HHYPERBOLIC MASS VIA HOROSPHERES
HYUN CHUL JANG AND PENGZI MIAO
Abstract.
We derive geometric formulas of the mass for asymptotically hyperbolic man-ifolds using large coordinate horospheres, and state relevant rigidity results. The formulasuse the difference between the mean curvatures with respect to a given metric and the hy-perbolic metric, so some rigidity results are stated in terms of the mean curvature inequalityon a certain portion of horospheres. We also present a way to characterize the region nearinfinity that does not contribute to the mass as an application.
Contents
1. Introduction 12. Preliminaries 62.1. The mass 1-form and the mean curvature difference 72.2. Example case: Anti-de Sitter Schwarzschild manifolds 83. Hyperbolic mass via horospheres 10Appendix A. The mass formulas using the mean curvature on large spheres 19References 221.
Introduction
The mass of an asymptotically hyperbolic Riemannian manifold introduced in [11, 23]serves as a global geometric invariant that measures the deviation from hyperbolic space. Inthis paper, we derive geometric formulas of the mass for asymptotically hyperbolic manifoldsusing large coordinate horospheres, and state some relevant rigidity results.Let ( H n , b ) be hyperbolic space as the upper sheet of the hyperboloid in the Minkowskispace: H n = { ( z, t ) ∈ R n, : z + · · · + z n − t = − , t > } . Let r = (cid:112) z + · · · + z n , then the metric b is written as b = dr r + r g S n − . Department of Mathematics, University of Miami
E-mail addresses : [email protected], [email protected] . a r X i v : . [ m a t h . DG ] F e b HYPERBOLIC MASS VIA HOROSPHERES
Following [11], a Riemannian manifold ( M n , g ) is said to be asymptotically hyperbolic ifthere exist a compact set K ⊂ M and a diffeomorphism Φ : M \ K → H n \ B R (0), where B R (0) = { r < R } , such that for h = (Φ − ) ∗ g − b , we have(1) as r → ∞ , | h | b + | ˚ ∇ h | b + | ˚ ∇ h | b = O ( r − q ) , q > n . (2) (cid:82) M r ( R g + n ( n − dµ g < ∞ where R g is the scalar curvature.The mass of asymptotically hyperbolic manifolds is defined as a ( n + 1)-vector using the so-called mass integral, which is defined as the following limit of the integral on large coordinatespheres:(1.1) H Φ ( V ) = lim r →∞ (cid:90) S r ( V div h − V d (tr b h ) + (tr b h ) dV − h (˚ ∇ V, · ))( ν ) dσ b for V ∈ C ( H n ). The components of the mass vector ( p , p , . . . , p n ) of ( M n , g ) are definedas p = H Φ ( √ r ) , p i = H Φ ( z i ) for i = 1 , . . . , n. Whereas its components may depend on the exterior coordinate chart Φ, the Minkowskianlength of the mass vector m ( g ) = (cid:118)(cid:117)(cid:117)(cid:116) p − n (cid:88) i =1 p i is a geometric invariant, which is often called the total mass of ( M, g ). The Riemannianpositive mass theorem for asymptotically hyperbolic manifolds states that if the scalar cur-vature satisfies R g ≥ − n ( n − m ( g ) is nonnegative, and m ( g ) = 0 if andonly if ( M, g ) is isometric to hyperbolic space. This theorem was first proved in [11, 23]under spinor assumption. In [1], the spinor assumption was replaced by the restriction ondimension and the geometry at infinity. These assumptions have recently been removed in[9, 10, 14]. We also remark that Sakovich [21] gave another proof for the 3-dimensional caseusing the Jang equation.To state our main theorems, let H L for large enough L > H L = { z ∈ H n : √ r − z = e L } . Note that H L = { y ∈ H n : y = e − L } where { y , . . . , y n } is the half-space-model coordinates(see Figure 1.1). Theorem 1.1.
Let ( M n , g ) , n ≥ , be an asymptotically hyperbolic manifold with metricfalloff rate q > n . Let ν g denote the unit normal vector to H L in ( M, g ) pointing toward { y = 0 } . Let H g be the mean curvature of H L with respect to ν g in ( M, g ) . Then, as L → ∞ , (1.2) p − p = 2 (cid:90) H L V ( H b − H g ) dσ g + o (1) YPERBOLIC MASS VIA HOROSPHERES 3 z = + ∞ z = −∞ H L ν g H L y ( y , . . . , y n ) ν g z = −∞ Figure 1.1.
Coordinate horosphere H L where p , p are the components of the mass vector, V = √ r − z = y on M \ K and H b = n − is the mean curvature of horospheres in hyperbolic space. The formula (1.2) can also be used to compute more general expression of the mass as thefollowing: let V = √ r − n (cid:88) i =1 a i z i , H L = { z ∈ H n : V = e L } where ( a ) + · · · + ( a n ) = 1. Note that H L represents the horospheres that are based at (cid:80) ni =1 a i z i = + ∞ in the conformal infinity. Then, as L → ∞ , (1.3) p − (cid:88) i =1 a i p i = 2 (cid:90) H L V ( H b − H g ) dσ g + o (1) . Especially, one can compute p + p i by using V = √ r + z i and the correspondinghorospheres which are based at the antipodal point of the base from the case of p − p i .Therefore, each component of the mass vector is computable from (1.2).Combining this formula and the positive mass theorem for asymptotically hyperbolic man-ifolds, the following rigidity can be obtained: Corollary 1.2.
Let ( M n , g ) , n ≥ , be an asymptotically hyperbolic manifold with metricfalloff rate q > n and scalar curvature R g ≥ − n ( n − . If ( M, g ) is isometric to hyperbolicspace outside a coordinate horoball, then ( M, g ) is isometric to hyperbolic space. Here, outside a coordinate horoball means a type of region { < y < a } for some positiveconstant a . Besides being a consequence of the positive mass theorem, the scalar curvaturerigidity of hyperbolic space with a compact set has been proved separately in the litera-ture, see [1, 2, 20]. If one consider horospheres in hyperbolic space as a natural analog ofhyperplanes in Euclidean space, it is known that Euclidean space does not have this kind HYPERBOLIC MASS VIA HOROSPHERES of rigidity. Indeed, it is proved by Carlotto and Schoen [6] that there exists a nontrivialasymptotically flat metric on R n with nonnegative scalar curvature such that the metric isisometric to Euclidean space in (0 , ∞ ) × R n − ⊂ R n .The key idea of the proof is to use a family of parabolic cylinders C L defined as C L = { ( y , ˆ y ) ∈ H n : e − L ≤ y ≤ e L , | ˆ y | ≤ σ ( L ) } where σ ( L ) is a positive, monotone increasing function of L such that σ ( L ) → ∞ as L → ∞ .In fact, we obtain a more refined mass formula which only uses one surrounding surface of C L : Theorem 1.3.
Let ( M n , g ) , n ≥ , be an asymptotically hyperbolic manifold with metricfalloff rate q > n . Define Σ L = { y = e − L , | ˆ y | < σ ( L ) } where (1.4) σ ( L ) n − − q = o ( e ( q − n +1) L ) as L → ∞ . Let ν g denote the unit normal vector to Σ L in ( M, g ) pointing toward { y = 0 } . Let H g bethe mean curvature of Σ L with respect to ν g in ( M, g ) . Then, as L → ∞ , (1.5) p − p = 2 (cid:90) Σ L V ( H b − H g ) dσ g + o (1) where V = t − z = y on M \ K .Remark . Theorem 1.3 implies that the quantity p − p is determined from the region (cid:91) L ≥ L Σ L = { ( y , ˆ y ) ∈ H n : 0 < y < e − L , | ˆ y | < f ( y ) } , for some large L >
0. Here, f ( y ) is any positive, monotone decreasing function of y such that f ( y ) → ∞ as y → f ( y ) n − − q = o ( y n − − q ). The latter condition is onlymeaningful if n < q < n −
1, so the shaded region in Figure 1.2 represents an example of (cid:83) L ≥ L Σ L in such case. On the other hand, if q ≥ n −
1, such region can be arbitrarily thinas | ˆ y | → ∞ since we do not need any additional assumption on σ ( L ) for Theorem 1.3. y = e − L y | ˆ y || ˆ y | = f ( y )Σ L Figure 1.2.
The region in Remark 1.4Using Theorem 1.3 and the mass rigidity of asymptotically hyperbolic manifolds in [14],we obtain the following rigidity property.
YPERBOLIC MASS VIA HOROSPHERES 5
Corollary 1.5.
Let ( M n , g ) , n ≥ , be an asymptotically hyperbolic manifold with the scalarcurvature lower bound R g ≥ − n ( n − . Define Σ L = { y = e − L , | ˆ y | < σ ( L ) } where σ ( L ) satisfies (1.4) . Suppose that there exists L > such that H g (Σ L ) ≥ n − for all L ≥ L .Then ( M n , g ) is isometric to hyperbolic space ( H n , b ) . In particular, if ( M, g ) is isometric tohyperbolic space in the region (cid:83) L>L Σ L , then ( M, g ) is isometric to hyperbolic space.Proof. By the assumption and Theorem 1.3, we have p − p = lim L →∞ (cid:90) Σ L e L ( n − − H g ) dσ g ≤ . Combining the positivity of the mass (see [9, 11, 23]), we have p ≥ p ≥ (cid:118)(cid:117)(cid:117)(cid:116) n (cid:88) i =1 p i ≥ | p | , which forces p = | p | = (cid:118)(cid:117)(cid:117)(cid:116) n (cid:88) i =1 p i . Hence by the mass rigidity result from [14], we obtain the first statement. The secondstatement follows from the fact that H g (Σ L ) = n − L ≥ L if ( M, g ) is isometric tohyperbolic space in the region (cid:83) L ≥ L Σ L . (cid:3) In fact, Corollary 1.5 implies Corollary 1.2 since the region (cid:83) L ≥ L Σ L lies outside a horo-sphere H L .The idea of using parabolic cylinders is motivated by the work in [17], which provides thegeometric mass formula for asymptotically flat metrics using large coordinate cubes. In arecent development, Bray et al. [5] presented a new proof of the Riemannian positive masstheorem for asymptotically flat 3-metrics that gives an explicit lower bound for the massin terms of linear growth harmonic functions and scalar curvature. Their approach was toapply Stern’s integral formula [22] on level sets of a harmonic function. In this context,the mass formula using coordinate cubes in [17] can be viewed as to compute the mass byusing the level sets of coordinate functions, which are linear growth harmonic functions onEuclidean space. This harmonic level set technique has been generalized for 3-dimensionalasymptotically flat initial data sets by Hirsch et al [13]. In particular, they defined spacetimeharmonic functions on a given intial data set ( M , g, k ) as a solution of the following equation∆ u + (Tr g k ) |∇ u | = 0 , and established a generalized integral formula for such functions. From this point of view, theformula (1.2) uses the level sets of a coordinate function y , which is a spacetime harmonicfunction on hyperbolic space as an initial data set ( H , b, − b ).It was also pointed out in [17] and [15] that the cubic mass formula for asymptoticallyflat manifolds has connection with Gromov’s scalar curvature comparison theory for cubicRiemannian polyhedra. In particular, Li [15, Section 5] observed that the so-called dihedral HYPERBOLIC MASS VIA HOROSPHERES rigidity phenomenon for a Euclidean polyhedron P is a localization of the positive masstheorem for asymptotically flat manifolds. Moreover, Li [16] extended the dihedral rigidityfor a collection of parabolic polyhedrons enclosed by horospheres in hyperbolic spaces. Theuse of parabolic cylinders in our proof is relatable to this, see Remark 3.2 for a relevantdiscussion.Another essential ingredient of the proof is the relation between the mass 1-form and themean curvature difference (see Proposition 2.1). Once we observed it, the remaining proofconsists of direct estimates of the integrals on the boundary of large parabolic cylinders C L .Also, we specify the region near infinity that does not contribute to the mass by examiningthe formula (1.5). Since the rigidity of the positive mass theorem forces that asymptoticallyhyperbolic metric cannot be localized inside such regions, it is related to seek the optimallocalization of an asymptotically hyperbolic metric g with R g ≥ − n ( n − Acknowledgements.
The authors would like to thank the organizers of the 2020 VirtualWorkshop on Ricci and Scalar Curvature, which provided a lot of inspiring talks. Especially,the plenary talk of Chao Li influenced the initial idea of this work. H. C. Jang was partiallysupported by NSF Grant DMS-1612049 for participating in the workshop and would liketo thank Christina Sormani for funding. P. Miao acknowledges the support of NSF GrantDMS-1906423. The authors are also grateful to Erwann Delay for his interest and carefulreading. 2.
Preliminaries
We recall various coordinates of hyperbolic space ( H n , b ).(1) Hyperbolic space can be obtained as the upper sheet of the hyperboloid in theMinkowski space: H n = { ( z, t ) ∈ R n, : z + · · · + z n − t = − , t > } . The set of functions { z , . . . , z n } restricted to H n are often called the hyperboloidalcoordinates. Let r = (cid:112) z + · · · + z n , then the metric b is written as b = dr r + r g S n − . YPERBOLIC MASS VIA HOROSPHERES 7 On H n , it satisfies that t = √ r , so we denote a function t = √ r throughoutthis paper.(2) Consider the upper half space model R n + = { y ∈ R n : y > } . Then the metric b is b = 1 y ( dy + · · · + dy n ) . (3) Define x = − ln y , x = y , . . . , x n = y n . Then { x , . . . , x n } becomes another set ofcoordinates of H n , and the metric b is b = dx + e x ( dx + · · · + dx n ) . The transformation between (1) and (2) is(2.1) y = 1 t − z , y i = z i t − z for i = 2 , . . . , n, so we get(2.2) e x = t − z , x i = z i e x for i = 2 , . . . , n. Using these transform, we can indicate a horosphere H L as H L = { x = L } = { y = e − L } = { t − z = e L } . For convenience, we use the notation ˆ x = ( x , . . . , x n ), and the same applies to ˆ y and ˆ z .By direct computation, it also implies that(2.3) r = (cid:18) cosh x + e x | ˆ x | (cid:19) − . In section 3, we will use the coordinates (3) to get essential estimates.2.1.
The mass -form and the mean curvature difference. It is well-known that theintegrands of the mass integral (1.1) can be viewed as the following one form:(2.4) U ( V ) = V div h − V d (tr b h ) + (tr b h ) dV − h (˚ ∇ V, · ) . Using this one form and the linearity of the mass integral H Φ , we can write p − p = lim r →∞ (cid:90) S r U ( t − z )( ν ) dσ b . To derive the mass formula in terms of geometric quantity, we use the following propositionshowing another expression of the one form U . Proposition 2.1.
Let Σ be any hypersurface in M . Let ν , ν g denote the unit normal vectorsto Σ pointing the same side in ( M, g ) , ( M, g ) , respectively. Then for | h | g sufficiently small,we have (2.5) U ( V )( ν ) = V [2( H g − H g )+ | A g | g O ( | h | g ) + O ( | ˚ ∇ h | g | h | g )]+ (tr Σ g h ) dV ( ν ) − V (cid:104) A g , h (cid:105) g | Σ − div Σ ( V X ) . HYPERBOLIC MASS VIA HOROSPHERES where H g , H g are the mean curvature of Σ with respect to ν , ν g in ( M, g ) , ( M, g ) , respec-tively, A g is the second fundamental form of Σ with respect to ν in ( M, g ) , and X is thevector field on Σ that is dual to the -form h ( ν , · ) with respect to g | Σ .Proof. By [18, Proposition 2.1], we have U ( V )( ν ) = V [2( H g − H g ) − div Σ X − (cid:104) A g , h (cid:105) g | Σ + | A g | g O ( | h | g )+ O ( | ˚ ∇ h | g | h | g )] + (tr g h ) dV ( ν ) − h (˚ ∇ V, ν ) . Then we get U ( V )( ν ) = V [2( H g − H g ) − div Σ X − (cid:104) A g , h (cid:105) g | Σ + | A g | g O ( | h | g ) + O ( | ˚ ∇ h | g | h | g )]+ ((tr Σ g h ) dV ( ν ) + h ( ν , ν ) dV ( ν )) − ( h (˚ ∇ Σ V, ν ) + h ( dV ( ν ) ν , ν ))= V [2( H g − H g ) − (cid:104) A g , h (cid:105) g | Σ + | A g | g O ( | h | g ) + O ( | ˚ ∇ h | g | h | g )]+ (tr Σ g h ) dV ( ν ) − (div Σ X + h (˚ ∇ Σ V, ν ))= V [2( H g − H g ) + | A g | g O ( | h | g ) + O ( | ˚ ∇ h | g | h | g )]+ (tr Σ g h ) dV ( ν ) − V (cid:104) A g , h (cid:105) g | Σ − div Σ ( V X ) . (cid:3) Remark . The mass integrals can be computed with a suitable exhaustion of M consistingof bounded domains to which the divergence theorem is applicable. See [4, Proposition 4.1]and [19, Section 2].2.2. Example case: Anti-de Sitter Schwarzschild manifolds.
Here, we compute themass of the anti-de Sitter (AdS) Schwarzschild manifolds by using the formula (1.2). For agiven m >
0, we call a Riemannian manifold [ r m , ∞ ) × S n − equipped with the metric g m = dr r − mr n − + r g S n − the AdS Schwarzschild manifold where r m is the largest zero of r n + r n − − m . By setting z = r cos θ for θ ∈ (0 , π ), we can write the metric as g m = dr r − mr n − + r ( dθ + (sin θ ) g S n − ) . Let M = − mr n − and V = t − z = √ r − r cos θ . By direct computation, we have(2.6) ∇ V = ˚ ∇ V + M V r ∂ r , |∇ V | g = | ˚ ∇ V | b + M ( V r ) , ∆ g V = ∆ b V − V r (cid:18) nM r (cid:19) + M V rr , ∇ ν ∇ ν V = ˚ ∇ ν ˚ ∇ ν V + (2 − n ) r ( V r ) M V + O ( r − ( n +1) ) , YPERBOLIC MASS VIA HOROSPHERES 9 where ∇ , ∆ g , ν are the gradient, Laplacian, normal vector to the level set { V = e L } pointingthe direction ∇ V with respect to the metric g , while ˚ ∇ , ∆ b , ν are the corresponding oneswith respect to the hyperbolic metric b . Also, the subscript r on V means the partialdifferentiation with respect to r .We observe that on { V = e L } e L = √ r − r cos θ ≤ √ r + r, thus it follows that r − = O ( e − L ) when L is large.By using these, we compute the mean curvature of the level set { V = e L } as L approachesinfinity:(2.7) H g = ∆ g V − ∇ ν ∇ ν V |∇ V | g = (cid:18) ∆ b V − ˚ ∇ ν ˚ ∇ ν V + V r (cid:18) nM r (cid:19) − M V rr − (2 − n ) r ( V r ) MV + O ( e − ( n +1) L ) (cid:19) × | ˚ ∇ V | b (cid:32) − M ( V r ) | ˚ ∇ V | b + O ( e − (2 n ) L ) (cid:33) , = H b − H b M ( V r ) V + 1 V (cid:18) V r (cid:18) nM r (cid:19) − (2 − n ) r ( V r ) M V (cid:19) + O ( e − ( n +2) L ) . Here, we used the fact that V = | ˚ ∇ V | b = ˚ ∇ ν ˚ ∇ ν V and V rr = √ r ) = O ( r − ).Therefore, we have(2.8) 2 V ( H b − H g ) = ( n − M ( V r ) V − V r (cid:18) nMr (cid:19) + (2 − n ) r ( V r ) MV + O ( e − ( n +1) L )= M V r r (cid:0) − n + O ( e − L ) + O ( e − L ) (cid:1) = 2 mr n − (cid:0) n − − ( n −
1) cos θ + O ( e − L ) (cid:1) = 2 mr n − (cid:16) n − − ( n − z r + O ( e − L ) (cid:17) . Hence, we obtain(2.9) (cid:90) H L V ( H b − H g ) dµ g = (cid:90) H L mr n − (cid:16) n − − ( n − z r + O ( e − L ) (cid:17) dµ b + o (1)= (cid:90) S n − (cid:90) ∞ mr n − (cid:16) n − − ( n − z r (cid:17) e L ( n − ρ n − dρdσ S n − + o (1)= 2 m ( n − ω n − + o (1) as L approaches infinity. In the second equality, we used the spherical coordinates on H L sothat b | H L = e L ( dρ + ρ g | S n − ) . Remark . From (2.8), the difference H b − H g is positive for sufficiently large r > R .Indeed, by using cos θ = √ r − e L r on H L , we have for large L − cos θ = r − √ r + e L r = 1 r (cid:18) e L − √ r + r (cid:19) > . This implies that for the AdS Schwarzschild manifold, the mean curvature on coordinatehorospheres H L for large L is less than n − Hyperbolic mass via horospheres
In this section, we derive the mass formula in terms of geometric quantities on coordinatehorospheres. Assume ( M n , g ) , n ≥
3, is asymptotically hyperbolic. By using the coordinates(3) in section 2, the parabolic cylinder C L is defined as(3.1) C L = { ( x , ˆ x ) ∈ H n : | x | ≤ L, | ˆ x | ≤ σ ( L ) } where σ ( L ) increases to infinity as L → ∞ , which will be determined later. Define thesurrounding surfaces of C L and their boundaries as the following: F ± ,L = { ( x , ˆ x ) ∈ H n : x = ± L, | ˆ x | ≤ σ ( L ) } ,S L = { ( x , ˆ x ) ∈ H n : | x | ≤ L, | ˆ x | = σ ( L ) } E ± ,L = { ( x , ˆ x ) ∈ H n : x = ± L, | ˆ x | = σ ( L ) } See Figure 3.1 to compare the shapes of C L in various coordinate charts.We first prove Theorem 1.3 by using the following formula (see Remark 2.2) p − p = lim L →∞ (cid:90) ∂C L U ( t − z )( ν ) dσ b . Let V = t − z = e x . We write the one form U on the surrounding surfaces using Lemma2.1. Lemma 3.1.
In the above setting, the following hold: (3.2) (cid:90) F ± ,L U ( V )( ν ) dσ b = (cid:90) F ± ,L V ( H b − H g ) dσ g + (cid:90) F ± ,L V O ( | h | b ) dσ b + (cid:90) E ± ,L V O ( | h | b ) ds b , YPERBOLIC MASS VIA HOROSPHERES 11 x | ˆ x | L − L σ ( L ) − σ ( L ) (a) In { x i } -coordinates y | ˆ y | e L e − L σ ( L ) − σ ( L ) x = Lx = − L (b) Upper half space model z | ˆ z | x = − Lx = L sinh( − L )sinh L (c) Hyperboloidal model x = − Lx = L (d) Poincar´e Ball model
Figure 3.1. C L in various coordinate charts and (3.3) (cid:90) S L U ( V )( ν ) dσ b = (cid:90) S L V ( H b − H g ) dσ g + (cid:90) S L (cid:2) V O ( | h | b ) + σ ( L ) − O ( | h | b ) (cid:3) dσ b + (cid:90) E ± ,L V O ( | h | b ) ds b , as L → ∞ .Proof. By direct computation, we have V = | ˚ ∇ V | b . Since F ± ,L ⊂ { V = e ± L } are level sets of V from (2.2), it follows that(tr F ± ,L b h ) dV ( ν ) − V (cid:104) A b , h (cid:105) b | F ± ,L = ± (tr F ± ,L b h )( V − V ) = 0 . By integrating it, we get (cid:90) F ± ,L U ( V )( ν ) dσ b = (cid:90) F ± ,L V ( H b − H g ) dσ g + (cid:90) F ± ,L V O ( | h | b ) dσ b + (cid:90) E ± ,L V O ( | h | b ) ds b , which proves (3.2).Now we consider the lateral surface S L . By writing the metric b = dx + e x n (cid:88) i =2 dx i = dx + e x ( dρ + ρ g S n − ) , we have Γ ρ = Γ α ρ = Γ αρ = 0 , Γ βαρ = 12 b βτ ( b ατ,ρ ) = ρ − δ βα , where the indices α, β represent local orthonormal coordinates on S n − . Thus, the secondfundamental form of S L in hyperbolic space is A = A α = A α = 0 , A αβ = e − x Γ ταρ b τβ = e x σ ( L ) δ αβ . It follows that V (cid:104) A b , h (cid:105) b | SL = b αβ b τσ A ατ h βσ ≤ ( n − σ ( L ) − | h | b , so by using Proposition 2.1, we get on S L U ( V )( ν ) = V [2( H b − H g ) + | A b | b O ( | h | b ) + O ( | ˚ ∇ h | b | h | b )]+ (tr S L b h ) dV ( ν ) − V (cid:104) A b , h (cid:105) b | SL − div S L ( V X )= V [2( H b − H g ) + O ( | h | b )] + σ ( L ) − O ( | h | b ) − div S L ( V X ) . Hence, we have (cid:90) S L U ( V )( ν ) dσ b = (cid:90) S L V ( H b − H g ) dσ g + (cid:90) S L (cid:2) V O ( | h | b ) + σ ( L ) − O ( | h | b ) (cid:3) dσ b + (cid:90) E ± ,L V O ( | h | b ) ds b . (cid:3) In what follows, we present the essential estimates for each surface and edge.
YPERBOLIC MASS VIA HOROSPHERES 13 • On F + ,L :(3.4) (cid:90) F + ,L V O ( | h | b ) dσ b ≤ C ω n − (cid:90) σ ( L )0 e L (cid:32)(cid:18) cosh L + e L ρ (cid:19) − (cid:33) − q e L ( n − ρ n − dρ ≤ C e L ( n − q ) (cid:90) ∞ (cid:0) e − L + ρ (cid:1) − q ρ n − dρ ≤ C e L ( n − q ) (cid:90) ∞ (1 + e − L + ρ ) − q + n − ρ n − − n +3 dρ ≤ ˜ Ce L ( n − q ) . Here, we need the condition 1 − q + n − <
0, i.e., q > n − to get the last inequality. Notethat σ ( L ) does not affect this estimate. • On F − ,L :We estimate with the integrand V O ( | h | b ) to show that the term V ( H b − H g ) on F − ,L has nocontribution to the mass.(3.5) (cid:90) F − ,L V O ( | h | b ) dσ b ≤ C ω n − (cid:90) σ ( L )0 e − L (cid:32)(cid:18) cosh L + e − L ρ (cid:19) − (cid:33) − q e − L ( n − ρ n − dρ ≤ C · q e L ( − n − q ) (cid:90) ∞ (cid:0) e − L + ρ e − L (cid:1) − q ρ n − dρ ≤ C e L ( − n − q ) (cid:90) ∞ (1 + e − L + ρ e − L ) − q + n − ρ n − − n +3 e L ( n − dρ ≤ ˜ Ce L ( − n − q ) e L ( n − = ˜ Ce L ( − − q ) . Similarly, we require the condition 1 − q + n − <
0, i.e., q > n +14 to get the last inequality.Again, σ ( L ) does not affect this estimate. • On E + ,L :(3.6) (cid:90) E + ,L V O ( | h | b ) dσ b ≤ C ω n − · e L (cid:32)(cid:18) cosh L + e L σ ( L ) (cid:19) − (cid:33) − q e L ( n − σ ( L ) n − ≤ C e L ( n − σ ( L ) n − (cid:18) e L + e − L + e L σ ( L ) (cid:19) q ≤ C e L ( n − σ ( L ) n − e − qL σ ( L ) − q ≤ ˜ Ce L ( n − − q ) σ ( L ) n − − q . Thus, to make the above integral vanish as L → ∞ , we need the following condition for σ ( L ):(3.7) σ ( L ) n − − q = o ( e L ( q − n +1) ) as L → ∞ . Note that if the falloff rate q ≥ n − σ ( L ) being increasing to infinity is sufficient to showthe above integral vanishes as L → ∞ . • On E − ,L :(3.8) (cid:90) E − ,L V O ( | h | b ) dσ b ≤ C ω n − · e − L (cid:32)(cid:18) cosh L + e − L σ ( L ) (cid:19) − (cid:33) − q e − L ( n − σ ( L ) n − ≤ C e − L ( n − σ ( L ) n − (cid:18) e L + e − L + e − L σ ( L ) (cid:19) q − n (cid:18) e L + e − L + e − L σ ( L ) (cid:19) n ≤ C e − L ( n − σ ( L ) n − o (1) (cid:18) e L + e − L + e − L σ ( L ) (cid:19) n ≤ C e − L ( n − σ ( L ) n − o (1) e n L σ ( L ) − n ≤ ˜ Ce − n − L σ ( L ) − o (1) . The third inequality follows from the condition q > n . Hence, it follows that the aboveintegral vanishes as L → ∞ regardless of σ ( L ). YPERBOLIC MASS VIA HOROSPHERES 15 • On S L :Similar to the case on F − ,L , we first estimate with the integrand V O ( | h | b ).(3.9) (cid:90) S L V O ( | h | b ) dσ b ≤ C ω n − (cid:90) L − L e x (cid:32)(cid:18) cosh x + e x σ ( L ) (cid:19) − (cid:33) − q e x ( n − σ ( L ) n − dx ≤ C (cid:90) L − L e x ( n − σ ( L ) n − (cid:18) e x + e − x + e x σ ( L ) (cid:19) q dx . We split the last integral into two parts: first, we have(3.10) (cid:90) L e x ( n − σ ( L ) n − (cid:18) e x + e − x + e x σ ( L ) (cid:19) q dx ≤ C (cid:90) L e x ( n − − q ) σ ( L ) n − − q dx ≤ ˜ Cσ ( L ) n − − q ( e L ( n − − q ) + 1) . Hence, we need an additional condition to make this integral converge to zero, which turnsout to be the same as (3.7).Next, we have(3.11) (cid:90) − L e x ( n − σ ( L ) n − (cid:18) e x + e − x + e x σ ( L ) (cid:19) q dx ≤ C (cid:90) L e − x ( n − σ ( L ) n − (cid:18) e x + e − x + e − x σ ( L ) (cid:19) q − n + n dx ≤ Co (1) (cid:90) L e − n − x σ ( L ) − dx ≤ ˜ Co (1) σ ( L ) − ( e − n − L + 1) . Therefore, the above integral vanishes as L → ∞ regardless of σ ( L ).Now, we consider the term σ ( L ) − O ( | h | b ).(3.12) (cid:90) S L σ ( L ) − O ( | h | b ) dσ b ≤ C ω n − (cid:90) L − L σ ( L ) − (cid:32)(cid:18) cosh x + e x σ ( L ) (cid:19) − (cid:33) − q e x ( n − σ ( L ) n − dx ≤ C (cid:90) L − L e x ( n − σ ( L ) n − (cid:18) e x + e − x + e x σ ( L ) (cid:19) q dx . Since σ ( L ) − O ( | h | b ) can be absorbed into V O ( | h | b ) for 0 ≤ x ≤ L , we only need to estimatefor − L ≤ x ≤ (cid:90) − L e x ( n − σ ( L ) n − (cid:18) e x + e − x + e x σ ( L ) (cid:19) q dx ≤ C (cid:90) L e − x ( n − σ ( L ) n − (cid:18) e x + e − x + e − x σ ( L ) (cid:19) q − n +1+ n − dx ≤ C (cid:90) L e − x ( n − σ ( L ) n − e − x ( q − n +1) σ ( L ) − n +1 dx ≤ ˜ Cσ ( L ) − ( e − ( q − L + 1) . For the third inequality, we used the following: (cid:18) e x + e − x + e − x σ ( L ) (cid:19) q − n +1+ n − ≤ (cid:18) e x (cid:19) q − n +1 (cid:18) e x + e − x σ ( L ) (cid:19) − n +1 ≤ q − n +1 e − x ( q − n +1) σ ( L ) − n +1 . Hence, the integral from (3.13) converges to zero as L → ∞ regardless of σ ( L ).Note that (3.6) and (3.10) are the only places that require an additional condition (3.7).Combining all, we obtain(3.14) p − p = (cid:90) ∂C L U ( V )( ν ) dσ b + o (1)= 2 (cid:90) F + ,L V ( H b − H g ) dσ g + o (1) , so it proves Theorem 1.3. Remark . From the above estimates, we observe that the metric falloff rate plays aninteresting role. If q ≥ n −
1, then the mass formula (1.5) holds regardless of σ ( L ). If n 1, then the integral on E + ,L may not converge to zero provided σ ( L ) does not increasefast enough. Similarly, if q ≥ n − 1, the mean curvature difference on the lateral surfacedoes not contribute to the mass by (3.9)-(3.13). As the mass formula for asymptoticallyflat manifolds via large coordinate cubes is considered in [17], one may attempt to use largerectangles {| x | ≤ L, | x i | ≤ σ ( L ) for i = 2 , . . . , n } instead of cylinders C L . However, ourestimates suggest that the dihedral angle deficit on edges and the mean curvature differenceson the lateral faces and { x = − L } will not affect the mass if q ≥ n − σ ( L ) = e kL for some k > q > n , one can reduce(3.7) as k ( n − − q ) + n − − q > ⇒ k ≥ n − , which is independent of q . Thus, we have the following corollary. YPERBOLIC MASS VIA HOROSPHERES 17 Corollary 3.3. Let ( M n , g ) , n ≥ , be an asymptotically hyperbolic manifold with metricfalloff rate q > n . Define Σ L = { y = e − L , | ˆ y | < e n − L } . Let ν g denote the unit normalvector to { y = e − L } in ( M, g ) pointing toward { y = 0 } . Let H g be the mean curvature of Σ L with respect to ν g in ( M, g ) . Then, as L → ∞ , (3.15) p − p = 2 (cid:90) Σ L V ( H b − H g ) dσ g + o (1) where V = t − z = y on M \ K . Now we conclude the proof of Theorem 1.1. Proof of Theorem 1.1. We only need to estimate the integral on { x = L } \ F + ,L :(3.16) (cid:90) { x = L }\ F + ,L V O ( | h | b ) dσ b ≤ C ω n − (cid:90) ∞ σ ( L ) e L (cid:32)(cid:18) cosh L + e L ρ (cid:19) − (cid:33) − q e L ( n − ρ n − dρ ≤ C e nL (cid:90) ∞ σ ( L ) (cid:18) e L + e − L + e L ρ (cid:19) q ρ n − dρ ≤ C e L ( n − q ) (cid:90) ∞ σ ( L ) ρ n − − q dρ ≤ ˜ Ce L ( n − q ) σ ( L ) n − − q . The last inequality implies that if q = n , then the above integral converges to zero as L → ∞ regardless of σ ( L ). If n < q < n , we let σ ( L ) = e kL with k satisfying(3.17) k ( n − − q ) + n − q < . Considering q > n , we can find the inequality independent of q :(3.18) k ≥ n . Let σ ( L ) = e n L . It is clear that σ ( L ) satisfies the assumptions in Theorem 1.3. Thus, byapplying Theorem 1.3, we get(3.19) p − p = 2 (cid:90) { x = L } V ( H b − H g ) dσ g − (cid:90) { x = L }\ F + ,L V O ( | h | b ) dσ b + o (1)= 2 (cid:90) H L V ( H b − H g ) dσ g + o (1) . (cid:3) We end this section by giving a characterization of the region that does not affect themass formula (1.5). Proposition 3.4. Let ( M n , g ) , n ≥ , be an asymptotically hyperbolic manifold with metricfalloff rate q > n . Let Σ L = { y = e − L , | ˆ y | ≤ σ ( L ) } where σ ( L ) satisfies (1.4) . For a givensubset U of M and large L > , define Θ( U, L ) = e − L ( n − | U ∩ Σ L | b . Suppose (3.20) Θ( U, L ) = o ( e L ( q − n ) ) as L → ∞ Then, as L → ∞ , (3.21) p − p = 2 (cid:90) Σ L \ U V ( H b − H g ) dσ g + o (1) . In particular, if q = n , the above holds if Θ( U ) := lim sup L →∞ Θ( U, L ) = 0 . Proof. By direct computation, we have(3.22) (cid:90) U ∩ Σ L V ( H b − H g ) dσ g ≤ C (cid:90) U ∩ Σ L e L (cid:18) e L + e − L + e L | ˆ x | (cid:19) q dσ b ≤ C e L (1 − q ) | U ∩ Σ L | b ≤ ˜ Ce L ( n − q ) o ( e L ( q − n ) ) . Hence, the proposition follows by Theorem 1.3. (cid:3) As pointed out in the introduction, a nontrivial asymptotically hyperbolic manifolds can-not be localized in a region appearing in the above proposition. One can interpret Θ( U ) as the asymptotic size of a subset U at infinity. Note that the related concept was consideredfor the asymptotically flat setting (see [6, Section 2.2]).For a conformally compact asymptotically hyperbolic manifold, the value of Θ( U ) is equiv-alent to the measure of ∂ ∞ U , where ∂ ∞ U is the boundary of U intersecting with the conformalinfinity. Since the mass of such manifolds can be defined as the integral on the conformalinfinity (see [23]), it is natural that any region U with Θ( U ) = 0 does not affect the totalmass.We also remark that an asymptotically hyperbolic metric is localizable in a subset U withΘ( U ) > 0. In other words, for such U in M , there exists an asymptotically hyperbolic metric( M n , g ) with R g ≥ − n ( n − 1) such that g is isometric to hyperbolic metric outside U and R g > − n ( n − 1) somewhere in U , proved by Chru´sciel and Delay [8].By the positive mass theorem and Proposition 3.4, we can state the following optimizedversion of Collorary 1.5. Corollary 3.5. Let ( M n , g ) , n ≥ , be an asymptotically hyperbolic manifold with metricfalloff rate q > n . Let U = (cid:40) p ∈ (cid:91) L ≥ L Σ L : H Σ L < n − (cid:41) . If Θ( U, L ) satisfies (3.20) , then ( M, g ) is isometric to hyperbolic space ( H n , b ) . YPERBOLIC MASS VIA HOROSPHERES 19 Appendix A. The mass formulas using the mean curvature on large spheres To state a new expression of the mass of both asymptotically flat and hyperbolic manifolds,we recall the definition of asymptotically flat manifolds here (see section 2 for asymptoticallyhyperbolic manifolds). Definition A.1. A Riemannian manifold ( M n , g ) is said to be asymptotically flat if thereexist a compact set K ⊂ M and a diffeomorphism Φ : M \ K → R n \ B R (0) such that(1) as r → ∞ , | g ij − δ ij | + r | ∂ k g ij | + r | ∂ l ∂ k g ij | = O ( r − q ) , q > n − . (2) (cid:82) M R g dµ g < ∞ where R g is the scalar curvature of g .Let h = (Φ − ) ∗ g − δ . Then the condition (1) above can be written equivalently as | h | δ + r | ˚ ∇ h | δ + r | ˚ ∇ h | δ = O ( r − q ) , q > n − . The ADM mass (or energy) [3] of ( M n , g ) is defined as(A.1) m ADM ( g ) = 12( n − ω n − lim r →∞ (cid:90) S r ( g ij,j − g jj,i ) ν i dσ = 12( n − ω n − lim r →∞ (cid:90) S r U (1)( ν ) dσ where the one form U is defined as (2.4) (with the background metric δ instead of thehyperbolic metric b ). As mentioned in the introduction, it is known that m ADM ( g ) is invariantunder the choice of coordinates. (R. Bartnik [4], P. Chru´sciel [7])Now, we give new geometric formulas of the mass of asymptotically flat and hyperbolicmanifolds respectively. Proposition A.2. If ( M n , g ) is an asymptotically flat manifold, then the ADM mass canbe computed as m ADM ( g ) = 1( n − ω n − lim r →∞ (cid:20)(cid:18)(cid:90) S r n − r − H g (cid:19) dσ g + 1 r ( | S r | δ − | S r | g ) (cid:21) where | S r | δ and | S r | g are the area of S r with respect to Euclidean metric and the metric g ,respectively. Similarly, if ( M n , g ) is an asymptotically hyperbolic manifold, then the massvector can be obtained from p = 2 lim r →∞ (cid:34)(cid:90) S r √ r (cid:32) √ r r ( n − − H g (cid:33) dσ g + 1 r ( | S r | b − | S r | g ) (cid:35) ,p i = 2 lim r →∞ (cid:90) S r z i (cid:32) √ r r ( n − − H g (cid:33) dσ g . Remark A.3 . One can derive the same mass formula for asymptotically locally hyperbolicmetrics. For example, if ( M n , g ) is an asymptotically locally hyperbolic manifold asymptoticto a model space [ r , ∞ ) × N n − equipped with the metric b = dr κ + r + r h , where r > (cid:112) | κ | and ( N n − , h ) is an ( n − κ (see[11] for the precise definition), then the mass p of ( M, g ) can be computed as the following: p = 2 lim r →∞ (cid:34)(cid:90) S r √ κ + r (cid:32) √ κ + r r ( n − − H g (cid:33) dσ g + κr ( | S r | b − | S r | g ) (cid:35) . Proof of Proposition A.2. First, suppose that ( M n , g ) is asymptotically flat and g = δ isEuclidean metric. On coordinate spheres S r , we have A δ = 1 r δ | S r . By Proposition 2.1, we have on S r , U (1)( ν ) = 2( H δ − H g ) + | A δ | δ O ( | h | δ ) + O ( | ˚ ∇ h | δ | h | δ ) − V (cid:104) A δ , h (cid:105) δ | Sr − div S r X = 2( H δ − H g ) − r tr S r δ h − div S r X + O ( r − q − ) . By integrating it on S r , we get (cid:90) S r U (1)( ν ) dσ δ = (cid:90) S r (cid:20) H δ − H g ) − r tr S r δ h + O ( r − q − ) (cid:21) dσ δ = (cid:90) S r H δ − H g ) dσ g − (cid:90) S r (cid:18) H δ − H g + 1 r (cid:19) tr S r δ h dσ δ + O ( r n − q − )= (cid:90) S r H δ − H g ) dσ g − r (cid:90) S r tr S r δ h dσ δ + O ( r n − q − )= (cid:90) S r H δ − H g ) dσ g + 2 r ( | S r | δ − | S r | g ) + O ( r n − q − )By taking the limit r → ∞ , we can conclude the formula for the ADM mass.Now we suppose that ( M n , g ) is asymptotically hyperbolic and g = b is the hyperbolicmetric. On coordinate spheres S r , we have A b = √ r r b | S r . We first show the formula for p ( g ). Let V = √ r . By direct computation, we have dV ( ν ) = r √ r dr (cid:18) √ r ∂∂r (cid:19) = r, (tr S r b h ) dV ( ν ) − V (cid:104) A b , h (cid:105)| S r = (tr S r b h ) (cid:32) r − √ r · √ r r (cid:33) = − r tr S r b h. YPERBOLIC MASS VIA HOROSPHERES 21 By Proposition 2.1, we have on S r , U ( V )( ν ) = 2 V ( H δ − H g ) − r tr S r δ h − div S r X + O ( r − q ) . By integrating it on S r , we get (cid:90) S r U ( V )( ν ) dσ b = (cid:90) S r V ( H b − H g ) dσ g + 2 r ( | S r | δ − | S r | g ) + O ( r n − q ) . So we conclude the formula of p ( g ). Let V = z i for i = 1 , . . . , n . Then we have dV ( ν ) = dz i (cid:18) √ r ∂∂r (cid:19) = √ r z i r , (tr S r b h ) dV ( ν ) − V (cid:104) A b , h (cid:105)| S r = (tr S r b h ) (cid:32) √ r z i r − z i · √ r r (cid:33) = 0 . By Proposition 2.1, we have on S r , U ( V )( ν ) = 2 V ( H δ − H g ) − div S r X + O ( r − q +1 ) . By integrating it on S r , we get (cid:90) S r U ( V )( ν ) dσ b = (cid:90) S r z i ( H b − H g ) dσ g + O ( r n − q ) . So the proof is done. (cid:3) Remark A.4 . A similar formula was used in the paper of Fan-Shi-Tam [12, Lemma 2.2]. Wecan obtain the analogous formula of the component p for asymptotically hyperbolic setting: √ r (cid:90) S r H g dσ g = (cid:18) ( n − r + n − r (cid:19) | S r | g + 1 r | S r | b − p O ( r n − q ) . Proof. In the proof of Proposition A.2, we have (cid:90) S r U ( V )( ν ) dσ δ = (cid:90) S r (cid:20) V ( H b − H g ) − r tr S r b h + O ( r − q ) (cid:21) dσ b where V = √ r . Note that on S r , we have V H b = 1 + r r ( n − , thus we get (cid:90) S r U ( V )( ν ) dσ δ = 2( n − (cid:18) r r (cid:19) | S r | b + (cid:90) S r (cid:20) − V H g − r tr S r b h + O ( r − q ) (cid:21) dσ b = 2( n − (cid:18) r r (cid:19) | S r | b − (cid:90) S r V H g dσ g + (cid:90) S r V H g (cid:18) 12 tr S r b h (cid:19) dσ b − r (cid:90) S r tr S r b h dσ b + O ( r n − q )= 2( n − (cid:18) r r (cid:19) | S r | b − (cid:90) S r V H g dσ g + (cid:18) V H g − r (cid:19) ( | S r | g − | S r | b ) + O ( r n − q )= 2 (cid:18) ( n − r + n − r (cid:19) | S r | g + 2 r | S r | b − √ r (cid:90) S r H g dσ g + O ( r n − q ) . By rearranging it and taking the limit, we conclude the proof. (cid:3) References [1] Lars Andersson, Mingliang Cai, and Gregory J. 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