R = 2 |E| , ∇ ij u + E i u j + E j u i − (cid:104)E , ∇ u (cid:105) g ij = 0 . Taking a divergence of the vanishing Hessian equation produces(8.18) u i R ij − |E| u j + (cid:104)E , ∇ u (cid:105)E j + u i ∇ i E j = 0 . Note that u may be chosen to asymptote to any one of the asymptotically flat coordinate functionsin M end , with the validity of (8.18) unchanged. We will denote the three charged harmonic functionsthat asymptote to x , y , and z by u x , u y , and u z . If ∇ u x , ∇ u y , and ∇ u z are linearly independent ateach point, it will follow that(8.19) R ij − |E| g ij + E i E j + ∇ i E j = 0 . To see that, indeed, these three gradient vector fields always form a basis for the tangent space,assume that there is a point a ∈ M where this is fails. Then there are constants c x , c y , and c z , notall zero, such that(8.20) V = c x ∇ u x + c y ∇ u y + c z ∇ u z PACETIME HARMONIC FUNCTIONS AND APPLICATIONS 23 vanishes at a . Let b ∈ M end be a point sufficiently far out in the asymptotic end, and connect a to b by a geodesic γ . Observe that ∇ ˙ γ V i = c x ( −∇ ˙ γ u x E i − ∇ i u x (cid:104)E , ˙ γ (cid:105) + (cid:104)E , ∇ u x (cid:105)(cid:104) ∂ i , ˙ γ (cid:105) )+ c y ( −∇ ˙ γ u y E i − ∇ i u y (cid:104)E , ˙ γ (cid:105) + (cid:104)E , ∇ u y (cid:105)(cid:104) ∂ i , ˙ γ (cid:105) )+ c z ( −∇ ˙ γ u z E i − ∇ i u z (cid:104)E , ˙ γ (cid:105) + (cid:104)E , ∇ u z (cid:105)(cid:104) ∂ i , ˙ γ (cid:105) ) . (8.21)Therefore if e , e are parallel fields along γ , such that ( e , e , ˙ γ ) forms an orthonormal basis, then ∇ ˙ γ (cid:104) V, ˙ γ (cid:105) = − (cid:104)E , ˙ γ (cid:105)(cid:104) V, ˙ γ (cid:105) + (cid:104)E , e (cid:105)(cid:104) V, e (cid:105) + (cid:104)E , e (cid:105)(cid:104) V, e (cid:105) , ∇ ˙ γ (cid:104) V, e i (cid:105) = − (cid:104)E , e i (cid:105)(cid:104) V, ˙ γ (cid:105) − (cid:104)E , ˙ γ (cid:105)(cid:104) V, e i (cid:105) , i = 1 , . (8.22)This is a first order homogeneous linear system of ODEs for V along γ . Since | V ( a ) | = 0 it followsthat V vanishes along γ , and in particular | V ( b ) | = 0. However, this contradicts the fact that thethree gradient fields are linearly independent in the asymptotically flat end. We conclude that (8.19)is verified.Next, notice that a consequence of (8.19) is that ∇E is symmetric, and hence E is closed as a1-form. Since the first cohomology H ( M ) is trivial, there exists a globally defined function h suchthat E = dh . The function h is harmonic, as E is divergence free, and according to the decay assumedon the electric field and the metric it follows that, after possibly adding a constant, h = c/r + O ( r − q )for some constant c . Therefore, the metric ˜ g = e − h g is asymptotically flat in M end . Moreover, withthe help of (8.19) we find that it is also flat(8.23) ˜ R ij = R ij + ∇ ij h + h i h j + (cid:0) ∆ h − |∇ h | (cid:1) g ij = (∆ h ) g ij = 0 . Consider now the behavior of h along an asymptotically cylindrical end. The decay along this end(8.1) yields h ∼ q s , and therefore each asymptotically cylindrical end is either conformally closedor opened in ˜ g depending on whether q is positive or negative. This may be seen more explicitly bymaking the change of radial coordinate r = e − q s . In the case that the end is conformally closed, ˜ g isasymptotically conical and flat, thus the removable singularity theorem of [72, Theorem 3.1] showsthat ˜ g smooth extends across the singular point. In the case that there are conformally opened endswe arrive at a contradiction, as such an end would give a positive contribution to the mass of M end ,yet since ˜ g is flat the mass of ( M end , ˜ g ) is zero. In particular, each q > g is complete, so that ( M, ˜ g ) ∼ = ( R \ ∪ Ii =1 p i , δ ).We then have g = φ δ where φ = e h . Finally, a computation shows that ∆ δ φ = 0 and φ = O ( r − i )where r i is the Euclidean distance to p i . Since in addition φ → φ = 1 + I (cid:88) i =1 q i r i , for some positive constants q i . The original initial data ( M, g, E ) is then isometric to the time sliceof a Majumdar-Papapetrou spacetime.9. Some Open Questions
Extension to higher dimensions.
One of the primary unresolved issues surrounding thelevel set techniques discussed here, is whether they can be generalized to higher dimensions in ameaningful way. Although independent workarounds for dealing with the singularities of stable minimal hypersurfaces have been found [53, 71], and the spinorial techniques are available in alldimensions for spin manifolds, it would be of significant interest for the study of scalar curvatureto extend the relatively simple harmonic level set approach to obtain an alternate proof of thepositive mass theorem in higher dimensions. One of the primary difficulties is the reliance, in threedimensions, on the Gauss-Bonnet theorem for regular level sets. In higher dimensions, the totalscalar curvature integral appears in its place but is not related to topological information. Perhapssome form of dimensional reduction can be carried in this context, similar to the strategy of Schoenand Yau [65].9.2.
Spacetime Penrose inequality.
Apparent horizons within initial data should contribute tothe ADM mass. The precise relationship expressing this conjecture is known as the Penrose inequal-ity. Namely, if the dominant energy condition is satisfied then the mass should satisfy the lower bound m ≥ (cid:112) A / π , where A is the minimal area required to enclose the outermost apparent horizon, andequality should be achieved only for slices of the Schwarzschild black hole. While this statement hasbeen confirmed [9, 43] in the time symmetric case k = 0, it remains open in general. In this articlewe have seen how to extend level set techniques for the Riemannian positive mass theorem to thespacetime setting, using spacetime harmonic functions. Since the proof of the Riemannian Penroseinequality given by Huisken and Ilmanen [43] is based on a level set characterization of inverse meancurvature flow, it remains an intriguing possibility that a similar extension may be possible thatis based on the level sets of some as yet undiscovered ‘spacetime inverse mean curvature flow’. Apotential candidate for this purpose is the inverse null mean curvature flow studied by Moore [61].9.3. Other geometric inequalities.
We have seen in this article how choosing different expressionsfor ∆ u , in combination with the primary identity (1.1), has led to several mass related inequalities.Surely there are more choices for equations to be found, with further applications. In particular,with each model stationary electro-vacuum black hole solution, there is a corresponding conjecturedgeometric inequality relating the total mass to the area of horizons, angular momentum, and chargeof the black holes contained within initial data. These relations are ultimately motivated by thegrand cosmic censorship conjecture [57, 63]. The classical Penrose inequality discussed above is onemember of this family. Very little is known for several of these inequalities, including the Penroseinequality with angular momentum and the hyperbolic Penrose inequality, of which there are twoversions depending on whether the apparent horizon is a minimal surface or a surface of constantmean curvature H = 2. Is it possible that the level set techniques discussed here can be developedfurther to address this larger class of inequalities?9.4. Some enigmatic features.
There are a few aspects of the level set method with spacetimeharmonic functions that are not quite well-understood. As discussed in Section 5, the spacetimeharmonic function equation naturally fits into a larger class of equations that are broken up into threegroups, depending on the causal character of the hyperplanes in Minkowski space which generatethe canonical solutions. The functions used in the proof of the spacetime and hyperboloidal positivemass theorem arise from the ‘null’ class of such equations. While the ‘spacelike’ version seemsto be undesirable due to the possibility of producing irregular slicings even for initial data withinMinkowski space, it is not clear why the ‘timelike’ spacetime harmonic functions should not be asuseful as their null counterparts.Another related aspect that deserves further investigation is the choice of asymptote for the space-time harmonic functions. In both the asymptotically flat and asymptotically hyperboloidal settings,
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Department of Mathematics, Duke University, Durham, NC 27708, USA
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