Asymptotically hyperbolic manifold with a horospherical boundary
aa r X i v : . [ m a t h . DG ] F e b ASYMPTOTICALLY HYPERBOLIC MANIFOLD WITH AHOROSPHERICAL BOUNDARY
XIAOXIANG CHAI
Abstract.
We discuss asymptotically hyperbolic manifold with a noncompactboundary which is close to a horosphere in a certain sense. The model case isa horoball or the complement of a horoball in standard hyperbolic space. Weshow some geometric formulas. Introduction
In the upper half-space model b = x n ) ((d x ) + · · · + (d x n ) ) , x n > n -space H n , we fix the horosphere H = { x ∈ H n : x n = 1 } . Definition 1.
We say that a manifold ( M n , g ) with a noncompact boundary isasymptotically hyperbolic with a horospherical boundary if outside a compact set K ⊂ M , M is diffeomorphic to { x n } ( { x n > } ) minus a compact set and themetric admits the decay rate (1) | e | b + | ¯ ∇ e | b + | ¯ ∇ ¯ ∇ e | b = O (e − τr ) where e = g − b , τ > n and r is the b -geodesic distance to a fixed point o (in H n ). We pick o to be (0 , . . . ,
1) without loss of generality, the b -distance to o is then(2) 2 cosh r = x n ( | ˆ x | + ( x n ) + 1) , where ˆ x = ( x , . . . , x n − ) and | ˆ x | is the Euclidean distance of ˆ x to the origin ˆ o . Seefor example [BP92, Chapter A] for the distance formula. Replacing e r with cosh r in (1) is somehow more convenient. We are concerned with behaviors near infinity,we assume in this article that the manifold M is diffeomorphic to { x n } with asmooth metric g . The other case { x n > } is handled in the same way, for clarityof presentation, we omit this case and leave the details to the reader.For ε ∈ (0 , C ε = { ε x n } ∩ {| ˆ x | ρ ( ε ) } . We also write C ( ε ) sometimes for clarity and also for others defined below. Wedenote A ε ,ε = C ε \ C ε with ε < ε . Inner boundary ∂ ′ C ε = ∂C ε \H of C ε is theunion of F ε = {| ˆ x | ρ ( ε ) , x n = ε } and S ε = { ε x n , | ˆ x | = ρ ( ε ) } . We deonte by ν the g -normal to ∂ ′ C ε . Along the horosphere H , c ε = { x n = 1 , | ˆ x | ρ ( ε ) } and a ε ,ε = c ε \ c ε , and s ε = { x n = 1 , | ˆ x | = ρ ( ε ) } . We denote by µ the normal to s ε in the horosphere.Here ρ ( ε ) is a smooth decreasing function on (0 ,
1) satisfying ρ ( ε ) → ∞ as ε → C ε is a parabolic cylinder in the region { x n } . Analogously, we candefine C ε = { x n ε − } ∩ {| ˆ x | ρ ( ε ) } . We also write C ( ε ) sometimes for clarity. We denote A ε ,ε = C ε \C ε with ε < ε .Inner boundary ∂ ′ C ε = ∂ C ε \H of C ε is the union of F ε = {| ˆ x | ρ ( ε ) , x n = ε − } and S ε = { x n ε − , | ˆ x | = ρ ( ε ) } . Let V = x n and V k = x k x n with k ranges from 1 to n −
1. We will show later that V and V k is a function satisfying the following condition. Let ¯ η = ∂ n , we see that(3) ¯ ∇ V = V b, ∂ ¯ η V = − V along H . The functions satisfying the condition ¯ ∇ V = V b is a static potential which isclosely related to the static spacetime. The scalar curvature R g admits the decayrate V ( R g + n ( n − ∇ i U i + O (e − τr + r )where U = V div e − V d(tr b e ) + tr b e d V − e ( ¯ ∇ V, · )is the mass integrands. See [CH03].We denote by d v , d σ and d λ respectively the n , n − n − M ( V )(4) M ( V ) = lim ε → (cid:20)Z ∂ ′ C ε U i ¯ η i d v − Z ∂c ε V e αn µ α d λ (cid:21) whenever exists. This is motivated by [Wan01], [CH03], [AdL20] and [Cha18b]. Wewill show the existence of M ( V ) in Theorem 3.Although not explicitly, [ACG08] show that Theorem 1. On T n − × [0 , , there does not exist a metric g with R g > − n ( n − , H g > − ( n − on the top face T n − × { } and H g > n − on the bottom face. However, it seems that a spinorial proof was not available explicitly in the litera-ture. One could use the boundary conditions [CH03, (4.25)] to write down a proof.We see that the quantities (4) we defined is a natural candidate to show a simi-lar positive mass theorem as in [Wan01, CH03, AdL20] thus giving a noncompactversion of Theorem 1. Although we have not shown the geometric invariance of M , that is the independence of M on the coordinate chart at infinity. The naturalconditions should be R g > − n ( n −
1) and H g + n − > SYMPTOTICALLY HYPERBOLIC MANIFOLD WITH A HOROSPHERICAL BOUNDARY 3 is to explore the spacetime version of M in (4). We will address these questions ina later work.We show that the M ( V ) can be evaluated via the Ricci tensor and the secondfundamental form the horosphere H similar to those in [Cha18b]. Theorem 2.
We assume that ρ ( ε ) = ε − α where α > .We have that M ( V ) = − n [ Z C ε G ( X, ν )d v + Z ∂C ε W ( X, µ )d λ ] + o (1) where G = Ric − Rg − ( n − n − g is the modified Einstein tensor and W = A − Hg − ( n − g . Here the pair V and X is given in (6). The important property of X is that itis a conformal Killing vector and tangent to the horosphere H .The article is organized as follows:In Section 2, we show some asymptotics which motivates the Definition 4. wecollect basics of conformal Killing vectors in standard hyperbolic space and provesome length estimates under the metric g . In Section 3, we give a proof of Theorem2. Acknowledgment
I would like to thank Prof. Kim Inkang for discussions ontetrahedron in hyperbolic 3-space. I would like to acknowledge the support of KoreaInstitute for Advanced study under the research number MG074401.2.
Background
The Christoffel symbols of the standard hyperbolic metric b is given by¯Γ jin = − ( x n ) − δ ji , ¯Γ γαβ = 0 , ¯Γ nαβ = ( x n ) − δ αβ for i , j possibly being n ranging from 1 to n and α, β, γ ranges from 2 to n . Weuse this convention later as well.Consider again the half space model and the slice { x n = 1 } , then (with allquantities evaluated on the slice { x n = 1 } )Γ nαβ = δ αβ + ( ¯ ∇ α e βn + ¯ ∇ β e αn − ¯ ∇ n e αβ ) . The outward normal is η i = ( g nn ) − g ni and the second fundamental form is A αβ = − ( g nn ) − Γ nαβ . Hence the mean curvature is H = − h αβ ( g nn ) − Γ nαβ . Note that h αβ = δ αβ − e αβ + O (e − τr ) and ( g nn ) − = 1 + e nn + O (e − τr ), wehave 2( H + n − − h αβ ( g nn ) − Γ nαβ = − X α (2 ¯ ∇ α e αn − ¯ ∇ n e αα ) − ( n − e nn + 2 e αα + O (e − τr ) . XIAOXIANG CHAI
Write U in coordinates, we have with E = tr b e U i = V g ik ¯ ∇ j e jk − V ¯ ∇ i E + E ¯ ∇ i V − g ik e jk ¯ ∇ k V. Along H , U i ¯ η i == V ¯ η i ¯ ∇ j e ji − V ¯ η i ¯ ∇ i e jj + E ¯ η i ¯ ∇ i V − ¯ η i e ji ¯ ∇ j V = V ¯ ∇ α e αn − e αn ¯ ∇ α V − V ¯ ∇ n e αα + e αα ¯ ∇ n V + O (e − τr + r ) . So we have that2 V ( H + n −
1) + U i ¯ η i = − V ¯ ∇ α e αn − e αn ¯ ∇ α V − ( n − V e nn + (2 V + ¯ ∇ n V ) e αα + O (e − τr + r )= − ¯ ∇ α ( V e αn ) − ( n − V e nn + (2 V + ¯ ∇ n V ) e αα + O (e − τr + r )= − ∂ α ( V e αn ) + V ¯Γ iαn e αi + V ¯Γ iαα e in − V ( n − e nn + (2 V + ¯ ∇ n V ) e αα + O (e − τr + r )= − ∂ α ( V e αn ) + O (e − τr + r ) . (5)2.1. Conformal Killing vectors.
Since the metric of the hyperbolic space usingthe upper half space model is conformal to standard metric of the Euclidean metric.We investigate ¯ ∇ i X j for such X which is a conformal Killing vector with respectto the Euclidean metric. We use repeatedly that the fact¯ ∇ i X j = ¯ ∇ i X j which follows easily from conformality to δ .The X = x i ∂ i X = x i ∂ i , then¯ ∇ i X j = ∂ i x j + x k Γ jik = δ ji + x k ( x n ) ( g ij,k + g kj,i − g ik,j )= δ ji + ( x n ) ( x n ∂∂x n (( x n ) − ) δ ij + x j g jj,i − x i g ii,j )= ( x n ) ( x j g jj,i − x i g ii,j ) . So ¯ ∇ i X j + ¯ ∇ j X i = 0. We see that vector x i ∂ i is actually a Killing vector.Now we consider the the translation vectors ∂ i . First, obviously ∂ i for i = n isobviously a Killing vector field. For X = ∂ n ,¯ ∇ i X j = X k Γ jik = Γ jin = − ( x n ) − δ ji . So ¯ ∇ i X j + ¯ ∇ j X i = − x n ) − δ ji and ∂ n is a conformal Killing vector. We remark here the recent article of Jangand Miao [JM21] uses x n to express the usual mass along horospheres convergingto infinity by taking ρ ( ε ) to grow fast. It might be possible to obtain a formulasimilar to Theorem 2 evaluating the their mass expression only along horospheresby exploiting the special properties of the vector − ∂ n .Now we consider the family X = X j ∂ j with X j being h x, a i δ x j − h x, x i δ a j SYMPTOTICALLY HYPERBOLIC MANIFOLD WITH A HOROSPHERICAL BOUNDARY 5 where a is nonzero constant vector in Euclidean space. We see first that ∂ i X j = a i x j + h x, a i δ δ ji − x i a j . We consider separately two cases of values of i, j . For i = n , we have that¯ ∇ i X j = ∂ i X j + X k Γ jik = ∂ i X j + X n Γ jin = a i x j + h x, a i δ ji − x i a j − [ h x, a i δ x n − h x, x i δ a n ]( x n ) − δ ji = a i x j − x i a j + h x, x i δ a n ( x n ) − δ ji . For i = n , we have that¯ ∇ n X j = ∂ n X j + X k Γ jnk = ∂ n X j − X k ( x n ) − δ jk = ∂ n X j − X j ( x n ) − = a n x j + h x, a i δ jn − x n a j − [ h x, a i δ x j − h x, x i δ a j ]( x n ) − = a n x j − x n a j − x n h x, a i δ x j + x n | x | δ a j . And for j = n , ¯ ∇ j X n = ¯ ∇ j X n = ∂ j X n + X k Γ njk = ∂ j X n + X α ( x n ) − δ jα = a j x n − x j a n + h x, a i δ x j x n − x n h x, x i δ a j . For j = n , we have that ¯ ∇ n X n = ∂ n X n + X k Γ nnk = h x, a i δ − X n ( x n ) − = h x, a i δ − [ x n h x, a i δ − | x | δ a n ]( x n ) − = | x | δ ( x n ) − a n . To summarize, we have that¯ ∇ i X j + ¯ ∇ j X i = x n h x, x i δ a n δ ji . We have that h x, ∂ n i δ x − h x, x i δ ∂ n is a conformal Killing vector and h x, ∂ i i δ x − h x, x i δ ∂ i is proper Killing vector for all i = n .We can construct a conformal Killing vector h x, ∂ n i δ x − h x, x i δ ∂ n − x which has no tangential component to H .Now we consider the the rotation vectors x i ∂ j − x j ∂ i . If both i and j is not n ,we see easily that x i ∂ j − x j ∂ i is a Killing vector. XIAOXIANG CHAI
We consider now the vector X = x n ∂ k − x k ∂ n with k = n , the components X j = x n δ jk − x k δ jn . We compute ¯ ∇ i X j . For i = n, j = n ,¯ ∇ i X j = ∂ i X j + X k Γ jik = δ ni δ kj − δ ki δ jn + X n Γ jin = δ ni δ kj − δ ki δ jn + x k ( x n ) − δ ji = x k ( x n ) − δ ji . So ¯ ∇ i X j + ¯ ∇ j X i = X k Γ jik + X k Γ ijk = 2 x k x n δ ji . We have for j = n , ¯ ∇ n X j = ∂ n X j + X l Γ jnl = δ jk − δ kn δ jn + X α Γ jnα = δ jk − δ kn δ jn + x n Γ jnk = δ jk − δ kn δ jn − δ jk = 0 . Similarly, ¯ ∇ j X n = 0. And¯ ∇ n X n = X j Γ njn = X n Γ nnn = x k x n . So we see that ¯ ∇ i X j + ¯ ∇ j X i = x k x n δ ji for all i and j .We consider the vectors Y = x − ∂ n and Y ( k ) = ( x n ∂ k − x k ∂ n ) + ( h x, ∂ k i x − h x, x i δ ∂ k ) . Both Y and Y ( k ) are tangent to H since x n = 1.The construction is by shifting a conformal Killing vector by a Killing vector.This is motivated by a recent work of the author [Cha21]. We have(6) div Y = nx n , div Y ( k ) = n x k x n by previous calculations. Along H , Y = ˆ x and Y ( k ) = ∂ k + h ˆ x, ∂ k i δ − h ˆ x, ˆ x i δ ∂ k . They are conformal Killing vectors along H , specifically ∂ α Y β + ∂ β Y α = 2 δ βα , div H Y = n − ∂ α ( Y ( k ) ) β + ∂ β ( Y ( k ) ) α = 2 h ˆ x, ∂ k i δ δ βα , div H Y ( k ) = ( n − h b x,∂ k i δ . Remark . The vectors Y and Y ( k ) constructed here can be used to prove similarresults as in [WX19] by considering free boundary hypersurfaces supported on thehorosphere. SYMPTOTICALLY HYPERBOLIC MANIFOLD WITH A HOROSPHERICAL BOUNDARY 7
Lemma 1.
These vectors admit the growth rate (7) | Y | b + | ¯ ∇ Y | b + | Y ( k ) | b + | ¯ ∇ Y ( k ) | b = O (cosh r ) as r → ∞ .Proof. The proof is by direct calculation for each term. The length of Y = x − ∂ n is | Y | b = x n p | ˆ x | + ( x n − √ x n ( | ˆ x | + | x n − | ) √ x n ( | ˆ x | + x n + 1)= O (cosh r ) , according to (2). For i = n , j = n , ¯ ∇ i Y j = δ ji x n ; ¯ ∇ i Y n = − ¯ ∇ n Y i = − x i x n ;¯ ∇ n Y n = x n . So the length of ¯ ∇ Y is | ¯ ∇ Y | b = x n ( n + 2 | ˆ x | ) = O ( | ˆ x | x n ) = O (cosh r ) . The length of Y ( k ) is | Y | b =( x n ) − [( x n ) + ( x k ) + h x, x i δ − h x, x i δ x n ]= x n ) [( x k ) + ( x n − h x, x i δ ) ] x n ) [ | ˆ x | + 2( x n ) + h x, x i δ ] x n ) [ | ˆ x | + 2( x n ) + | ˆ x | + ( x n ) ] . We see then that | Y | b = O (cosh r ). We write U = Y ( k ) , we have that for i = n , j = n , ¯ ∇ i U j = a i x j − x i a j + x k x n δ ji . For j = n , ¯ ∇ j U n = − ¯ ∇ n U j = a j x n + x j x k x n − x n h x, x i δ a j and ¯ ∇ n U n = x k x n . Here a is the vector ∂ k . Summing up these entries of ¯ ∇ U , wehave that | ¯ ∇ U | b = ( x k ) ( x n ) + X i = n,j = n ( a i x j − x i a j + x k x n δ ji ) + 2 X j = n ( a j x n + x j x k x n − x n h x, x i δ a j ) = ( x k ) ( x n ) + [ | ˆ x | + | ˆ x | + ( n − ( x k ) ( x n ) − x k ) ]+ 2[( x n ) + | ˆ x | ( x k ) ( x n ) + h x,x i δ x n ) + 2( x k ) − h x, x i δ − x n ) ( x k ) h x, x i δ ] . Using the relation that h x, x i δ = ( x n ) + | ˆ x | , the last line reduces to | ¯ ∇ U | b = ( x n ) − ( h x, x i δ + ( x k ) ) . We see then | ¯ ∇ U | b = O (cosh r ). (cid:3) XIAOXIANG CHAI Proof
Finiteness of M ( V ) . First, we show that the quantity M ( V ) is well definedunder natural conditions. Theorem 3. If ( M, g ) is an asymptotically hyperbolic manifold with a horosphericalboundary and if e r ( R g + n ( n − ∈ L ( M ) and e r ( H + n − ∈ L ( M ) , then thequantity M ( V ) defined in (4) exists and is finite. We are concerned only with behavior near infinity, so we can assume that M isdiffeomorphic to H n minus a horoball. We then set up notations.Before going to the proof of Theorem 3, we have the following elementary lemmaconcerning the integral of cosh − τ +1 r on the regions defined as I = { ε x n } ∩ { ρ ( ε ) | ˆ x | ρ ( ε ) } ,I = { ε x n ε } ∩ {| ˆ x | ρ ( ε ) } ,I = { x n ε − } ∩ { ρ ( ε ) | ˆ x | ρ ( ε ) } ,I = { ε − x n ε − } ∩ {| ˆ x | ρ ( ε ) } . Lemma 2.
Assume that ρ ( ε ) → ∞ as ε → , we have that for k ∈ { , , , } , Z I k cosh − τ +1 r d¯ v → if ε → and ε → .Proof. We deal with I first. From (2), Z I cosh − τ +1 r d¯ v = Z ε d x n Z ρ ( ε ) | ˆ x | ρ ( ε ) ( x n ) − n +1 cosh − τ +1 r dˆ x = Z ε ( x n ) − n +1 d x n Z ρ ( ε ) | ˆ x | ρ ( ε ) ( | ˆ x | + ( x n ) + 12 x n ) − τ +1 dˆ x − τ Z ε ( x n ) − n +2 τ d x n Z ρ ( ε ) | ˆ x | ρ ( ε ) | ˆ x | − τ dˆ x C Z ρ ( ε ) ρ ( ε ) s − τ + n − d s Note that − n + 2 τ > − τ + n − < − n < −
1, this is o (1) obviously aslong as as ε → ρ ( ε ) ր ∞ . Similarly, on the region I , Z I cosh − τ +1 r d¯ v − τ Z ε ε ( x n ) − n +2 τ d x n Z ρ ( ε )0 ( ρ + 1) − τ +1 ρ n − d s which is also o (1) as ε → ε → I , Z ε − d x n Z ρ | ˆ x | ρ ( x n ) − n +1 cosh − τ +1 r dˆ x SYMPTOTICALLY HYPERBOLIC MANIFOLD WITH A HOROSPHERICAL BOUNDARY 9
We have that Z ε − Z ρ | ˆ x | ρ (cid:16) | ˆ x | +( x n ) +12 x n (cid:17) − τ +1 ( x n ) − n +1 dˆ x d x n =2 − τ Z ε − Z ρ ( ε ) ρ ( ε ) ( x n ) − n +2 τ ( s + ( x n ) + 1) − τ +1 d s d x n C Z ε − Z ρ ( ε ) ρ ( ε ) ( x n ) − n +2 τ ( s + ( x n ) + 1) n − τ − p − τ +1 − n − τ − p s n − d s d x n C Z ε − ( x n ) − p d x n Z ρ ( ε ) ρ ( ε ) ( s + 1) − τ +1 − n − τ − p s n − d s C Z ∞ t − p d t Z ∞ ρ ( ε ) ( s + 1) − τ +1 − n − τ − p s n − d s. We fix some p with 1 < p < τ −
1. Then power n − − τ + 1 − n − τ − p )is less than −
1. We see the integral is o (1) as ε →
0. With a similar argument, onthe region I , Z ε − ε − Z | ˆ x | ρ ( x n ) − n +1 cosh − τ +1 r dˆ x d x n C Z ε − ε − t − p d t Z ρ ( ε )0 ( s + 1) − τ +1 − n − τ − p s n − d s C Z ∞ ε − t − p d t Z ∞ ( s + 1) − τ +1 − n − τ − p s n − d s we fix the same p as before, this integral is also o (1). (cid:3) Proof of Theorem 3.
The proof is basically a restatement of the expansion we de-rived earlier. See [CH03]. We have the expansion of the scalar curvature R g near b that R g = − n ( n −
1) + DR ( e ) + O (e − τr ) . The specific form of O ( e − τr ) is O ( | e | + | ¯ ∇ e | + | e || ¯ ∇ e | ) (See for example [Cha18a]).Here, DR ( e ) = div(div e − d E ) + ( n − E is the linearization operator of the scalar curvature. We have V DR ( e ) = h D ∗ R ( V ) , e i + ¯ ∇ i U i where D ∗ R = ¯ ∇ V − V b is the formal L adjoint of DR . Since V is the staticpotential (3), we have(8) V ( R g + n ( n − ∇ i U i + O (e − τr + r ) . Now we integrate (8) over the region A = A ε ,ε , we see that Z A V ( R g + n ( n − Z ∂ ′ C ( ε ) U i ¯ η i − Z ∂ ′ C ( ε ) U i ¯ η i + Z A O (e − τr + r ) + Z a ε ,ε U i ¯ η i . Using (5), so Z A V ( R g + n ( n − Z ∂ ′ C ( ε ) U i ¯ η i − Z ∂ ′ C ( ε ) U i ¯ η i + Z A O (e − τr + r )+ Z a ε ,ε [ − ∂ α ( V e αn ) − V ( H g + n − . Using divergence theorem on a ε ,ε , we have that( Z ∂ ′ C ( ε ) U i ¯ η i − Z ∂c ε V e αn θ α ) − ( Z ∂ ′ C ( ε ) U i ¯ η i − Z ∂c ε V e αn θ α )= Z A V ( R g + n ( n − Z a ε ,ε V ( H g + n −
1) + Z A O (e − τr + r )= Z A O (e − τr + r ) + Z a ε ,ε O (e − τr + r ) = o (1)(9)by Lemma 2, (2), and the integrability of V ( R g + n ( n − V ( H g + n − M ( V ) exists and is finite. (cid:3) Now we turn to the proof of Theorem 2.
Proof of Theorem 2.
We use the method of [Her16]. We define a cutoff function χ which vanish inside C ( ε ), equals 1 outside C ( ε ) andˆ g = χg + (1 − χ ) b The cutoff function is a product of two cutoff functions χ = χ ( x n ) χ ( | ˆ x | ). Thefunction χ ( x n ) = f ( − log x n ) where f ( t ) is the cutoff vanish inside [0 , − log ε ]and equal to 1 in [ − log ε, ∞ ) with the estimate f + (log ε ) | f ′ | + (log ε ) | f ′′ | C. We find then |∇ χ | b = x n | ∂χ ∂x n | = x n · x n f ′ ( − log x n ) C log ε C and similarly |∇ χ | b C (log ε ) C . We define χ ( t ) to be the standard cutoffwhich vanishes inside [0 , ε − α ] and is equal to 1 in [ ε − α , ∞ ) with the estimate0 χ , | χ ′ | C ( ε − α − ε − α ) − , | χ ′ | C ( ε − α − ε − α ) − . It is easy to check that | χ | + | ¯ ∇ χ | + | ¯ ∇ χ | C SYMPTOTICALLY HYPERBOLIC MANIFOLD WITH A HOROSPHERICAL BOUNDARY 11 when x n ∈ (0 , ε − ] due to the condition α > . From the construction of the cutofffunction, ˆ g is an asymptotically hyperbolic metric, that is the difference ˆ e = ˆ g − b satisfies also the decay rate (1).The reader might want to visit the the model e t | dˆ x | +d t where t = − log x n fora better understanding of the construction of the cutoff function. We use mostly theupper half space model because of the conformality used in calculation of conformalKilling vectors.We shall also denote by ˆ g the complete metric obtained by gluing the hyperbolicmetric inside C ( ε ) and the metric g outside C ε .By divergence theorem, we have on A = C ε \ C ( ε ) by the divergence theoremand the second Bianchi identity that(10) Z ∂A ˆ G ( X, ˆ ν )dˆ σ = Z A ˆ ∇ i ( ˆ G ij X j )dˆ v = Z A ˆ G ij ˆ ∇ i X j dˆ v. We first analyze the term R A ˆ G ij ˆ ∇ i X j dˆ v . We useˆ ∇ i X j = h il ( ¯ ∇ l X j + X k (ˆΓ lik − ¯Γ lik ))= ¯ ∇ i X j + (ˆ g il − b il ) ¯ ∇ l X j + h il X k (ˆΓ lik − ¯Γ lik ) . In fact R A ˆ G ij (ˆ g il − b il ) ¯ ∇ l X j dˆ v is o (1) by noting that the decay of ˆ G ij (ˆ g il − b il ) ¯ ∇ l X j = O (cosh − τ r | ¯ ∇ X | ) and the growth rate (7). Similarly R A ˆ G ij ˆ g il X k (ˆΓ lik − ¯Γ lik )dˆ v is o (1). Z A ˆ G ij ¯ ∇ i X j dˆ v = Z A ˆ G ij ( ¯ ∇ i X j + ¯ ∇ j X i )dˆ v = n Z A div b X ˆ G ij b ij dˆ v = n Z A div b X ˆ G ij ˆ g ij dˆ v − n Z A div b X ˆ G ij (ˆ g ij − b ij )dˆ v = − n n Z A div b X ( ˆ R + n ( n − v − n Z A div b X ˆ G ij ( h ij − b ij )dˆ v We have that similarly the two terms R A div b X ˆ G ij ( h ij − b ij )dˆ v and R A div b X ( ˆ R + n ( n − v − d¯ v ) are o (1). Therefore,(11) Z A ˆ G ij ˆ ∇ i X j dˆ v = − n n Z A div b X ( ˆ R + n ( n − v + o (1) . We are concerned about R a ˆ G ( X, ˆ ν )dˆ σ . Denote a = c ε \ c ( ε ). Since X is tangentto the H , we use Gauss-Codazzi equation, we find that Z a ˆ G ( X, ν )dˆ σ = Z a X β ˆ D α ( ˆ A αβ − ˆ H ˆ ρ αβ )dˆ σ. We add an extra term to the tensor ˆ A − ˆ H ˆ ρ so that it is trace free if g is just thehyperbolic metric b . The modification is in spirit similar to the modified Einsteintensor G . So Z a ˆ G ( X, ˆ ν )dˆ σ = Z a X β ˆ D α ( ˆ A αβ − ˆ H ˆ ρ αβ − ( n − h αβ )dˆ σ. We apply the divergence theorem, we obtain(12) Z a ˆ G ( X, ˆ ν )dˆ σ = Z ∂a ˆ W ( X, ˆ µ )dˆ λ − Z a ˆ W αβ ˆ D α X β dˆ σ. Now we analyze the term R a ˆ W αβ ˆ D α X β dˆ σ . We useˆ D α X β =ˆ ρ αγ ( ¯ D γ X β + X ξ (ˆΛ βγξ − ¯Λ βγξ ))= ¯ D α X β + (ˆ ρ αγ − ¯ ρ αγ ) ¯ D γ X β + ˆ ρ αγ X ξ (ˆΛ βγξ − ¯Λ βγξ ) . The terms R a ˆ W αβ (ˆ ρ αγ − ¯ ρ αγ ) ¯ D γ X β dˆ σ and R a ˆ ρ αγ X ξ (ˆΛ βγξ − ¯Λ βγξ )dˆ σ are o (1).We know that Z a ˆ W αβ ¯ D α X β dˆ σ = Z a ˆ W αβ ( ¯ D α X β + ¯ D β X α )dˆ σ = n − Z a div b H X ¯ ρ αβ ˆ W αβ dˆ σ = − nn − Z a div b H X ˆ ρ αβ ˆ W αβ dˆ σ − n − Z a div b H X (ˆ ρ αβ − ¯ ρ αβ ) ˆ W αβ dˆ σ We have that R a div b H X (ˆ ρ αβ − ¯ ρ αβ ) ˆ W αβ dˆ σ and R a div b H X ˆ ρ αβ ˆ W αβ (dˆ σ − d¯ σ ) are o (1). Therefore,(13) Z a ˆ W αβ ˆ D α X β dˆ σ = − nn − Z a div b H X ( ˆ H + n − σ + o (1) . We have from (10), (11) , (12) and (13) that Z ∂A ˆ G ( X, ˆ ν )dˆ σ = Z ( ∂A ) \ a ˆ G ( X, ˆ ν )dˆ σ + Z a ˆ G ( X, ˆ ν )dˆ σ = Z ( ∂A ) \ a ˆ G ( X, ˆ ν )dˆ σ + Z ∂a ˆ W ( X, ˆ µ )dˆ λ − − nn − Z a div b H X ( ˆ H + n − σ = − n n Z A div b X ( ˆ R + n ( n − v + o (1) . Since div b X = nV and div b H X = ( n − V along H , by the same argument arriving(9), we have (2 − n ) M ( V ) + o (1)= − n n Z A div b X ( ˆ R + n ( n − v + − nn − Z a div b H X ( ˆ H + n − σ = Z ( ∂A ) \ a ˆ G ( X, ˆ ν )dˆ σ + Z ∂a ˆ W ( X, ˆ µ )dˆ λ. SYMPTOTICALLY HYPERBOLIC MANIFOLD WITH A HOROSPHERICAL BOUNDARY 13
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Korea Institute for Advanced Study, Seoul 02455, South Korea
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