The Penrose inequality for asymptotically locally hyperbolic spaces with nonpositive mass
aa r X i v : . [ m a t h . DG ] O c t The Penrose inequality for asymptotically locallyhyperbolic spaces with nonpositive mass
Dan A. Lee Andr´e NevesJune 13, 2018
Abstract
In the asymptotically locally hyperbolic setting it is possible to havemetrics with scalar curvature ≥ − negative mass when the genusof the conformal boundary at infinity is positive. Using inverse meancurvature flow, we prove a Penrose inequality for these negative massmetrics. The motivation comes from a previous result of P. Chru´scieland W. Simon, which states that the Penrose inequality we prove im-plies a static uniqueness theorem for negative mass Kottler metrics.
The Penrose inequality for asymptotically flat 3-manifolds M with mass m and nonnegative scalar curvature states that if ∂M is an outermost minimalsurface ( i.e. , there are no compact minimal surfaces separating ∂M frominfinity), then m ≥ r A π , where A is the area of ∂M .G. Huisken and T. Ilmanen [12] first proved this inequality for A equal tothe largest area of a connected component of ∂M . H. Bray [3] later provedthe more general inequality described above using a different method, andthis result was later extended to dimensions less than 8 by Bray and thefirst author [4].We are interested in an analog of this theorem for a class of asymptoti-cally locally hyperbolic manifolds which we now define. Definition.
We say that a C i Riemannian metric g on a smooth manifold M is C i asymptotically locally hyperbolic if there exists a compact set K ⊂ and a constant curvature surface ( ˆΣ , ˆ g ), called the conformal infinity of( M, g ), such that M r K is diffeomorphic to (1 , ∞ ) × ˆΣ with the metricsatisfying g = (ˆ k + ρ ) − dρ + ρ ˆ g + ρ − h + Q, where • ˆ k is the constant curvature of ( ˆΣ , ˆ g ); • ρ is the coordinate on (1 , ∞ ); • Q is a C i symmetric two-tensor on M so that | Q | b + ρ | ¯ ∇ Q | b + · · · + ρ i | ¯ ∇ i Q | b = o ( ρ − ) , where b is the hyperbolic metric (ˆ k + ρ ) − dρ + ρ ˆ g and ¯ ∇ are deriva-tives taken with respect to b . We will use the notation Q = o i ( ρ − ) asa convenient abbreviation; • h is a C i symmetric two-tensor on ˆΣ depending on ρ in such a waythat there exists a function µ on ˆΣ, called the mass aspect function ,such that lim ρ →∞ tr ˆ g h = µ , where the convergence is in C i .For the sake of convenience, we assume that ˆ k is 1, 0, or −
1, and in the caseˆ k = 0, we further assume that | ˆΣ | ˆ g = 4 π . (These assumptions simply servethe purpose of normalization.)Finally, we define the mass to be m = ˆΣ µ d ˆ g = 1 | ˆΣ | ˆ g ˆ ˆΣ µ d ˆ g, and we also define ¯ m = sup ˆΣ µ. An important class of asymptotically locally hyperbolic manifolds isgiven by the Kottler metrics (see [8] for instance), which are static met-rics with cosmological constant Λ = − Definition.
Let ( ˆΣ , ˆ g ) be a surface with constant curvature ˆ k equal to 1, 0,or −
1, with area equal to 4 π in the ˆ k = 0 case. Let m ∈ R be large enoughso that the function V ( r ) = r r + ˆ k − mr r m be the largest zero of V , and define themetric g = V − dr + r ˆ g on ( r m , ∞ ) × ˆΣ. Define ( M, g ) to be the metric completion of this Rieman-nian manifold. We say that (
M, g ) is a
Kottler space with conformal infinity ( ˆΣ , ˆ g ) and mass m . Remark. • The Kottler metrics have scalar curvature R = − . • The most familiar situation is when ˆΣ is a sphere (ˆ k = 1) , in whichcase the metric is also called an anti-de Sitter–Schwarzschild metric in the literature. • As long as V ( r ) has a positive largest zero r m , we obtain M = [ r m , ∞ ) × ˆΣ with ∂M = { r m } × ˆΣ as an outermost minimal surface boundary.One can see that these metrics are asymptotically locally hyperbolic byperforming a substitution, in which case one has h = m ˆ g . • In order for V ( r ) to have a positive zero , we must have m > when ˆ k = 0 or ˆ k = − . However, when ˆ k = − , the parameter m need notbe positive but only greater than a critical mass m crit = − √ . • When the largest zero of V ( r ) is exactly , we say that ( M, g ) is a critical Kottler space. When ˆ k = − and m = m crit , the metric g on (0 , ∞ ) × ˆΣ is a two-ended complete Riemannian manifold, with oneend asymptotically locally hyperbolic and the other end asymptotic tothe cylindrical metric dt + ˆ g on R × ˆΣ . When ˆ k = 0 and m = 0 ,the metric g can be written as dt + e t ˆ g on R × ˆΣ after a coordinatechange, and of course, when ˆ k = 1 and m = 0 , the completed metric ( M, g ) is just hyperbolic space. We can now state our main theorem.
Theorem 1.1 (Penrose Inequality for nonpositive mass) . Let ( M , g ) bea C asymptotically locally hyperbolic manifold with ¯ m ≤ and conformalinfinity ( ˆΣ , ˆ g ) , whose genus is g .Assume that R ≥ − , ∂M is an outermost minimal surface, and thereis a boundary component ∂ M of genus g . Then ¯ m ≥ γ r A π (cid:18) − g + A π (cid:19) , (1)3 here A is the area of ∂ M , and γ = (max { , g − } ) / is a topologicalconstant.Furthermore, equality occurs if and only if ( M, g ) is isometric to theKottler space with infinity ( ˆΣ , ˆ g ) and mass ¯ m . If one replaces ¯ m by m and removes the ¯ m ≤ one obtains the natural analog of Huisken and Ilmanen’s Penroseinequality [12] in the asymptotically locally hyperbolic setting. Versions ofthis statement have been conjectured by P. Chru´sciel and W. Simon [8, Sec-tion VI], and in the g = 0 case, by X. Wang [19, Section 1]. The graph caseof the conjecture has recently been established by L. de Lima and F. Gir˜aowhen m ≥ Corollary 1.2.
Let ( M , g ) be a C asymptotically locally hyperbolic man-ifold with conformal infinity of genus g .Assume that R ≥ − , ∂M is an outermost minimal surface, and thereis a boundary component ∂ M of genus g . Then ¯ m > for g = 0 or , ¯ m > − √ for g > . Proof. If g = 0 or g = 1 and ¯ m ≤ | ∂ M | = 0, which is a contradiction.If g >
1, then it is an easily verifiable fact that r x π (cid:16) − g + x π (cid:17) ≥ − ( g − / √ x ≥ . This inequality and Theorem 1.1 imply at once that ¯ m ≥ − √ . If equalityholds then we are in the equality case of Theorem 1.1 and so M must beisometric to a critical Kottler space of mass − √ . But this is impossiblebecause a critical Kottler space has no compact minimal surfaces. (Theyare foliated by strictly mean convex surfaces.)Of course, the g = 0 case of Corollary 1.2 is just the positive masstheorem for asymptotically hyperbolic manifolds, proved by Chru´sciel andM. Herzlich [7] and by Wang [19], for the case of a manifold with minimal Although we are unable to replace ¯ m by m in general, we do obtain a slightly strongerinequality than (1). See Lemma 3.13. g = 1, no boundary, and negative mass.We note that the inverse mean curvature flow technique allows us to agive a new proof of a weakened version of the positive mass theorem forasymptotically hyperbolic manifolds that was mentioned above. Theorem 1.3 (Positive ¯ m theorem for asymptotically hyperbolic mani-folds) . Let ( M , g ) be a complete C asymptotically hyperbolic manifold(with or without a minimal boundary) with scalar curvature R ≥ − . Then ¯ m ≥ . Moreover, if ¯ m = 0 , then ( M, g ) must be hyperbolic space. We prove Theorem 1.1 following the inverse mean curvature flow theorydeveloped by Huisken and Ilmanen in [12]. The general idea is to flow ∂ M outward with speed inversely proportional to the mean curvature and obtaina (weak) flow of surfaces (Σ t ) t ≥ (where Σ = ∂ M ). To each compactsurface one considers its Hawking mass to be m H (Σ) := r | Σ | π (cid:18) − g − π ˆ Σ ( H − (cid:19) , (2)where | Σ | denotes the area of Σ, and H is its mean curvature. Observethat our notation m H (Σ) leaves out the dependence on g . The key propertyof inverse mean curvature flow is that m H (Σ t ) is non-decreasing in time.Therefore, since m H (Σ ) coincides with the right-hand side of the inequalityin Theorem 1.1, the desired result follows iflim t →∞ m H (Σ t ) ≤ ¯ m · (max { , g − } ) / . In the asymptotically locally hyperbolic setting this inequality is subtle,and in fact the second author constructed well-behaved examples (with g =0) where the above inequality does not hold [15]. The central observationof this paper is that this inequality holds if ¯ m ≤
0. The difference betweenour result here and the one in [15] can be traced to the fact that if the massaspect is positive, then a desired inequality goes in the wrong direction,whereas for nonpositive mass aspect, it goes in the right direction. See theend of the proof of Lemma 3.13 to see the exact place where the condition¯ m ≤ Here we define this to mean asymptotically locally hyperbolic with infinity equal tothe round sphere. cknowledgements: The authors would like to thank Piotr Chru´sciel andWalter Simon for bringing this problem to our attention and for their interestin this work. We also thank Richard Schoen for some helpful conversations.
Chru´sciel and Simon proved in [8] that Theorem 1.1 implies a uniquenesstheorem for static metrics with negative mass that we now explain.
Definition.
We say that ( M , g, V ) is a complete vacuum static data setwith cosmological constant Λ = − M is a smooth manifold(possibly with boundary) equipped with a complete C Riemannian metric g and a nonnegative C function V such that ∂M = { V = 0 } and∆ g V = − Λ V (3)Ric( g ) = 1 V Hess g V + Λ g. (4)If ( M , g, V ) is a vacuum static data set as above, then the Lorentzianmetric h = − V dt + g on R × M is a solution to Einstein’s equations withcosmological constant Λ. In the Λ = 0 case a 1987 result of G. Bunting andA. Masood-ul-Alam [6] shows that Schwarzschild spaces are the only asymp-totically flat vacuum static data sets (with the case of connected boundaryoriginally proved by H. M¨uller zum Hagen, D. Robinson, and H. Seifert in1973 [14]).We are interested in a similar characterization of the Kottler metricsdefined in the previous section, which are known to be vacuum static datasets with cosmological constant Λ = −
3. A static uniqueness theorem forthe hyperbolic space was proved in work of Boucher-Gibbons-Horowitz [2],Qing [18], and Wang [20].The following definition is equivalent to the one given in [8, SectionIII.A].
Definition.
Let i ≥
2. We say that a complete vacuum static data set( M , g, V ) with cosmological constant Λ = − C i conformally compactifi-able if g and V are C i and there exists a smooth compact manifold M ′ withboundary and a C i +1 embedding of M into M ′ such that M ′ ∼ = M ∪ ∂ ∞ M , • the function V − extends to a C i function on M ′ with d ( V − ) = 0 at ∂ ∞ M , and • the formula ˆ g = V − g near ∂ ∞ M defines a Riemannian metric.6f ( M, g, V ) is a complete vacuum static data set, the fact that Hess g V = 0on ∂M implies that |∇ V | is constant on each component of ∂M , and we callthis constant the surface gravity κ of that component.The surface gravity is strictly positive for the following reason: Given p ∈ ∂M , let γ be the unit speed geodesic that starts at p perpendicular to ∂M and set f ( t ) = V ( γ ( t )). From the static equation we see the existenceof c > t > f ′′ ≤ cf for all 0 ≤ t ≤ t . Standard o.d.e.comparison shows that if f ′ (0) = f (0) = 0 then f ( t ) ≤ ≤ t ≤ t , which is impossible because V is strictly positive on the interior of M .The (non-critical) Kottler metrics have exactly one component of ∂M and, for fixed ˆ k , there is a bijection between possible surface gravities in(0 , ∞ ) and possible masses in ( − √ , ∞ ) when ˆ k = −
1, or in (0 , ∞ ) when ˆ k is 0 or 1. Therefore, for fixed ˆ k we can define a bijection m ( κ ) according tothe fixed relationship between mass and surface gravity for Kottler metricswhose infinities have curvature equal to ˆ k . For this bijection, when ˆ k = − κ → m ( κ ) = − √ and m (1) = 0 (see [8, Section II] for details).We can now state a static uniqueness theorem for Kottler metrics ofnegative mass, which will follow from Theorem 1.1 combined with some ofthe results of [8] that we will describe later. Theorem 2.1 (Static uniqueness with nonpositive mass) . Let ( M , g, V ) be a complete vacuum static data set with cosmological constant Λ = − ,and assume that it is C conformally compactifiable with conformal infinity ∂ ∞ M of constant curvature ˆ k = − .Assume that there is a component ∂ M of ∂M such that ∂ M is home-omorphic to ∂ ∞ M and ∂ M has the largest surface gravity κ of any compo-nent.If m ( κ ) ≤ (or equivalently κ ≤ ), then ( M, g ) must be isometric tothe Kottler metric with infinity ∂ ∞ M and mass m ( κ ) , while V is equal tothe usual static potential of the Kottler metric, up to a constant multiple. Before we present the proof some comments are in order. • It follows from Lemma 2.4 and the proof of Lemma 3.3 that ∂ M cannever have larger genus than ∂ ∞ M . It would be nice to also rule outthe possibility that ∂ M has strictly smaller genus. • The Horowitz-Myers AdS solitons [11] described in the Introductiondo not only have have negative mass, no boundary, and ˆ k = 0, butthey are also static . A static uniqueness theorem for these exampleswas proved by G. Galloway, S. Surya, and E. Woolgar [10].7 It would be interesting to remove the condition on m ( κ ). The following proposition is a consequence of Theorem I.1 and PropositionIII.7 of [8], together with a coordinate change.
Proposition 2.2.
Let i ≥ . Let ( M , g, V ) be a C i conformally compact-ifiable complete vacuum static data set with cosmological constant Λ = − .Further assume that the induced metric ˆ g on ∂ ∞ M (as defined in the previ-ous section) has locally constant Gauss curvature ˆ k equal to , , or − .Then ∂ ∞ M is connected, ( M, g ) is asymptotically locally hyperbolic (asdefined in the Introduction) with conformal infinity ( ∂ ∞ M, ˆ g ) , and V = ρ + ˆ k − µρ + o ( ρ − ) , where ρ is the coordinate used in the definition of asymptotically locallyhyperbolic, and µ is the mass aspect. In particular, static data sets with Λ = − m and ¯ m .The key theorem of [8] for the purposes of this article is Theorem I.5: Theorem 2.3 (Chru´sciel-Simon) . Let ( M , g, V ) be a C conformally com-pactifiable complete vacuum static data set with cosmological constant Λ = − , conformal infinity ( ∂ ∞ M, ˆ g ) of constant curvature ˆ k = − , and ∂M = ∅ .Let ∂ M denote the boundary component with the largest surface grav-ity κ and suppose m := m ( κ ) ≤ (i.e., < κ ≤ ).If ( M , g , V ) denotes the Kottler space with infinity ( ∂ ∞ M, ˆ g ) and mass m (i.e., the one with surface gravity κ ), then χ ( ∂ M ) | ∂ M | ≥ χ ( ∂M ) | ∂M | g and ¯ m ≤ m , where | ∂M | g is the area with respect to g . In the case where ∂ M has the same genus as ∂ ∞ M , this theorem pro-vides a simple comparison between the masses and boundary areas of avacuum static data set and its so-called reference solution ( M , g , V ). Forthe sake of completeness we provide the proof of this theorem in Section 4.8 emma 2.4. Let ( M , g, V ) be a C asymptotically locally hyperbolic, com-plete vacuum static data set with cosmological constant Λ = − . If ∂M = ∅ ,then ∂M is an outermost minimal surface. In fact, there are no compactminimal surfaces in the interior of M .Proof. First note that the static equations imply that ∂M is totally geodesic,so we need only show that there are no other compact minimal surfaces.Consider the trapped region K of M , which is the union of all compactminimal surfaces in M , together with all regions of M that are boundedby these minimal surfaces. The boundary of the trapped region, ∂K , mustitself be a smooth compact minimal surface. (See the proof of Lemma 4.1(i)of [12].) Following [12], we define the exterior region M ′ of M to be themetric completion of M r K . Thus ( M ′ , g, V ) is a vacuum static dataset, except for the requirement that { x ∈ M ′ | V ( x ) = 0 } = ∂M ′ . Theexterior region M ′ has a strictly outward minimizing minimal boundaryand no interior compact minimal surfaces (see [12]), where strictly outwardminimizing means that | ∂M ′ | is strictly less than the area of any othersurface that encloses it.Suppose that M has a compact minimal surface other than ∂M . Since V > ∂M , it follows from the definition of M ′ that V does notvanish identically on ∂M ′ . We consider the outward normal flow of surfaces(Σ t ) t ≥ with initial condition Σ = ∂M ′ that flows with speed V . Since V ≥ ∂M ′ , this flow is nontrivial. According to theformula for the variation of mean curvature, we see that the mean curvatureof Σ t evolves according to ∂H∂t = − ∆ Σ t V − (Ric( ν, ν ) + | A Σ t | ) V = − (∆ g V − ∇ ν ∇ ν V + h H, ∇ V i ) − ( ∇ ν ∇ ν V − V + | A Σ t | V )= −h H, ∇ V i − | A Σ t | V, where ν is the outward unit normal, and A Σ t is the second fundamentalform. Since V ≥
0, it follows that Σ t must have H ≤ t . Bythe first variation of area formula, Σ t must have area less than or equal tothat of Σ = ∂M ′ . But this contradicts the strictly outward minimizingproperty of ∂M ′ .We can now prove Theorem 2.1 following the description in [8].Let ( M, g, V ) be as in the statement of the theorem. In particular, allof the hypotheses of Chru´sciel-Simon’s Theorem (Theorem 2.3) are satisfied9nd so ¯ m ≤
0. Hence Proposition 2.2 and Lemma 2.4 imply that all of thehypotheses of our Penrose inequality (Theorem 1.1) are also satisfied.Since we are assuming that ∂ M is homeomorphic to ∂ ∞ M , Theorem2.3 tells us that A ≥ A and ¯ m ≤ m , where A is the area of ∂ M and A is the area of ∂M in the referencesolution. Moreover, since the curvature ˆ k of ˆ g is equal to − m ( g − / ≥ r A π (cid:18) − g + A π (cid:19) . (5)It is convenient to define constants r := s A π ( g −
1) and r := s A π ( g − , so that we have r ≥ r . Inequality (5) then becomes¯ m ≥ r ( − r ) = ⇒ m + r − r ≥ . Meanwhile, on the reference space we know that r is the largest root of2 m + r − r = 0 . and so r ≥ √ using elementary reasoning. Thus0 ≤ m + r − r = (2 ¯ m + r − r ) − (2 m + r − r )= 2( ¯ m − m ) + ( r − r ) − ( r − r )= 2( ¯ m − m ) + ( r − r )[1 − ( r + r r + r )] ≤
2( ¯ m − m ) + ( r − r )(1 − r ) ≤ . Therefore all of the inequalities must be equalities and so it follows from therigidity part of Theorem 1.1 that (
M, g ) is isometric to the Kottler spacewith infinity ∂ ∞ M and mass m .It is simple to check that any two static potentials on the Kottler space( M, g ) must be proportional and so V is the usual static potential, up to aconstant multiple. 10 Proof of Theorem 1.1
Assume that ( M , g ) be a C asymptotically locally hyperbolic manifoldwith R ≥ − ∂M is an outermost minimal surface. In order toestablish existence of the weak inverse mean curvature flow, we first need tofind a weak subsolution. Lemma 3.1.
Let ( M , g ) be a C asymptotically locally hyperbolic metricwith radial coordinate ρ as in the definition of asymptotically locally hyper-bolic. There exists r so that for all r ≥ r , Σ − t = { ρ = ( r + 1) e t/ − } and Σ + t = { ρ = ( r − e t/ + 1 } , t ≥ are, respectively, subsolutions and supersolutions for inverse mean curvatureflow with initial condition { ρ = r } . Proof.
We will prove that Σ + t is a supersolution. (The proof for Σ − t issimilar.) Using the asymptotics of g , one can see that the inverse meancurvature of the constant ρ sphere in ( M, g ) is H − = ρ q ˆ k + ρ + O ( ρ − ) . Since that Σ + t is just the constant ρ sphere with ρ = ( r − e t/ + 1, we seethat the speed of the flow is just dρdt | ∂ ρ | g = ( r − e t/ [(ˆ k + ρ ) − / + O ( ρ − )] = ρ q ˆ k + ρ − q ˆ k + ρ + O ( ρ − ) . Clearly, for sufficiently large ρ , this speed is less than H − + t , showing thatΣ + t is a supersolution.Given Lemma 3.1, we may now apply Huisken and Ilmanen’s Weak Exis-tence Theorem 3.1 of [12] to find a weak solution u for inverse mean curvatureflow with initial condition ∂M . More precisely, u is a proper, locally Lips-chitz nonnegative function u defined on M with u = 0 on ∂M that satisfiesa certain variational property (defined on page 365 of [12]). The surfacesΣ t := ∂ { u < t } are C ,α and strictly outward minimizing, as defined in theproof of Lemma 2.4. In particular, each Σ t is mean convex. There are only Huisken and Ilmanen instead describe the region enclosed by Σ t as a strictly minimiz-ing hull [12]. t for which Σ t := ∂ { u < t } does not equal Σ + t := ∂ (int { u ≤ t } ). In a nonrigorous sense, Σ t may beregarded as flowing by smooth inverse mean curvature flow, except whenit ceases to be strictly outward minimizing, at which time it “jumps” to astrictly outward minimizing surface Σ + t of equal area.In case ∂M is not connected, Huisken and Ilmanen explained how one cansingle out a component ∂ M of ∂M as the initial surface while treating theother components of ∂M as “obstacles.” See Section 6 of [12] for details.Essentially, we arbitrarily “fill in” all other components ∂ M, . . . , ∂ n M of ∂M to obtain a new space ˜ M and then run the weak inverse mean curvatureflow in ˜ M with initial condition ∂ M , except that whenever the surface Σ t isabout to enter the filled-in region, we jump to a connected strictly outwardminimizing surface F enclosing both Σ t and one or more of the filled-inregions. We then restart the flow with initial condition F .There is another important alteration introduced by Huisken and Ilma-nen [12, Section 4]. We consider the exterior region M ′ of M , as defined inthe proof of Lemma 2.4. Since ∂M was the outermost minimal surface of M , it follows that ∂M is still part of the boundary of M ′ , but now theremight be more minimal boundary components. The exterior region M ′ isan improvement over M because it is completely free of compact minimalsurfaces in its interior. We will actually run the weak inverse mean curva-ture flow in the exterior region of M rather than in M itself. So for ourproof of Theorem 1.1, we may assume without loss of generality that M isan exterior region. Fix an integer g and recall our definition of the Hawking mass of a surface Σin ( M, g ) to be m H (Σ) := r | Σ | π (cid:18) − g − π ˆ Σ ( H − (cid:19) . The proof of the Geroch Monotonicity Formula 5.8 in [12] adapts straight-forwardly to the locally hyperbolic setting to show the following:
Theorem 3.2 (Huisken-Ilmanen) . Let ( M , g ) be a complete, one-ended, C asymptotically locally hyperbolic manifold with outermost minimal boundary,and let ∂ M be one of its boundary components. Let Σ t be a weak solutionto inverse mean curvature flow (possibly with obstacles, as described above), ith initial surface ∂ M . Then for ≤ ξ < η , if there are no obstaclesbetween Σ ξ and Σ η , then m H (Σ η ) − m H (Σ ξ ) ≥
12 (16 π ) − / ˆ ηξ | Σ t | / " π ( ˆ χ − χ (Σ t ))+ ˆ Σ t (cid:0) R + 6) + | ˚ A | + 4 H − |∇ H | (cid:1) dt, (6) where ˆ χ := 2 − g , and ˚ A is the trace-free part of the second fundamentalform. To make use of this theorem we need the following lemma.
Lemma 3.3.
Let ( M , g ) be a complete, one-ended, C asymptotically lo-cally hyperbolic manifold, which is an exterior region. Let ∂ M be a compo-nent of ∂M with genus g , and let Σ t be a weak solution to inverse mean cur-vature flow (possibly with obstacles), with initial surface ∂ M . For all t , thesurface Σ t is connected and has genus at least g . In particular, ˆ χ ≥ χ (Σ t ) .Proof. Let ˜ M be the manifold M with the obstacles filled in. Let u be thefunction defining the weak flow (possibly with obstacles). For each t > M t be the closure of { x ∈ ˜ M | u ( x ) < t } , so that ∂ ˜ M t = Σ t ∪ ∂ M . Weclaim that ˜ M t is connected. If it were not connected, one of the componentsΩ of ˜ M t would be disjoint from ∂ M . By the variational property thatcharacterizes u , one can deduce that u must be constant over Ω. (See theproof of [12, Connectedness Lemma 4.2(i)].) Since ˜ M is connected, Ω mustmeet Σ t , and thus u = t on Ω, which is a contradiction to the definitionof ˜ M t . Since ˜ M t is connected, it follows that M t := ˜ M t ∩ M is connected.Note that ∂M t = Σ t ∪ ∂ M ∪· · ·∪ ∂ k M , where ∂ M, . . . , ∂ k M is some labelingof the other components of ∂M that touch M t .The rest of the proof does not use inverse mean curvature flow. It isessentially a topological argument that relies only on the following factsabout Σ t : There exists a connected manifold M t whose boundary is Σ t ∪ ∂ M ∪ · · · ∪ ∂ k M , Σ t is mean convex, each ∂ i M is minimal, and the ∂ i M ’sare the only compact minimal surfaces in M t . This last part is where weuse the assumption that M is an exterior region.Let Σ , . . . , Σ ℓ be the connected components of Σ t . We minimize area inthe isotopy class of Σ in M t . Note that Theorem 1 of Meeks, Simon, andYau applies because M t has mean convex boundary, as explained in Section 6of [13]. According to Theorem 1 and Remark 3.27 of [13], there exists some13urface ˜Σ obtained from Σ via isotopy and a series of γ -reductions suchthat each component of ˜Σ is a parallel surface of a connected minimalsurface, except for one component that may be taken to have arbitrarilysmall area. Recall that a γ -reduction is a surgery procedure that deletes anannulus and replaces it with two disks in such a way that the annulus andtwo disks bound a ball in M t . (See [13, Section 3] for details.)Note that γ -reduction preserves homology class. Since the only compactminimal surfaces in M t are the ∂ i M ’s, and because a surface of small enougharea must be homologically trivial, it follows that[Σ ] = [ ˜Σ ] = k X i =1 n i [ ∂ i M ] in H ( M t , Z ) , (7)for some integers n i . Using the long exact sequence for the pair ( M t , ∂M t ),we have exactness of H ( M t , ∂M t ) ∂ −→ H ( ∂M t , Z ) ι ∗ −→ H ( M t , Z ) . Since M t is connected, ker i ∗ must be generated by ∂ [ M t ] = ℓ X i =1 [Σ i ] − k X i =1 [ ∂ i M ] , where Σ i and ∂M are oriented using the outward normal in M as usual.Since equation (7) says that [Σ ] − P ki =1 n i [ ∂ i M ] ∈ ker i ∗ , it follows that Σ t must be connected and hence equal to Σ . In particular,[Σ ] = k X i =1 [ ∂ i M ] in H ( M t , Z ) . Since each component of ˜Σ is either isotopic to one of the ∂ i M ’s (withsome orientation) or is null homologous, and since there are no relationsamong [ ∂ i M ] in H ( M t , Z ), the previous equation implies that at least onecomponent of ˜Σ is isotopic to ∂ M . Finally, since γ -reduction can onlyreduce the total genus of all components of a surface, we know that Σ t hasgenus at least as large as that of ∂ M .Note that if we apply the reasoning in the proof of Lemma 3.3 above tothe “conformal infinity” ˆΣ of M , we see that the genus of ˆΣ is at least aslarge as the genus of ∂ M . 14 orollary 3.4 (Geroch monotonicity) . Let ( M , g ) be a complete, one-ended, C asymptotically locally hyperbolic manifold, which is an exteriorregion. Let ∂ M be a component of ∂M with genus g , and let Σ t be a weaksolution to inverse mean curvature flow (possibly with obstacles), with initialsurface ∂ M . Then the Hawking mass of Σ t is nondecreasing in t .Proof. The result follows immediately from Theorem 3.2 and Lemma 3.3 inthe absence of obstacles. Since there are only finitely many obstacles, allthat is left to show is that the mass cannot drop when we jump over anobstacle. Let t be the first time that we have to jump over an obstacle,and let F denote the strictly outward minimizing surface that Σ t jumps to.Then we know from [12, Equation 6.1] that | Σ t | ≤ | F | and ˆ Σ t H t ≥ ˆ F H F , where the second inequality essentially follows from the fact that the strictlyminimizing hull of a surface should be minimal away from where it agreeswith the original surface. In the case g <
2, these inequalities combine withnonnegativity of m H (Σ t ) (from monotonicity in the absence of obstacles) toimmediately show that m H (Σ t ) ≤ m H ( F ), just as in the asymptotically flatcase [12, Section 6].To handle the case g ≥ m H ( F ) − m H (Σ t ) = | F | / m H ( F ) | F | / − | Σ t | / m H (Σ t ) | Σ t | / = ( | F | / − | Σ t | / ) m H (Σ t ) | Σ t | / + | F | / (cid:18) m H ( F ) | F | / − m H (Σ t ) | Σ t | / (cid:19) = ( | F | / − | Σ t | / ) m H (Σ t ) | Σ t | / + | F | / (16 π ) − / (cid:18) − ˆ F ( H F −
4) + ˆ Σ t ( H t − (cid:19) ≥ ( | F | / − | Σ t | / ) m H (Σ t ) | Σ t | / + | F | / (16 π ) − / (4 | F | − | Σ t | ) ≥ ( | F | / − | Σ t | / ) m H (Σ t ) | Σ t | / + | Σ t | / (16 π ) − / (4 | F | − | Σ t | )= ( | F | / − | Σ t | / ) (cid:18) m H (Σ t ) | Σ t | / + 4(16 π ) − / | Σ t | / ( | F | / + | Σ t | / ) (cid:19) ≥ ( | F | / − | Σ t | / ) (cid:18) m H (Σ t ) | Σ t | / + 8(16 π ) − / | Σ t | (cid:19) . m H (Σ t ) ≥ − π ) − / | Σ t | / . Notethat for any A ≥ r A π (cid:18) − g + 14 π A (cid:19) ≥ − (cid:18) g − (cid:19) / . Taking A = | ∂ M | and using monotonicity in the absence of obstacles, wethen have m H (Σ t ) ≥ m H ( ∂ M ) ≥ − (cid:18) g − (cid:19) / . On the other hand, observe that a stable compact minimal surface of genus g in a 3-manifold with R ≥ − π ( g − π ) − / | Σ t | / ≥ π ) − / | ∂ M | / ≥ π ) − / (cid:20) π g − (cid:21) / ≥ (cid:18) g − (cid:19) / , which completes the proof. First we compute the asymptotics of Ricci curvature for asymptotically lo-cally hyperbolic manifolds. Proceeding as in Lemma 3.1 of [16], we deducethe following:
Lemma 3.5.
Let ( M , g ) be a asymptotically locally hyperbolic, with radialcoordinate ρ . If ∇ ρ |∇ ρ | , e , e is an orthonormal frame at a point in M , then Ric (cid:18) ∇ ρ |∇ ρ | , ∇ ρ |∇ ρ | (cid:19) = − − µρ − + o ( ρ − )Ric( e i , e j ) = − δ ij + O ( ρ − )Ric (cid:18) ∇ ρ |∇ ρ | , e i (cid:19) = o ( ρ − ) R = − o ( ρ − ) .
16n order to make certain computations easier, we consider a conformalcompactification ˜ g = ρ − g of the exterior region of ( M, g ). Then if we set s = ρ − , we have ˜ g = ds + ˆ g + ˜ Q, on the space (0 , s ) × ˆΣ for small enough s , where | ˜ Q | + s | ˜ ∇ ˜ Q | + s | ˜ ∇ ˜ Q | = O ( s ) . Lemma 3.6.
There is a constant C such that for sufficiently large t , the ρ coordinate on Σ t lies in ( C e t/ , Ce t/ ) and the s coordinate on Σ t must liein ( C e − t/ , Ce − t/ ) .Proof. This follows immediately from the subsolutions and supersolutionsof inverse mean curvature flow described in Lemma 3.1.We will use the following area bound repeatedly.
Lemma 3.7.
Let ˜Σ t denote the surface Σ t endowed with the metric inducedfrom ˜ g . The area of ˜Σ t is uniformly bounded in time.Proof. This follows immediate from the fact that | Σ t | = | Σ | e t , the defini-tion of ˜ g , and the previous Lemma. Lemma 3.8. ´ Σ t ( H − is uniformly bounded above and below.Proof. The upper bound follows easily from Corollary 3.4: Since m H (Σ ) ≤ m H (Σ t ) = r | Σ t | π (cid:18) − g − π ˆ Σ t ( H − (cid:19) , we have ˆ Σ t ( H − ≤ π − g + m H (Σ ) s π | Σ t | ! = 16 π (1 − g ) + O ( e − t/ ) . To prove the lower bound, we consider the Gauss-Codazzi equations in(
M, g ): 2 | ˚ A | + 4 K = H + 2 R − ν, ν ) (8)Performing the same computation in the compactified metric ˜ g and usingthe fact that integral of the left-hand side is conformally invariant, we seethat ˆ Σ t ( H + 2 R − ν, ν )) = ˆ ˜Σ t ( ˜ H + 2 ˜ R − g Ric(˜ ν, ˜ ν )) .
17y Lemma 3.5, we see that ˆ Σ t ( H −
4) = ˆ ˜Σ t ( ˜ H + 2 ˜ R − g Ric(˜ ν, ˜ ν )) + O ( e − t/ ) ≥ − C + ˆ ˜Σ t ˜ H ≥ − C, (9)for some constant C independent of t , where we used the fact that thecurvature of ˜ g is bounded. Lemma 3.9.
We have ˆ ˜Σ t | ˜ ∇ T s | = O ( e − t/ ) . Proof. ˆ ˜Σ t | ˜ ∇ T s | = − ˆ ˜Σ t s ∆ ˜Σ t s = ˆ ˜Σ t s (∆ ˜ g s − ˜ ∇ ˜ ν ˜ ∇ ˜ ν s + h ˜ H, ∂ s i ) ≤ ˆ ˜Σ t s | ∆ ˜ g s − ˜ ∇ ˜ ν ˜ ∇ ˜ ν s | + ˆ ˜Σ t s | ˜ H | · | ˜ ∇ s | = ˆ ˜Σ t O ( s ) + ˆ ˜Σ t | ˜ H | · O ( s ) ≤ O ( e − t ) + O ( e − t/ ) (cid:18) ˆ ˜Σ t ˜ H (cid:19) / , where we used the H¨older inequality in the last line. The result now followsbecause inequality (9) states that ˆ ˜Σ t ˜ H ≤ C + ˆ Σ t ( H − , while Lemma 3.8 implies that the right-hand side is bounded. Lemma 3.10. | ˜Σ t | = | ˆΣ | + O ( e − t/ ) . Proof.
Choose ˜ ν to be the inward pointing normal of ˜Σ t . We will first showthat | ˜Σ t | ≤ | ˆΣ | + O ( e − t/ ) . Using the fact that s is approximately a distance function with respect to ˜ g , | ˜Σ t | = ˆ ˜Σ t ( | ˜ ∇ s | + O ( s )) = ˆ ˜Σ t ( | ˜ ∇ T s | + | ˜ ∇ N s | ) + O ( e − t )18 ˆ ˜Σ t | ˜ ν ( s ) | + O ( e − t/ ) , where the last line follows from Lemma 3.9. So it suffices to estimate ´ ˜Σ t | ˜ ν ( s ) | in terms of | ˆΣ | .We now divide ˜Σ t into three parts: A t = { x ∈ ˜Σ t | ˜ ν ( s ) ≤ − e − t/ } , B t = { x ∈ ˜Σ t | − e − t/ < ˜ ν ( s ) ≤ } and C t = { x ∈ ˜Σ t | < ˜ ν ( s ) } . Then ˆ ˜Σ t | ˜ ν ( s ) | = ˆ A t | ˜ ν ( s ) | + ˆ B t | ˜ ν ( s ) | + ˆ C t | ˜ ν ( s ) | ≤ (1 + O ( e − t )) | A t | + e − t/ | B t | + (1 + O ( e − t )) ˆ C t ˜ ν ( s ) ≤ | A t | + ˆ ˜Σ t ˜ ν ( s ) + O ( e − t/ )If we take ˜ M t to be the region of (0 , s ) × ˆΣ so that ∂M t = { } × ˆΣ ∪ Σ t ,then the divergence theorem tells us that ˆ ˜Σ t ˜ ν ( s ) = ˆ { }× ˆΣ ˜ ν ( s ) + ˆ ˜ M t ˜∆ s = | ˆΣ | + O ( e − t ) . Thus ˆ ˜Σ t | ˜ ν ( s ) | ≤ | A t | + | ˆΣ | + O ( e − t/ ) . (10)It remains to estimate | A t | . The mean curvature changes under underthe conformal change g = s − ˜ g according to the formula H = s ˜ H + 2˜ ν ( s ) . Since Σ t evolves by inverse mean curvature flow, we know that H ≥ ν ( s ) ≥ − s ˜ H . So the definition of A t tells us that − e − t/ ≥ ˜ ν ( s ) ≥ − s ˜ H on A t . Squaringthis and integrating over A t gives ˆ A t e − t/ ≥ ˆ A t s ˜ H ⇒ | A t | ≤ O ( e − t/ ) ˆ ˜Σ t ˜ H .
19s mentioned in the proof of Lemma 3.9, ´ ˜Σ t ˜ H is bounded, and thereforethis inequality combined with inequality (10) gives us ˆ ˜Σ t | ˜ ν ( s ) | ≤ | ˆΣ | + O ( e − t/ ) , completing the proof of one side of the inequality.The reverse inequality follows from the fact that | ˜Σ t | is within error O ( e − t ) from the area of ˜Σ t as measured in the product metric ds + ˆ g on(0 , s ) × ˆΣ, and the projection map of the product metric onto ˆΣ is area-nonincreasing. Lemma 3.11.
There exists a sequence of times t i such that lim i →∞ ˆ Σ ti | ˚ A | = 0 . Proof.
We use formula (5.22) from [12]: For ξ < η , we have ˆ Σ ξ H ≥ ˆ Σ η H + ˆ ηξ ˆ Σ t (cid:18) | DH | H + 2 | A | + 2Ric( ν, ν ) − H (cid:19) dt ≥ ˆ Σ η H + ˆ ηξ ˆ Σ t (2 | ˚ A | + 2Ric( ν, ν ))Since ddt | Σ t | = | Σ t | , it follows that ˆ Σ ξ ( H − ≥ ˆ Σ η ( H −
4) + ˆ ηξ ˆ Σ t (2 | ˚ A | + 2(Ric( ν, ν ) + 2)) dt. The last term is easily bounded: ˆ ηξ ˆ Σ t ν, ν ) + 2) dt = ˆ ηξ ˆ Σ t O ( ρ − ) dt = ˆ ηξ O ( e − t/ ) dt = O ( e − ξ/ )Hence, ˆ ηξ ˆ Σ t | ˚ A | dt ≤ ˆ Σ ξ ( H − − ˆ Σ η ( H −
4) + O ( e − ξ/ ) . Since ´ Σ ξ ( H −
4) is bounded above and below by Lemma 3.8, the integralon the left-hand side is bounded as η → ∞ , completing the proof.20 emma 3.12. Using the same sequence as in Lemma 3.11, we have χ (Σ t i ) = χ ( ˆΣ) for sufficiently large i .Proof. Note that by Lemma 3.3, we already know that χ (Σ t i ) ≤ χ ( ˆΣ), sowe need to show that χ (Σ t i ) ≥ χ ( ˆΣ) for all i large.The Gauss-Codazzi equations in the compactified metric tell us that2 | ˚˜ A | + 4 ˜ K = ˜ H + 2 ˜ R − g Ric(˜ ν, ˜ ν ) , where ˜ ν is the normal vector to the surface. If we apply this to the surfaces { s } × ˆΣ we obtain thatlim s → (cid:16) R − g Ric( ˜ ∇ s, ˜ ∇ s ) (cid:17) = 4ˆ k, (11)where ˆ k is the constant Gaussian curvature of ( ˆΣ , ˆ g ).Decompose ˜ ν = a ˜ ∇ s | ˜ ∇ s | + v, so that v is orthogonal to ˜ ν , we have | v | = | ˜ ∇ T s || ˜ ∇ s | = | ˜ ∇ T s | + O ( s ).Integrating Gauss-Codazzi equations on ˜Σ t and recalling Lemma 3.9 weobtain (cid:18) ˆ ˜Σ t | ˚˜ A | (cid:19) + 8 πχ ( ˜Σ t ) = ˆ ˜Σ t ˜ H + 2 ˜ R − g Ric(˜ ν, ˜ ν ) ≥ ˆ ˜Σ t R − g Ric(˜ ν, ˜ ν )= ˆ ˜Σ t (2 ˜ R − a g Ric( ˜ ∇ s, ˜ ∇ s ) − g Ric( ˜ ∇ s, v ) − g Ric( v, v )) + O ( s ) ≥ ˆ ˜Σ t (2 ˜ R − g Ric( ˜ ∇ s, ˜ ∇ s ) − C ( | ˜ ∇ T s | + | ˜ ∇ T s | )) + O ( s )= ˆ ˜Σ t (2 ˜ R − g Ric( ˜ ∇ s, ˜ ∇ s )) + O ( e − t/ ) . Thus, using (11), Lemma 3.10, and Lemma 3.11, we obtainlim i →∞ πχ ( ˜Σ t i ) ≥ k | ˆΣ | = 8 πχ ( ˆΣ) . This implies χ ( ˜Σ t i ) ≥ χ ( ˆΣ) for all i sufficiently large. Lemma 3.13. If ¯ m ≤ , then lim t →∞ m H (Σ t ) ≤ − (cid:18) ˆΣ µ / (cid:19) / | ˆΣ | π ! / . roof. Using Gauss-Codazzi equations, Gauss-Bonnet Theorem, and Lemma3.5, we have m H (Σ t ) = r | Σ t | π (cid:18) − g − π ˆ Σ t ( H − (cid:19) = s | Σ t | (16 π ) (cid:18) π (1 − g ) − ˆ Σ t ( − R + 4 K Σ t + 4Ric( ν, ν ) + | ˚ A | − (cid:19) = s | Σ t | (16 π ) (cid:18) πχ ( ˆΣ) − ˆ Σ t (4 K Σ t + 4(Ric( ν, ν ) + 2) + | ˚ A | + o ( ρ − )) (cid:19) = s | Σ t | (16 π ) (cid:18) πχ ( ˆΣ) − πχ (Σ t ) − ˆ Σ t | ˚ A | − ˆ Σ t (Ric( ν, ν ) + 2) + o ( e − t/ ) (cid:19) If we choose t to be one of the times from the sequence described in Lemma3.12, we have m H (Σ t ) ≤ −
12 (4 π ) − / p | Σ t | ˆ Σ t (Ric( ν, ν ) + 2) + o (1) . (12)Decompose ν = a ∇ ρ |∇ ρ | + v so that v is orthogonal to ∇ ρ . We have | v | = ∇ T ρ |∇ ρ | while a = 1 − | v | . Itfollows from Lemma 3.5 thatRic( ν, ν ) = a ( − − µρ − ) + | v | ( − O ( ρ − )) + o ( ρ − )= − − µρ − + | v | O ( ρ − ) + o ( ρ − ) . (13)Note that ˆ Σ t | v | ρ − = ˆ Σ t (cid:18) |∇ T ρ ||∇ ρ | (cid:19) ρ − = ˆ Σ t (cid:18) |∇ T s ||∇ s | (cid:19) s = ˆ ˜Σ t | ˜ ∇ T s || ˜ ∇ s | ! s = O ( e − t ) , where we used Lemma 3.9 in the last line. Putting this together with for-mulas (12) and (13) we obtain m H (Σ t ) ≤ (4 π ) − / p | Σ t | ˆ Σ t µρ − + o (1)22 (4 π ) − / (cid:18) ˆ ˜Σ t s − (cid:19) / (cid:18) ˆ ˜Σ t µs (cid:19) + o (1) . (14)We use the H¨older inequality with p = 3 and q = 3 / ˆ ˜Σ t µ / = ˆ ˜Σ t s − / ( µs ) / ≤ (cid:18) ˆ ˜Σ t s − (cid:19) / (cid:18) ˆ ˜Σ t | µ | s (cid:19) / . Taking the 3 / µ ≤
0, we have − (cid:18) ˆ ˜Σ t µ / (cid:19) / ≥ (cid:18) ˆ ˜Σ t s − (cid:19) / (cid:18) ˆ ˜Σ t µs (cid:19) . Combining this with inequality (14) to conclude that m H (Σ t ) ≤ − (cid:18) π ˆ ˜Σ t µ / (cid:19) / + O ( e − t/ ) , for any sequence of t ’s as in the previous lemma. Now the result follows fromtaking the limit as this sequence approaches infinity and applying mono-tonicity of ˜Σ t and Lemma 3.10. Proof of Theorem 1.1.
Recalling that ˆ g has constant curvature −
1, 0, or 1on ˆΣ and our normalizaton of area in the ˆ k = 0 case, we have | ˆΣ | = 4 π max(1 , g − / . Using Geroch Monotonicity (Corollary 3.4) and the previous lemma, we cannow conclude m H ( ∂ M ) ≤ lim t →∞ m H (Σ t ) ≤ − (cid:18) ˜Σ t µ / (cid:19) / | ˆΣ | π ! / ≤ ¯ m (max { , g − } ) / , where we used the definition ¯ m = sup µ .All that remains is to prove the rigidity. We assume all of the hypothesesof Theorem 1.1, as well as equality in the Penrose inequality. By Corol-lary 3.4, the Hawking mass m H (Σ t ) must be constant, and moreover, byequation (6), ˆ Σ t (cid:0) R + 6) + | ˚ A | + 4 H − |∇ H | (cid:1) = 0 . (15)23s argued on page 422 of [12], it follows that H is a positive constant oneach Σ t , each Σ t is smooth, and there are no jump times. By the SmoothStart Lemma 2.4 of [12], Σ t is a classical solution of inverse mean curvatureflow foliating the manifold M . This allows us to think of inverse meancurvature flow as defining a diffeomorphism from [0 , ∞ ) × ∂ M to M . (Inparticular, ∂M must be connected.) We can write the metric as g = H − dt + g Σ t . Note that equation (15) implies that R = − A = 0 oneach Σ t . In particular, A = H g Σ t and thus ∂ t g Σ t = H A = g Σ t . Hence g = H − dt + e t g Σ . (16)Recall that dHdt = ∆ H − − ( | A | + Ric( ν, ν )) H − , which implies Ric( ν, ν ) is constant on each Σ t and so, by the Gauss-Codazziequations, so is the Gauss curvature K . In particular, Σ = ∂M has constantcurvature. From here it is clear that ( M, g ) must be the Kottler space, sinceKottler spaces are the unique asymptotically locally hyperbolic manifoldswith R = − g = f ( r ) dr + r g Σ , where g Σ is a constantcurvature metric on Σ . Proof of Theorem 1.3.
Assume the hypotheses of Theorem 1.3. If there ex-ists at least one compact minimal surface in M , then the result follows fromCorollary 1.2 applied to the exterior region of M .So let us assume that M contains no compact minimal surfaces, andsuppose that ¯ m <
0. In particular, the mass aspect µ is everywhere negative.We can consider a weak inverse mean curvature flow whose initial conditionis effectively a point, just as Huisken and Ilmanen did in [12, Section 8].Then the same arguments used to prove Theorem 1.1 show that0 = lim t → m H (Σ t ) ≤ lim t →∞ m H (Σ t ) ≤ ¯ m, where the Hawking mass here has g = 0. Rigidity follows according to anargument similar to the one used in Theorem 1.1 above. Assume the hypotheses of Theorem 2.3. In particular, we have a static dataset (
M, g, V ) and a reference Kottler space ( M , g , V ) that has the same24urface gravity. We also have the important assumption that m ≤
0. Wedefine W = |∇ V | , and also a reference function W on M as follows: On the reference space M , |∇ V | is constant on each level set of V and, because of this, one may regardit as a function composed with V . That is, there exists a single-variablefunction ω with the property that |∇ V | = ω ( V ) as functions on M . Wedefine the function W on M to be the function W = ω ( V ) . Recall the formula V = q r + ˆ k − mr for an appropriate coordinate r on M . Inverting this formula, we may think of r as some function of V .In a manner similar to the way we defined W , we can define a referencefunction r on M by composing this function with V . (It might be logical tocall this function r , but that is unnecessary because there is no ambiguityhere.) Lemma 4.1.
Under the hypotheses of Theorem 2.3 and using the notationintroduced above, if we consider an open set in M on which W does notvanish, then on that open set W − W satisfies the elliptic inequality ∆( W − W ) + h ξ, ∇ ( W − W ) i + α ( W − W ) ≥ , where ξ is a smooth vector field and α is a smooth function whose sign isthe opposite of m . This is the critical ingredient of the proof of Theorem 2.3 and it corre-sponds to equation (VII.15) in [8].
Proof.
Since ∇ V = 0, the level sets of V are smooth surfaces which wedenote Σ V . We introduce the notation ∂ V = ∇ V |∇ V | = ∇ VW .
Note that ∂ V has the nice property that if f is a single-variable function,then ∂ V ( f ( V )) = f ′ ( V ) . Recall the static equations∆ g V = 3 V and Ric( g ) = 1 V Hess( V ) − g. ∇ W = 2 Hess V ( ∇ V, · ) and thus ∂ V W = 2 Hess V (cid:18) ∇ V |∇ V | , ∇ V |∇ V | (cid:19) = 2 W Hess V ( ∇ V, ∇ V ) . Choose an orthonormal frame e , e , e such that e = ∇ V |∇ V | . Starting withthe B¨ochner formula and the static equations above, we have∆ W = 2 | Hess V | + 2Ric( ∇ V, ∇ V ) + 2 h∇ (∆ V ) , ∇ V i = 2 X i,j =1 | Hess V ( e i , e j ) | + V Hess V ( ∇ V, ∇ V )= 2 | Hess V ( e , e ) | + 4 X i =1 | Hess V ( e , e i ) | + 2 X i,j =1 | Hess V ( e i , e j ) | + V Hess V ( ∇ V, ∇ V )= 2 (cid:12)(cid:12)(cid:12) Hess V (cid:16) ∇ V |∇ V | , ∇ V |∇ V | (cid:17)(cid:12)(cid:12)(cid:12) + |∇ V | X i =1 | Hess V ( ∇ V, e i ) | + 2 |∇ V | | A Σ V | + |∇ V | V Hess V (cid:16) ∇ V |∇ V | , ∇ V |∇ V | (cid:17) = ( ∂ V W ) + W |∇ T W | + 2 W ( | ˚ A Σ V | + H V ) + WV ∂ V W = ( ∂ V W ) + W H V + WV ∂ V W + W |∇ T W | + 2 W | ˚ A Σ V | = ( ∂ V W ) + [Hess V ( e , e ) + Hess V ( e , e )] + WV ∂ V W + W |∇ T W | + 2 W | ˚ A Σ V | = ( ∂ V W ) + [∆ V − Hess V ( e , e )] + WV ∂ V W + W |∇ T W | + 2 W | ˚ A Σ V | = ( ∂ V W ) + [3 V − Hess V ( e , e )] + WV ∂ V W + W |∇ T W | + 2 W | ˚ A Σ V | = ( ∂ V W ) + [3 V − ∂ V W ] + WV ∂ V W + W |∇ T W | + 2 W | ˚ A Σ V | = ( ∂ V W ) − V ∂ V W + 9 V + WV ∂ V W + W |∇ T W | + 2 W | ˚ A Σ V | W = div( ∇ ( ω ( V ))) = div( ω ′ ( V ) ∇ V )= ω ′′ ( V ) W + ω ′ ( V )∆ V = ω ′′ ( V ) W + 3 ω ′ ( V ) V. Notice that both of the computations above provide a way to compute∆ ( ω ( V )) in the reference space M . Since ω ( V ) is constant on level sur-faces of V and since those level surfaces are umbilic, the first computationyields∆ ( ω ( V )) = | ∂ V ( ω ( V )) | − V ∂ V ( ω ( V )) + 9 V + ω ( V ) V ∂ V ( ω ( V ))= | ω ′ ( V ) | − V ω ′ ( V ) + 9 V + ω ( V ) V ω ′ ( V ) . Meanwhile, if we do the second computation in the reference space, we find∆ ( ω ( V )) = ω ′′ ( V ) ω ( V ) + 3 ω ′ ( V ) V . Equating the right-hand sides of the two equations above, we actually obtaina differential equation for the single-variable function ω , which means thatwe can replace the V by V to obtain the following equation on the originalspace M : ω ′′ ( V ) ω ( V ) + 3 ω ′ ( V ) V = | ω ′ ( V ) | − V ω ′ ( V ) + 9 V + ω ( V ) V ω ′ ( V ) ω ′′ ( V ) W + 3 ω ′ ( V ) V = | ∂ V W | − V ∂ V W + 9 V + W V ∂ V W . Picking up from our expression for ∆ W above, we now have∆ W = ω ′′ ( V ) W + 3 ω ′ ( V ) V = ω ′′ ( V ) W + 3 ω ′ ( V ) V + ω ′′ ( V )( W − W )= | ∂ V W | − V ∂ V W + 9 V + W V ∂ V W + ω ′′ ( V )( W − W ) . Now we can subtract this expression from our expression for ∆ W (exceptfor the last two nonnegative terms which we ignore) to find∆( W − W ) ≥ ( ∂ V W ) − V ∂ V W + 9 V + WV ∂ V W − (cid:2) | ∂ V W | − V ∂ V W + 9 V + W V ∂ V W (cid:3) − ω ′′ ( V )( W − W )= ∂ V ( W + W ) ∂ V ( W − W ) − V ∂ V ( W − W )27 WV ∂ V W − WV ∂ V W + WV ∂ V W − W V ∂ V W − ω ′′ ( V )( W − W )= (cid:2) ∂ V ( W + W ) − V + WV (cid:3) ∂ V ( W − W )+( W − W ) ω ′ ( V ) V − ω ′′ ( V )( W − W )= (cid:2) ∂ V ( W + W ) − V + WV (cid:3) ∂ V ( W − W ) + [ ω ′ ( V ) V − ω ′′ ( V )]( W − W ) . It only remains to compute the sign of the zero order coefficient. Using thethe formula V = (cid:18) r + ˆ k − m r (cid:19) / in M , we can perform simple computations using the chain rule to showthat ω ( V ) = (cid:16) r + m r (cid:17) . and dVdr = p ω ( V ) V .
Therefore ω ′ ( V ) = 2 V (cid:18) − m r (cid:19) , and ω ′′ ( V ) = 2 V (cid:18) − m r (cid:19) + 2 V m r drdV = ω ′ ( V ) V + 12 V p ω ( V ) m r . Thus ω ′ ( V ) V − ω ′′ ( V ) = − V √ W m r . So we see that as long as m ≤
0, the coefficient of W − W is nonnegative. Corollary 4.2.
Under the hypotheses of Theorem 2.3 and using the notationintroduced above, we have W ≤ W on M .Proof. By the definition of W , we know that W = W on ∂ M , and W ≤ W on all of ∂M since ∂ M was assumed to have the largest surface grav-ity of any boundary component. We also know from Proposition 2.2 that W − W = 0 at the conformal infinity. (We will see a more detailed calcu-lation below.) Suppose that W > W somewhere in M . Since W ≤ W at the boundary and also “at infinity,” the function W − W must achieve28ts positive maximum value at some point p in the interior of M . Since W ( p ) > W ( p ) >
0, we see that the previous lemma applies to some openset U containing p . By the maximum principle for the elliptic inequalitygiven by the Lemma, W − W cannot have a local maximum in U , which isa contradiction. Proof of Theorem 2.3.
We claim that for any point p ∈ ∂ M , K ∂ M ( p ) ≥ K ∂M , where K ∂ M ( p ) is the Gaussian curvature of ∂ M at p , while K ∂M is theconstant Gaussian curvature of the ∂M in the reference space. Integrat-ing this inequality and using the Gauss-Bonnet Theorem, the claim clearlyimplies that χ ( ∂ M ) | ∂ M | ≥ χ ( ∂M ) | ∂ M | , so we now focus on proving the claim.For the following, we use the same notation as in the proof of Lemma 4.1.Since W = 0 at ∂M , the vector field ∂ V is well-defined near ∂ M , and we canconsider the flow ϕ v generated by ∂ V near ∂ M . Choose a point p ∈ ∂ M .Applying Taylor’s Theorem to the function W restricted to the flow linestarting at p , we have W ( ϕ v ( p )) = W ( p ) + [( ∂ V W )( p )] v + [( ∂ V W )( p )] v + O ( v ) . We know that W ( p ) = κ , where κ is the surface gravity of ∂ M . Next, ∂ V W = 2 Hess V ( ∇ V, ∂ V ) = 2 Hess V ( e , e )= V (2Ric( e , e ) + 6) = 0 , where the last identity follows because V vanishes on ∂M . Following samereason and using the Gauss-Codazzi equations plus the fact that ∂M istotally geodesic, we have ∂ V W = (2Ric( e , e ) + 6) + V ∂ V (2Ric( e , e ) + 6)= 2Ric p ( e , e ) + 6 = R − K ∂ M ( p ) + 6 = − K ∂ M . Thus W ( ϕ v ( p )) = κ − K ∂ M ( p )] v + O ( v ) . Performing the same computation in the reference solution, we obtain W ( ϕ v ( p )) = κ − K ∂M v + O ( v ) . µ ≤ m . Recall fromProposition 2.2 that we have V = ρ + ˆ k − µρ − + o ( ρ − ) , where ρ is a coordinate as in the definition of asymptotically locally hyper-bolic. Differentiating this, we find2 V ∇ V = (2 ρ + µρ − ) ∇ ρ + o ( ρ − ) . Taking the norm-square of both sides, and using the asymptotically locallyhyperbolic property, V W = ( ρ + µρ − ) |∇ ρ | + o ( ρ )= ( ρ + µρ − )( ρ + ˆ k ) + o ( ρ )= ρ + ˆ kρ + µρ + o ( ρ ) . Then W = ρ + ˆ kρ + µρ + o ( ρ ) ρ + ˆ k − µρ − + o ( ρ − )= ρ kρ − + µρ − kρ − − µρ − + o ( ρ − )= ρ + µρ − + o ( ρ − ) . Recall that we also have r + ˆ k − m r = V = ρ + ˆ k − µρ − + o ( ρ − ) , by definition of the function r . In particular, r − = ρ − + o ( ρ − ) . By changing variables from r to ρ , we have W = (cid:16) r + m r (cid:17) = r + 2 m r − + O ( r − )= ( r + ˆ k − m r − ) − ˆ k + 4 m r − + O ( r − )= ρ − µρ − + 4 m ρ − + o ( ρ − ) . Comparing these asymptotic expansions for W and W and using Corollary4.2, we see that as ρ → ∞ , we have µ ≤ − µ + 4 m , from which the result follows. 30 eferences [1] Vincent Bonini and Jie Qing. A positive mass theorem on asymptot-ically hyperbolic manifolds with corners along a hypersurface. Ann.Henri Poincar´e , 9(2):347–372, 2008.[2] W. Boucher, G. W. Gibbons, and Gary T. Horowitz. Uniqueness theo-rem for anti-de Sitter spacetime.
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