On the long time existence and convergence of inverse curvature flow in anti-desitter- Schwarzschild space

Transcription

1 The University of Toledo The University of Toledo Digital Repository Theses and Dissertations 2013 On the long time existence and convergence of inverse curvature flow in anti-desitter- Schwarzschild space Hanyang Liu The University of Toledo ollow this and additional works at: Recommended Citation Liu, Hanyang, "On the long time existence and convergence of inverse curvature flow in anti-desitter-schwarzschild space" (2013). Theses and Dissertations This Dissertation is brought to you for free and open access by The University of Toledo Digital Repository. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of The University of Toledo Digital Repository. or more information, please see the repository's About page.

2 A Dissertation entitled On the Long Time Existence and Convergence of Inverse Curvature low in Anti-deSitter-Schwarzschild Space by Hanyang Liu Submitted to the Graduate aculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Mathematics Dr. Mao-Pei Tsui, Committee Chair Dr. Biao Ou, Committee Member Dr. Gerard Thompson, Committee Member Dr. Patricia R. Komuniecki, Dean College of Graduate Studies The University of Toledo August 2013

3 Copyright 2013, Hanyang Liu This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author.

4 An Abstract of On the Long Time Existence and Convergence of Inverse Curvature low in Anti-deSitter-Schwarzschild Space by Hanyang Liu Submitted to the Graduate aculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Mathematics The University of Toledo August 2013 We study the long time existence and convergence of the the inverse curvature flow in Anti-deSitter-Schwarzschild space. If the initial hypersurface is compact, starshaped and strictly k + 1-convex in Anti-deSitter-Schwarzschild space, then the solution for the inverse curvature flow exists for all time and the hypersurfaces converge to infinity while maintaining star-shapedness and strictly k +1-convex. Moreover, the hypersurfaces become strictly convex exponentially fast and more and more totally umbilical in the sense that the principal curvatures of the hypersurface are uniformly bounded and converge exponentially fast to one. iii

5 Acknowledgments This dissertation is a milestone in my academic career. I have been fortunate to learn theories and concepts which would have been impossible if I had not extensively carried out the needed research. I am grateful to a number of people who have guided and supported me throughout the research process and provided assistance for my venture. I would like to express the deepest appreciation to my advisor Professor Mao-Pei Tsui, who guided me in selecting the final theme for this research. My advisor was there throughout my preparation of the proposal and the conceptualization of its structure. I would not have been able to do the research and achieve learning in the same manner without his help and support. His recommendations and instructions have enabled me to assemble and finish the dissertation effectively. Also, I would like to thank my committee members, Professor Gerard Thompson and Professor Biao Ou for reading previous drafts of this dissertation and providing many valuable comments that improved the presentation and contents of this dissertation. I would also like to thank all my instructors and teachers, who throughout my educational career have supported and encouraged me to believe in my abilities. They have directed me through various situations, allowing me to reach this accomplishment. inally, my family has supported and helped me along the course of this dissertation by giving encouragement and providing the moral and emotional support I needed to complete my thesis. To them, I am eternally grateful. iv

6 Contents Abstract iii Acknowledgments iv Contents v 1 Introduction 1 2 The Main Result and Previous Research The Main Result Previous research Background and Preliminary Results Geometry of Submanifolds Rotationally symmetric space and Anti-deSitter-Schwarzschild Space Properties of Symmetric unction Inverse Curvature low Maximum Principle Proof of the Main Result Long-time Existence of Inverse Curvature low in Euclidean Space Long-time Existence and Convergence in Anti-deSitter-Schwarzschild Space v

7 4.2.1 Preliminaries Long-time Existence of the low and Convergence of the Second undamental orm References 54 vi

8 Chapter 1 Introduction There have been extensive interests on flows of hypersurfaces evolving by functions of their principal curvatures in the past 30 years. Brakke [3] used geometric measure theory to study surfaces driven by their mean curvature. Huisken [9] studied the mean curvature flow and proved that a compact, convex hypersurface contracts into a round point under the mean curvature flow. In contrast, a different type of flow, the inverse curvature flow, is an expanding flow introduced by Gerhardt [5] and Urbas [14], which expands star-shaped mean convex hypersurfaces out to a round sphere after rescaling. Hypersurface flows can be used to prove geometric inequalities. One of the most important applications of inverse curvature flow is the use of inverse mean curvature flow to prove the Riemannian Penrose Inequality by Huisken and Ilmanen [10]. Guan and Li [8] used the inverse curvature flows to prove Alexandrov-enchel quermassintegral inequalities for starshaped k-convex domains. Recently, Brendle, Hung and Wang ([4]) obtained a generalization of Minkowski s inequality using the inverse mean curvature flow in the Anti-deSitter-Schwarzschild space. In this thesis, we prove two results about the inverse curvature flow. The first one is a different proof of the long time existence of inverse curvature flow in Euclidean space. The proof relies on the lower bound estimate of the speed function and the 1

9 second fundamental form estimate using a matrix type maximum principle. The second result is the proof of the long time existence and convergence result of inverse curvature flow in Anti-deSitter-Schwarzschild. The proof depends on the estimates of various geometric quantities associated with the second fundamental form. In the first chapter, we describe the background and motivation of the problem. In the second chapter, we state the main results and discuss previous relevant research. In the third chapter, we provide the details of the proof. The classical Minkowski s inequality for a closed convex surface Σ in R 3 states that Σ Hdµ 16π Σ, where H is the mean curvature, the trace of the second fundamental form, and Σ is the area of Σ. Minkowski s inequality can be generalized to the so-called Alexandrov-enchel inequalities [1, 2] for closed convex hypersurface in Euclidean space in R n+1. We will assume that Σ is a smooth hypersurface in R n+1. Let κ(x) = (κ 1 (x),, κ n (x)) be the principal curvatures of x Σ, and let σ k (κ) the kth elementary function in κ = (κ 1,, κ n ) R n (with σ 0 (κ) 1). If Σ is closed and convex, the celebrated Alexandrov-enchel quermassintegral inequality states that, for 0 k n 1, ( S n σ k+1 (1,, 1)dµ) 1 n k 1 ( S n σ k (1,, 1)dµ) 1 n k ( σ Σ k+1(κ 1,, κ n )dµ) ( σ Σ k(κ 1,, κ n )dµ) 1 1 n k 1 n k, (1.0.1) where S n is the standard unit sphere in R n+1. The equality holds if and only if Σ is a sphere. The case k = 0 and n = 2 is the classical Minkowski s inequality. There have been some interests in extending the original Alexandrov-enchel in- 2

10 equality to non-convex domains (e.g., [8],[13], [7]). In [8], Guan and Li extend this inequality to starshaped and k convex hypersurfaces in R n+1. To state their results, We introduce the following definitions. Definition Let σ k (κ) be the kth elementary function in κ = (κ 1,, κ n ) R n (with σ 0 (κ) 1) defined by σ k (κ) = 1 i 1 <i 2 < <i k n κ i 1 κ i2 κ ik. Definition or Σ R n+1 a smooth hypersurface in R n+1, we say Σ is k-convex if κ(x) Γ k for all x Σ, where Γ k is the Garding s cone Γ k = {κ R n σ p (κ) > 0, 0 p k} and Γ k = {κ R n σ p (κ) 0, 0 p k}. We say Σ is strictly k-convex if κ(x) Γ k for all x Σ. We note that n-convex is convex in usual sense and 1-convex is sometimes referred as mean convex. Definition A set S is called a star-shape domain if there exists x 0 such that for all x in S, the line segment from x 0 to x is in S. In geometry, for any submanifold Σ, we introduce the support function ω := X, ν where X is the position vector for any x Σ and ν is the outer normal vector. We can see that the submanifold Σ is star-shape if and only if ω > 0. In ([8]), Guan and Li prove the following results. Theorem ([8]) Suppose Σ is a p + 1-convex starshaped hypersurface in R n+1, then inequality (1.0.1) is true for 0 k p. The equality holds if and only if Σ is a sphere. 3

11 Their proof is a parabolic one, using the flow studied by Gerhardt [5] and Urbas [14]. They only use a special case of their result for the following evolution equation on a hypersurface Σ 0 in R n+1, X t = σ k(κ) σ k+1 (κ) ν where ν is the unit outer normal vector on Σ t. (1.0.2) Theorem (Gerhardt [5], Urbas [14]) If Σ 0 is a starshaped strictly k +1-convex hypersurface, then solution for flow (1.0.2) exists for all time t > 0 and it converges to a round sphere after a proper rescaling. Let Σ t denote the solution to (1.0.2) with Σ 0 = Σ. The key observation is to show that the scale invariant quantity ( Σ t σ k+1 (κ 1,, κ n )dµ) 1 n k 1 ( Σ t σ k (κ 1,, κ n )dµ) 1 n k (1.0.3) is monotone decreasing along expanding flow of (1.0.2). Using the monotonicity of (1.0.3) and the convergence of the flow as t after scaling, one has the Alexandrov- enchel inequality ( Σ 0 σ k+1 (κ 1,, κ n )dµ) 1 n k 1 ( Σ 0 σ k (κ 1,, κ n )dµ) 1 n k ( lim t Σ t σ k+1 (κ 1,, κ n )dµ) 1 n k 1 ( Σ t σ k (κ 1,, κ n )dµ) 1 n k = ( σ S n k+1 (1,, 1)dµ) ( σ S n k (1,, 1)dµ) 1 1 n k 1 n k. Recently, Brendle, Hung and Wang ([4]) obtained a generalization of Minkowski s inequality using inverse mean curvature flow in the Anti-deSitter-Schwarzschild space. Let us recall the definition of the Anti-deSitter-Schwarzschild space. We fix a real number m > 0, and let s 0 denote the unique positive solution of the equation 1 + s 2 0 m s 1 n 0 = 0. We then consider the manifold M = S n [s 0, ) equipped with the 4

12 Riemannian metric ḡ = s 2 m s 1 n ds ds + s2 g S n, where g S n is the standard round metric on the unit sphere S n. The sectional curvatures of (M, ḡ) approach 1 near infinity, so ḡ is asymptotically hyperbolic. Moreover, the scalar curvature of (M, ḡ) equals n(n+1). The boundary M = S n {s 0 } is referred to as the horizon. Let f = 1 + s 2 m s 1 n. (1.0.4) Brendle, Hung and Wang prove the following result. Theorem [4] Let Σ be a compact mean convex, star-shaped hypersurface Σ in the Anti-deSitter-Schwarzschild space, and let Ω denote the region bounded by Σ and the horizon M. Then f H dµ n(n + 1) f dvol Σ Ω ( n S n 1 n 1 n 1 ) n Σ n M n. Moreover, equality holds if and only if Σ is a coordinate sphere, i.e. Σ = S n {s} for some number s [s 0, ). The key observation is that they show that Q(t) = Σ t n 1 n ( f H dµ n(n + 1) Σ t Ω ) f dvol + n s n 1 0 S n, (where f is the static potential defined above) is monotone decreasing along the inverse mean curvature flow in Anti-deSitter-Schwarzschild space. It is natural to ask if one can derive Alexandrov-enchel type inequality involving σ k+1 in Anti-deSitter-Schwarzschild space. Therefore it is important to study the 5

13 inverse curvature flow X t = σ k(κ) σ k+1 (κ) ν. in Anti-deSitter-Schwarzschild space. The goal of this thesis is to study the long time existence and convergence of the the inverse curvature flow (1) in Anti-deSitter- Schwarzschild space. 6

14 Chapter 2 The Main Result and Previous Research We study the inverse curvature flow X t = σ k(κ) σ k+1 (κ) ν (2.0.1) in Euclidean Space and Anti-deSitter-Schwarzschild space. We state the main results of the thesis in this chapter and discuss the relation with earlier results. 2.1 The Main Result The goal of this thesis is to study the inverse curvature flow (1) X t = σ k(κ) σ k+1 (κ) ν in Euclidean space and Anti-deSitter-Schwarzschild space. The first result is about the curvature estimate of inverse curvature flow in Euclidean space. We have the following results. Theorem Let X : Σ n [0, T ) R n+1 be a smooth solution of (1) in R n+1 satisfying 7

15 uniform bounds 0 < 0 < = tensor M ij σ k(κ) σ k+1 (κ) < 1. Then the largest eigenvalue µ n of the = h ij, and the largest eigenvalue κ n of the second fundamental form {h ij } satisfy the estimates: µ n 2 1 2t, κ n t, everywhere on M n [0, T ). Hence,the second fundamental form satisfies an estimate of the form A c n H 2 1 H 0 1 t. Corollary Let X : Σ n [0, T ) R n+1 be a smooth solution of (1) with > 0, 0 < T <. If remain bounded from below by a constant 0 > 0 for all t [0, T ), then the solution can be extended beyond T. In Anti-deSitter-Schwarzschild space, we obtain the following existence and convergence result about the inverse curvature flow. Theorem If the initial hypersurface is star-shaped and strictly k + 1-convex in Anti-deSitter-Schwarzschild space, then the solution for the flow (1) exists for all time t > 0 and the hypersurfaces converge to infinity while maintaining star-shapedness and strictly k+1-convex. Moreover, the hypersurfaces become strictly convex exponentially fast and more and more totally umbilical in the sense of h j i δj i Ce t n k, t > 0, i.e., the principal curvatures are uniformly bounded and converge exponentially fast to one. 8

16 2.2 Previous research In 1990 s, Gerhardt [5] and Urbas [14] introduced the following inverse curvature flow given by X t (x, t) = 1 ν(x, t) (2.2.1) X(, 0) = X 0 where (h ij ) = f(κ 1,..., κ n ) > 0 is a smooth and symmetric function and κ 1,, κ n are the principal curvatures of a hypersurface. We assume satisfies the following conditions: (1) f is homogeneous of degree one on Γ; (2) f κ i > 0; (3) f is concave on Γ; (4) f = 0 on Γ, where Γ {(κ 1,..., κ n ) R n : n i=1 κ i > 0}. Gerhardt [5] and Urbas [14] independently proved the following results. Theorem (Gerhardt and Urbas) Let Σ 0 be a smooth, closed, compact hypersurface in R n+1, n 2, given by a smooth embedding X 0 : S n R n+1, and suppose that Σ 0 is starshaped with respect to a point P 0. Let Γ be as above and suppose that f is a positive, symmetric function satisfying above hypothesis. Suppose that for each point ξ Σ 0, we have f(κ 1 (ξ),..., κ n (ξ)) > 0 where κ 1,...κ n are the principal curvature of M 0 relative to the inner unit normal. Then the inverse curvature flow has a unique smooth solution X defined on the time interval [0, ). or each t [0, ), X(,t) is a parametrization of a smooth, closed, 9

17 compact hypersurface Σ t in R n+1, which is starshaped with respect to P 0. urthermore, if Σ t is the hypersurface parameterized by X(,t) = e βt X(,t), where β = f(1,..., 1), then Σ t converges to a sphere centered at P 0 in the C topology as t. A special example is the case when = 1 = σ k(κ) f σ k+1. Then, in 2002, Ilmanen (κ) and Huiskein [11] used a different way to prove the inverse mean curvature flow has long-time existence. Theorem (Ilmanen and Huiskein) Let X : M n [0.T ) R n+1 be a smooth solution of the inverse mean curvature flow with H > 0, 0 < T <. If the mean curvature H remains bounded from below by a constant H 0 > 0 for all t [0, T ), then the solution can be extended beyond T. In 2011, Gerhardt [6] proved that the inverse curvature flow (3.4.1) has long-time existence and convergence in hyperbolic space. Theorem (Gerhardt [6]) The inverse curvature flow with smooth and admissible initial hypersurface Σ 0 exists for all time. The flow hypersurfaces in hyperbolic space converge to infinity, become strongly convex exponentially fast and also more and more totally umbilical. In fact there holds h i j δ i j ce t n i.e.,the principal curvature are uniformly bounded and converge exponentially fast to 1. Recently, Brendle, Hung and Wang [4] proved that the inverse mean curvature flow has long-time existence and convergence in AdS-Schwartzchild space. In this thesis, we show that the inverse curvature flow with = 1 f = σ k(κ) σ k+1 (κ) existence and convergence in AdS-Schwartzchild space. has long-time 10

18 Chapter 3 Background and Preliminary Results In this chapter, we include the basic results about the geometry of submanifolds, the introduction of AdS-Schwarzchild space, properties of symmetric function, the evolution equations of inverse curvature flow and the maximum principle for parabolic equation. 3.1 Geometry of Submanifolds Let (M n+1, ḡ) be a smooth complete Riemannian manifold without boundary. We denote by a bar for all quantities on M, for example by ḡ = {ḡ αβ }, 0 α, β n, the metric, by ȳ = {ȳ α } coordinates, by Γ = { Γ γ αβ } the Levi-Civita connection, by the covariant derivative and by Riem = {Riem αβγδ } the Riemann curvature tensor. Components are sometimes taken with respect to the tangent vector fields { ȳ α }, 0 α n associated with a local coordinate chart ȳ = {ȳ α } and sometimes with respect to a moving orthonormal frame {e α }, 0 α n, where ḡ(e α, e β ) = δ αβ. We write ḡ 1 = {ḡ αβ } for the inverse of the metric and use the Einstein summation convention for the sum of repeated indices. The Ricci curvature Ric = R αβ and scalar 11

19 curvature R of (M n+1, ḡ) are then given by R αβ = ḡ γδ Rαγβδ, and the sectional curvatures (in an orthonormal frame) are given by σ αβ = R αβαβ. Now let X : Σ n M n+1 be a smooth hypersurface immersion. or simplicity we restrict our attention to closed surfaces, i.e., compact without boundary. The induced metric on Σ n will be denoted by g, in local coordinates we have g ij (p) = X X (p), xi x (p) j M = ḡ αβ (X(p)) Xα x i (p) Xβ x j (p), p Σn. (3.1.1) urthermore, {Γ i jk }, and Riem = {R ijkl} with Latin indices i, j, k, l ranging from 1 to n describe the intrinsic geometry of the induced metric g on the hypersurface. If ν is a local choice of unit normal for X(Σ n ), we often work in an adapted othonormal frame e 0 (= ν), e 1,, e n in a neighborhood of X(Σ n ) such that e 1 (p),, e n (p) T p Σ n T p M n+1 and g(p)(e i (p), e j (p)) = δ ij for p Σ n, 1 i, j n. The second fundamental form A = {h ij } as a bilinear form A(p) : T p Σ n T p Σ n R and the Weingarten map W = {h i j} = {g ik h kj } as an operator W : T p Σ n T p Σ n are then given by h ij = ei ν, e j = ν, ei e j. In local coordinates x i, 1 i n, near p Σ n and {ȳ α }, 0 α n, near X(p) M n+1 these relations are equivalent to the Weingarten equations: 2 X α x i x X α j Γk ij x + Γ α X β X δ k βδ = h x i x j ij ν α ; (3.1.2) ν α x + Γ α X β i βδ x i νδ jl Xα = h ij g (3.1.3) x l 12

20 Recall that A(p) is symmetric, i.e., W is self-adjoint, and the eigenvalues κ 1 (p),..., κ n (p) are called the principal curvatures of X(Σ) at X(p). Also note that at a given point p Σ n by choosing normal coordinates and then possibly rotating them we can always arrange that at this point g ij = δ ij, ( ei e j ) T = 0, h ij = h i j = diag(κ 1,, κ n ). Here ( ei e j ) T means the projection of the vector ei e j from T M (p) to T Σ (p). The mean curvature and the norm of the second fundamental form are given by H := tr(w ) = h i i = κ κ n and A 2 := κ κ 2 n. The commutator of second derivatives of a vector field V and a one-form ω on Σ n are given by i j V k j i V k = R ijlp g kl V p and i j ω k j i ω k = R ijkl g lp ω p. More generally, the commutator of second derivatives for an arbitrary tensor involves one curvature term as above for each of the indices of the tensor. The corresponding laws of course also hold for the metric ḡ. The curvature of the hypersurface and ambient manifold are related by the equations of Gauss: R ijkl = R ijkl + h ik h jl h il h jk, 1 i, j, k, l n, 13

21 the equations of Codazzi-Mainardi: i h jk k h ij = R 0jki. the second derivatives of the second fundamental form satisfies the identities: k l h ij = i j h kl + h kl h im h mj h km h mj h il + h kj h im h ml h km h ml h ij + R kilm h mj + R kijlm h ml + R mjil h km + R 0i0j h kl R 0k0l h ij + R mljk h im + k R 0jil + i R 0ljk. 14

22 3.2 Rotationally symmetric space and Anti-deSitter-Schwarzschild Space Recall that the metric in Anti-deSitter-Schwarzschild Space is ḡ = s 2 m s 1 n ds ds + s2 g S n, The following result are from [4] and we include the proof here for reader s convenience. Lemma By a change of variable, the Anti-deSitter-Schwarzschild metric can be rewritten as ḡ = dr dr + λ(r) 2 g S n where κ(r) satisfies the ODE λ (r) = 1 + λ 2 mλ 1 n and the asymptotic expansion λ(r) = sinh(r) + m 2(n + 1) sinh n (r) + O(sinh n 2 (r)). Proof. We define r(s) = s dt b, 1 + t2 mt1 n s 0 1 where b = s 0 ( 1 s0 1 + t2 mt 1 n t 2 )dt s t 2 dt With this understood, the metric g can be written as g = dr dr + λ(r) 2 g S n, where λ(r(s)) = s. 15

23 The function r(s) can be written as r(s) = s 0 1 dt 1 s0 ( 1 + t t2 mt 1 n = arcsinh(s) = arcsinh(s) Hence, by Taylor expansion, we have s s ( m 2 t n 2 + O(t n 4 ))dt m 2(n + 1) s n 1 + O(s n 3 ). m sinh(r(s)) = s 2(n + 1) s n + O(s n 2 ) = s t 2 ) m 2(n + 1) sinh n (r(s)) + O(sinh n 2 (r(s))). rom this, the assertion follows. Next,we can write the initial star-shape hypersurface Σ 0 as the graph of a function r 0 defined on the unit sphere: Σ 0 = {(r 0 (θ), θ) : θ S n }. If each Σ t is star-shaped, it can be parametrized as the graph Σ t = {(r(θ, t), θ) : θ S n }. Normally, we have the metric g = g ij dx i dx j of Σ induced from M, let, and be the Levi-Civita connections of S n, Σ and M, respectively. Γ k ij denote the Christoffel symbols of S n with respect to the tangent basis { θ i} n i=1 and Γ γ αβ denote the Christoffel symbols of M n+1 with respect to the tangent basis { θ i = θ i } n i=1 { r = r } which has the form e i = r i r + θ i. We next define a new function ϕ : S n R by ϕ(θ) = Φ(r(θ)), 16

24 where Φ(r) is a positive function satisfying Φ (r) = 1 λ(r). Let ϕ i = i ϕ and ϕ ij = j i ϕ denote the covariant derivatives of ϕ with respect to the round metric g S n(same for r). Moreover, let σ ij = θ i, θ j gs n, σ ij is the inverse of σ ij which is defined by σ ij σ jk = δ i k and v = 1 + ϕ 2 g S n. Proposition Let Σ be a star-shaped hypersurface in Anti-deSitter-Schwarzschild Space, g ij be the induced metric on Σ and h ij be the second fundamental form in term of the coordinates θ j. Then g ij = λ 2 (σ ij + ϕ i ϕ j ), h ij = λ v ( λ (σ ij + ϕ i ϕ j ) ϕ ij ). and h i j = g ik h kj = 1 λv (λ δj i (σ ik ϕi ϕ k v 2 )ϕ kj) where ϕ i = σ ij ϕ j. Proof. Recall that the metric of Anti-deSitter-Schwarzschild Space is ḡ = dr 2 + λ(r) 2 g gs n. We have r, r ḡ = 1, θ i, r ḡ = 0 and θ i, θ j ḡ = λ(r) 2 σ ij. Also ϕ i = ϕ θ i = Φ (r(θ)) r θ i = 1 λ(r) r i and r i r j = λ 2 (r)ϕ i ϕ j. The induced metric g ij with respect to the basis {e i = r i r + θ i} n i=1 is g ij = r i r + θ i, r j r + θ j ḡ = λ 2 σ ij + r i r j = λ 2 (σ ij + ϕ i ϕ j ), (3.2.1) Then we have g ij = {g ij } 1 = λ 2 (σ ij ϕi ϕ j ), (3.2.2) v2 17

25 and the unit normal vector ν is given by ν = 1 v ( r rj λ 2 θj) (3.2.3) Here r i = σ ij r j and ϕ i = σ ij ϕ j. Meanwhile, we may calculate the Christoffel symbols: Γ k ij = Γ k ij; Γ 0 ij = 1 2ḡ0l ( ḡ il θ + g jl g ij j θ i θ ) l = 1 2ḡ00 ( ḡ i0 θ j = λλ σ ij ; + ḡ j0 θ i ḡ ij θ 0 ) Γ k 0i = 1 2ḡkl ( ḡ 0l θ i = λ λ δk i ; Γ 0 0i = 1 2ḡ0l ( ḡ 0l θ i = 1 2ḡ00 ( ḡ 00 θ i + ḡ il θ 0 ḡ 0i θ l ) + ḡ il θ 0 ḡ 0i θ l ) + ḡ i0 θ 0 ḡ 0i θ 0 ) (3.2.4) = 0; Γ k 00 = 1 2ḡkl ( ḡ 0l θ 0 + ḡ 0l θ 0 ḡ 00 θ k ) = 0; Γ 0 00 = 1 2ḡ0l ( ḡ 0l θ 0 + ḡ 0l θ 0 ḡ 0l θ 0 ) = 0. where θ 0 = r. 18

26 We may calculate the second fundamental form of Σ in M. h ij = ej e i, ν = 1 v r rk λ 2 θ k, r j r (r i r + θ i) + θ j (r i r + θ i) = 1 v r rk λ 2 θ k, r j r θ i + r ij r + r i θ j r + θ j θ i = 1 v {r j r, r θ i + r ij + r, θ j θ i rk r j λ 2 θ k, r θ i rk r i λ 2 θ k, θ j r rk λ 2 θ k, θ j θ i } By using Γ k 00 = 0 and r, k = 0, we have: h ij = 1 v {r ij + r, θ j θ i + rk r j λ 2 r, θi θ k + rk r i λ r, 2 θ rk j θk λ 2 θ k, θ j θ i } = 1 v {r ij λλ σ ij 2λ λ r ir j rk λ 2 Γ l ijλ 2 σ kl } = 1 v {r ij λλ σ ij 2λ λ r ir j } = λ v (λ (σ ij + ϕ i ϕ j ) ϕ ij ) Also, we may calculate that: h i j = g ik h kj = λ λv δi j 1 λv σik ϕ kj where σ ij = σ ij ϕi ϕ j v 2. Under this metric, Brendle,Hung and Wang derive the following lemma, I include the proof for reader s convenience: Lemma Let e α, α = 0, 1,..., n be a orthonormal frame and R αβγµ is the 19

27 Riemannian curvature tensor of the Anti-deSitter-Schwarzschild metric. Then R αβγµ = δ βµ δ αγ + δ βγ δ αµ + O(e (n+1)r ) (3.2.5) and D ρ R αβγµ = O(e (n+1)r ). (3.2.6) Proof. Each level set of r is a round sphere with induced metric λ(r) 2 g S n and second fundamental form λ(r)λ (r)g S n. Applying the Gauss equation, we compute R( θ i, θ j, θ k, θ l) = λ(r) 2 (1 λ (r) 2 )(σ ik σ jl σ il σ jk ). Since the level set of r is umbilical, from the Codazzi equation, we derive R( θ i, θ j, θ k, θ l) = 0. The remaining components of the curvature tensors are: R( θ i, θ j, θ k, θ l) = ( i r r i ) r, θ j = r i r, θ j = r ( λ λ θ i), θ j = λ(r)λ (r)σ ij. rom this, the results follow easily. 20

28 3.3 Properties of Symmetric unction later. In this section, we include the properties of symmetric function which will be used Theorem or any k = 0,..., n, i = 1,..., n and κ R n n i=1 κ 2 i σ k+1 κ i = σ k+1 κ i = σ k;i (κ) (3.3.1) σ k+1 (κ) = σ k+1;i (κ) + κ i σ k;i (κ) (3.3.2) n σ k;i (κ) = (n k)σ k (κ) (3.3.3) i=1 n κ i σ k;i (κ) = (k + 1)σ k+1 (κ) (3.3.4) i=1 n κ 2 i σ k;i (κ) = σ 1 (κ)σ k+1 (κ) (k + 2)σ k+2 (κ) (3.3.5) i=1 where σ k;i (κ) is the sum of the terms of σ k (κ) not containing the factor κ i. Proof. We may easily derive the first two equations according to the definition, while (2.3.4) is a consequence of (2.3.1) and Euler s theorem on homogeneous functions. Taking sums over i in (2.3.2) and applying (2.3.4) we obtain (2.3.3). By (2.3.2) we also obtain, for any κ R n, σ k+2 (κ) σ k+2:i (κ) = κ i σ k+1;i (κ) = κ i σ k+1 (κ) κ 2 i σ k;i (κ) Taking sums over i and using (2.3.3) we find (k + 2)σ k+2 (κ) = n (κ i σ k+1 (κ) κ 2 σ k;i (κ)), i=1 which implies the identity (2.3.5). Next, we introduce Newton s Inequality: 21

29 Theorem If κ = (κ 1,, κ n ) Γ k+1 then σ k 1 σ k+1 σ 2 k σ k 1(I)σ k+1 (I) σ 2 k (I) = k(n k) (n k + 1)(k + 1) (3.3.6) To prove it, we first show a lemma. Lemma or real κ 1,..., κ n, there exists real κ 1,..., κ n 1 with the same symmetric average d 0,..., d n 1 where d k := σ k ( n k) Proof. We consider the derivative of P = n i=1 (x + κ i) and we can see that κ 1,..., κ n are the roots of P = 0. are κ 1,..., κ n. Without loss of generosity, we assume that κ 1... κ n Now for any i [1, n 1], P must have a root between κ i and κ i+1 by Rolle s theorem if κ i κ i+1 and if κ i = κ i+1 = = κ i+k, then κ i is a root of P for k + 1 times, so it must be a root of P (t) for k times. It follows that P must have n 1 real roots, i.e., for some reals κ 1,..., κ n 1, So now, we have P = n n 1 i=1 (x+κ i) = nx n 1 + n 1 j=1 hand, since P = x n + n j=1 ( n j we compare both P, we have n ( n 1 j ( n 1 ) j d j x n 1 j. On the other ) dj x n j, so P = nx n 1 + n 1 j=1 (n j)( n ) d j = (n j) ( n j ) dj and since n ( n 1 j j) di x n 1 j. If ) ( = (n j) n ) j, we get d j = d j. So now we find n 1 real numbers to replace previous n real numbers such that d j = d j where 1 j n 1. Then we use this lemma to prove Theorem Proof. Based on the previous lemma, it is sufficient to prove d 2 n 1 d n 2 d n for any n. If κ i = 0 for some i, then we can see that the inequality is obviously true. If κ i 0 for all i, then since this is a homogenous inequality, we may normalize it so that d n = n i=1 κ i = 1. The inequality then becomes n(n 1) 2 ( n l=1 1 κ l ) 2 n 2 1 i<j n 1 κ i κ j 22

30 i.e., (n 1)( n l=1 1 κ l ) 2 2n 1 i<j n 1 κ i κ j n 1 (n 1) l=1 κ 2 l + (n 1) 1 i<j n 2 κ i κ j 2n 1 i<j n 1 κ i κ j n 1 n l=1 κ 2 l n 1 κ 2 l=1 l i<j n 1 κ i κ j = ( n k=1 1 κ l ) 2 We may notice that the final inequality is the mean square inequality and so the Inequality d 2 n 1 d n 2 d n holds. Then since d k := σ k, we may derive the Newton ( n k) Inequality: σ k 1 σ k+1 σ 2 k σ k 1(I)σ k+1 (I) σ 2 k (I) = k(n k) (n k + 1)(k + 1) 23

31 3.4 Inverse Curvature low In this section, we introduce the inverse curvature flow and derive evolution equations of several geometric quantities. The inverse curvature flow is given by: X t (x, t) = 1 ν(x, t) (3.4.1) X(, 0) = X 0 where (h ij ) = f(κ 1,..., κ n ) > 0 is a smooth and symmetric function and κ i is the eigenvalues of the second fundamental form {h ij }. We assume satisfies the following conditions: (1) f is homogeneous of degree one on Γ; (2) f κ i > 0; (3) f is concave on Γ; (4) f = 0 on Γ, where Γ {(κ 1,..., κ n ) R n : n i=1 κ i > 0}. Later, we will only consider the case where 1 f = σ k(κ) σ k+1 (κ). or a fixed point x 0 Σ n and a fixed time t 0, then computation are best done in a local coordinate system {x i }, 1 i n near x 0 in Σ n, and {ȳ α }, 0 α n for M n+1 are normal coordinates at X(x 0, t 0 ). We can also arrange that in these coordinates ν α = δ α 0 and Xα x i = δ α i at X(x 0, t 0 ). Then all Christoffel symbols of the connection Γ vanishes at X(x 0, t 0 ) and we have only to take derivative s of the Christoffel symbols into account, which will lead to curvature terms eventually. Using Gauss-Weingarten relations (2.1.2), (2.1.3) and that fact that ḡ αβ / ȳ γ vanishes at X(x 0, t 0 ) for 0 γ n in our coordinates. irst, we derive the evolution equations of the induced metric and the outward unit normal vector. 24

32 Lemma The metric of Σ t satisfies the evolution equation g ij t = 2 h ij (3.4.2) Proof. The vectors X x i are tangential to Σ t and h ij = ν, 2 X x i x j from the Weigarten equations in (3.1.2), and thus ν, X x i = 0 and h ij = ν x i, X x j = X x i, ν x j Using g ij = X, X and X x i x j t = 1 ν, we have g ij t = t X x i, X x j = x ( 1 X ν), i x + X j x, i x ( 1 j )ν = 1 ν x, X i x + 1 j X x, ν i x j = 2 h ij. Lemma The outward unit normal to Σ t satisfies ν t = 1. (3.4.3) Proof. Since ν t is a tangent vector, we have ν t = ν t, X x X i = ν, X x i X x j gij x j gij = ν, t x ( 1 i ν) X x j gij = x ( 1 i ) X x j gij = 1 Now, we derive the evolution equations of the second fundamental form and the velocity function : 25

33 Lemma The second fundamental form and satisfy the evolution equations h ij t = Lh ij 2 + kl (h 2 ) kl h ij 2 2 i j 3 + kl,rs i h rs j h kl 2 + kl (h il h km h mj h kj h im h ml ) kl (R kilm h mj + R kijm h ml + R mijl h km + R mljk h im + R oioj h kl R okol h ij ) kl ( k R ojil + i R oljk )) R oioj (3.4.4) and t = L 2 ij i j 2 3 ij h k i h kj ij Roioj. (3.4.5) Here we adopt the notation Lu = kl k l u and ij = h ij Proof. h ij t By using (3.1.2), we have = t (ḡ 2 X α αβ x i x j νβ ) 1 ḡαβ ȳ Γ α iδν γ Xδ γ x j νβ 2 = ḡ αβ x i x ( 1 j να ) ν β 2 X α ( 1 + ḡ ) lm Xβ αβ g x i x j x l x 1 m ḡαβ ȳ Γ α iδν γ Xδ γ x j νβ (3.4.6) where we use the fact that {ȳ α }, 0 α n, is a normal coordinates. Then using (3.1.3), the first term of the right hand side of (3.4.6) is equal to 2 ( 1 ) x i x 1 2 ν α j ḡαβ x i x j νβ = 2 ( 1 ) x i x 1 j ḡαβh jl g lm 2 X α x i x m νβ + 1 ḡαβ x Γ α X ρ i ρσ x j να ν β. 26

34 By using (3.1.2), again, the second term of the right hand side of (3.4.6) is equal to Γ k ( 1 ) ij x k Thus the right hand side of (3.4.6) becomes 2 ( 1 ) x i x + ( 1 ) j Γk ij x k 1 ḡαβ ȳ Γ α iδν γ Xδ γ x j νβ 1 ḡαβ( ȳ Γ α δ ij ȳ Γ α jδ)ν δ ν β i = i j h ikh k j 1 R 0i0j (3.4.7) where we use the chosen coordinates satisfying ν α = δ α 0 and Xα x i = δ α i at that point and definition of Riemannian curvature. inally the conclusion follows from the identities of the second derivatives of the second fundamental form. Based on (3.4.7), we may continue to derive the evolution equation for : t = ( ) hi j h i j t = ( )g li h lj t (gik h kj ) = 2 lj h k l h kj + ij ( t h ij) = L 2 ij i j 2 3 ij h k i h kj ij R 0i0j. Under Anti-deSitter-Schwarzschild metric, by using (3.2.5),and (3.2.6),we may rewrite the evolution of and h ij. Here we adopt the notation i i = tr( kl ) t = L 2 ij i j 2 3 = L 2 ij i j 2 3 = L 2 ij i j 2 3 ij h k i h kj ij h k i h kj ij h k i h kj 27 ij R 0i0j ij ( δ ijδ 00 + δ i0 δ 0j + O(e (n+1)r )) + i i + i i O(e (n+1)r ) (3.4.8)

35 h ij t space. = Lh ij 2 + kl (h 2 ) kl h ij 2 2 i j 3 + kl,rs i h rs j h kl 2 + kl (h il h km h mj h kj h im h ml ) kl (R kilm h mj + R kijm h ml + R mijl h km + R mljk h im + R 0i0j h kl R 0k0l h ij ) kl ( k R 0jil + i R 0ljk )) R 0i0j 2 = Lh ij + kl (h 2 ) kl h ij 2 i j kl,rs i h rs j h kl kl (h il h km h mj h kj h im h ml ) { kl ( δ kl δ im + δ ki δ lm + O(e (n+1)r ))h mj + kl ( δ kj δ im + δ ki δ jm + O(e (n+1)r ))h ml + kl ( δ mj δ il + δ mi δ jl + O(e (n+1)r ))h km + kl ( δ mj δ lk + δ ml δ jk + O(e (n+1)r ))h im 2 + kl ( δ 00 δ ij + δ 0i δ 0j + O(e (n+1)r ))h kl kl ( δ 00 δ kl + δ 0k δ 0l + O(e (n+1)r ))h ij } kl ( k R 0jil + i R 0ljk )) δ ijδ 00 + δ i0 δ 0j + O(e (n+1)r ) 2 = Lh ij + kl (h 2 ) kl h ij 2 i j + kl,rs i h rs j h kl il h lj + jl h li i i + 2 i i h ij δ ij + 2 i h ij 2 2 O(e (n+1)r ) kl (h il h km h mj h kj h im h ml ) 2 i 2 O(e (n+1)r ) + O(e (n+1)r ) (3.4.9) Then we will derive evolution for sphere coordinates in Anti-deSitter-Schwarzschild If the hypersurface is starshaped, the position vector at time t can be written as X = (r(θ(t), t), θ(t)). Then X t = dr dt r + dθi dt θ i rom X t = 1 ν and ν = 1 v ( r rj λ 2 θ j), we have: dr dt = 1 v and dθi dt = ri λ 2 v (3.4.10) 28

36 Recall that dr(θ(t),t) dt = r t + r j dθj dt. Thus r t =dr dt r dθ j j dt = 1 v + r jr j λ 2 v = 1 v (1 + r jr j λ 2 ) (3.4.11) = v We have used the fact that 1 + r jr j λ 2 = v 2. If we choose = σ k+1 σ k r t = σ k σ k+1 v., we get 29

37 3.5 Maximum Principle Here we state and prove the scalar Maximum Principle for parabolic equation from that will be used later for our various estimate for inverse curvature flow. Theorem Suppose that g(t) is a family of metrics on a closed manifold M and u : M [0, T ) R satisfies t u Lu + X(t), u + (u), (3.5.1) where X(t) is a time-dependent vector field and is a Lipschitz unction. If u c at t = 0 for some c R, then u(x, t) U(t) for all x M n and t 0, where U(t) is the solution to the ODE du dt = (U) with U(0)=c. Proof. By (3.5.1), we have (u U) L(u U) + X(t), (u U) + (u) (U), t then according to the Lipschitz property of,we have: (u U) L(u U) + X(t), (u U) + C u U t This implies that v =: e Ct (u U) satisfies v L(u U) + X(t), v + C( v v) t 30

38 Hence v ε =: v ε(1 + t) satisfies v ε (0) ɛ and t v ε Lv ε + X(t), v ε + C( v v) ε. We claim v ε < 0 for all t 0. If not, then there exists a first time t 0 at which there is a point x 0 such that v ε (x 0, t 0 ) = 0. Then v = v = ε(1 + t 0 ) at (x 0, t 0 ) and 0 t v ε Lv ε + X(t), v ε + C( v v) ɛ ɛ. at (x 0, t 0 ), which is a contradiction. Hence v ε < 0 for all t 0 and ɛ > 0. The result follows from taking ε 0. 31

39 Chapter 4 Proof of the Main Result In this chapter, we prove our main results. 4.1 Long-time Existence of Inverse Curvature low in Euclidean Space In this section, we study the inverse curvature flow in Euclidean Space. irst, we show that the star-shaped condition is preserved under the inverse curvature flow. or any submanifold in this space, we may show that star-shape will be maintained under the inverse curvature flow (3.4.1). Also, we will show that the inverse curvature flow (3.4.1) has long-time existence in Euclidean Space. ist, we consider star-shape will be maintained under the inverse curvature flow (3.4.1). Theorem If the submanifold Σ is star-shaped at the initial time, then it remains star-shaped for all time. Proof. We compute the evolution equation of ω. t ω = t X, ν = t X, ν + X, t ν = 1 + X, 1 = X, 2.

40 On the other hand, let s compute Lω : Lω = kl k ( l X, ν + X, l ν ) = kl k X, h m l m X = kl ( k X, h m l m X + X, k h m l m X + X, h m l k m X ) = kl h m l g km + X, kl m h kl m X X, h m l h km ν = kl h kl + X, kl m h kl m X X, h m l h km ν = + X, kl h km h m l ω. So, we get: t ω = Lω + kl h km h m l ω = Lω + kl h km h m l ω 2 2 Since kl h km h m l 2 f = kl κ k κ l > 0, f 2 we get ( t L )ω > 0 and ω > 0 for any time t > 0. 2 We can see that the positivity of support function will be preserved by the flow and so the manifold will be star-shaped for all time. Next we prove the inverse curvature flow (3.4.1) has long-time existence under Euclidean Space. To prove it, we only need to show the norm of the second fundamental form is bounded by some time t, then we may extend the solution beyond the time t. Theorem Let X : M n [0, T ) R n+1 be a smooth solution of (3.4.1) satisfying uniform bounds 0 < 0 < < 1. Then the largest eigenvalue µ n of the tensor M ij = h ij, and the largest eigenvalue κ n of the second fundamental form 33

41 {h ij } satisfy the estimates: µ n 2 1 2t, κ n t, everywhere on M n [0, T ). Hence,the second fundamental form satisfies an estimate of the form A c n H 2 1 H 0 1 t. Proof. Let s compute t ( h ij ) t ( h ij ) = ( t )h ij + ( t h ij ) and = (L )h ij + (Lh ij ) 2 2 kl k l h 3 ij 2 i j 2 + kl,rs i h rs j h kl So, L( h ij ) = kl k l ( h ij ) = kl k [( l )h ij + ( l h ij )] =( kl k l )h ij + ( kl k l h ij ) + 2 kl k l h ij =(L )h ij + (Lh ij ) + 2 kl k l h ij =(L )h ij + (Lh ij ) + 2 kl k [ l( h ij ) ( l )h ij ] =(L )h ij + (Lh ij ) + 2 kl k l ( h ij ) 2 kl k l h ij t ( h ij ) = L( h ij) 2 2 i j 2 2 kl k l ( h ij ) 3 + kl,rs i h rs j h kl 34

42 We let M ij = h ij and consider t M i j t M i j = t (g ik M kj ) = ( t g ik )M kj + g ik ( t M kj ) = L(M i j) 2 2 kl k l Mj i 3 2 i j 3 + kl,rs i h rs j h kl 2 2 M ik M kj Then we use maximum principle (3.5.1),if v j is the eigenvector for M i j, we have: 2 i j 3 v i v j 0, Also, because is a concave function, then kl,rs is semi-negative definite, i.e., kl,rs i h rs j h kl v i v j 0 Then,we get: t (M i jv i v j ) = L(M i jv i v j ) 2 2 kl k l (Mjv i i v j ) 2 i j v 3 3 i v j + kl,rs i h rs j h kl v i v j 2 M ik M 2 kj v i v j 2 M ik M 2 kj v i v j Thus, if we suppose that 0 < 0 1, then the eigenvalue follows by comparison with the ODE: d dt φ = 2 φ that the largest eigenvalue of M satisfies κ n 2 1 2t 35

43 or the largest principal curvature µ n = κn, we have, and so µ n = λ n t t A c(n) t Corollary Let X : M n [0, T ) R n+1 be a smooth solution of (3.4.1) with > 0, 0 < T <. If remain bounded from below by a constant 0 > 0 for all t [0, T ), then the solution can be extended beyond T. Proof. By the evolution equation of, we observe that is uniformly bounded above by it initial value 1 = sup N0 on M n [0, T ). Given additional uniform lower bound 0 > 0 for, theorem above implies that the second fundamental form is bounded by A c(n)1 2 / 0 t, which is bounded for t T. The regularity results of Krylov guarantees higher regularity of the solution and convergence to a smooth limit surface N T as t T, satisfying 0 > 0. The short-time existence of solution to (3.4.1) yield the desired extension. 36

44 4.2 Long-time Existence and Convergence in AntideSitter-Schwarzschild Space In this section, we prove that the inverse curvature flow (3.4.1) has long-time existence in the Anti-deSitter-Schwarzschild space with = σ k+1 σ k and k + 1-convex. Also, the second fundamental form will converge to identity matrix as time passes. We outline the ideas of the proof here. Recall that the inverse curvature flow exists as long as the second fundamental form h j i = 1 λv (λ δj i (σ ik ϕi ϕ k )ϕ v 2 kj ) remains bounded. In order to prove these results, we will need to prove some lemmas. In Lemma 4.2.2, we estimate λ and λ. The uniform estimate of = σ k+1 σ k is proved in Lemma In Lemma and , we estimate v and ϕ Preliminaries In this section, we use the properties of symmetric function to derive some results about the derivative of function = σ k+1 σ k. Lemma If κ = (κ 1,, κ n ) Γ k+1 then Proof. n i=1 i i = n k k + 1 i i n k and kl h km h m l k + 1 n k 2. By using (3.3.1) and (3.3.3), we have n i=1 κ i σ k+1 = (n k)σ k and κ i σ k = (n k + 1)σ k 1. It follows that n i=1 κ i σ k+1 σ k = ( κ i σ k+1 )σ k σ k+1 ( κ i σ k ) σ 2 k = (n k) (n k + 1) σ k+1σ k 1. σk 2 = (n k)σ2 k (n k + 1)σ k+1σ k 1 σ 2 k 37

45 Then by [?], we know if Σ is a connected and closed hypersurface in M and σ p > 0 on Σ, then σ l > 0 for all 1 l p 1. Since the manifold is strictly k + 1-convex, so we conclude that σ k+1, σ k, σ k 1 are all positive. So, i i < n k. On the other hand, by using Newton Inequality (3.3.2) σ k+1 σ k 1 σ 2 k k(n k) (n k + 1)(k + 1), we derive i i k(n k) (n k) (n k + 1) (n k + 1)(k + 1) = n k k + 1. Now, we can see that i i is scalinginvariant. is bounded and also we may notice that i i We may assume that the second fundamental form is diagonalized as h km = λ k δ km. Thus kl h km h m l = kl λ k δ km λ l δ m l = n i=1 κ2 i κ i ( σ k+1 σ k ). Using n i=1 κ2 σ p+1 i κ i = σ 1 (κ)σ p+1 (κ) (p + 2)σ p+2 (κ) from (3.3.5), we derive next result by the following calculation. kl h km h m l = n i=1 κ 2 i ( σ k+1 ) = 1 κ i σ k σk 2 n i=1 κ 2 i ( κ i (σ k+1 )σ k κ i (σ k )σ k+1 ) rom Newton inequality = 1 [σ σk 2 k (σ 1 σ k+1 (k + 2)σ k+2 ) σ k+1 (σ 1 σ k (k + 1)σ k+1 )] = 1 [(k + 1)σ 2 σk 2 k+1 (k + 2)σ k σ k+2 ] = 1 [(k + 2)( k + 1 σk 2 k + 2 σ2 k+1 σ k σ k+2 )] σ k+2 σ k σ 2 k+1 (k + 1)(n k 1), (n k)(k + 2) 38

46 we derive kl h km h m l 1 (k + 2){[ k + 1 σk 2 k + 2 = k + 1 σk+1 2 n k σk 2 = k + 1 n k 2. (k + 1)(n k 1) ]σ 2 (k + 2)(n k) k+1} Then, we will derive the estimate for the function λ. Lemma Let r = sup S n 1 r(, t) and r(t) = inf S n 1 r(, t). Then λ( r(t)) e k+1 n k t λ( r(0)) and In particular, λ(r(t)) e k+1 n k t λ(r(0)). R 1 e k+1 n k t λ R 2 e k+1 n k t Proof. irst, we prove that at the point where the function r(, t) attains its maximum, we have v = 1 and λ n k. λ k+1 When r(, t) attains its maximum, we have i r = 0 and [ i j r] 0 (nonnegative definite as a matrix). Recall that φ(θ) = Φ(r(θ)) and d Φ = 1. We have dr λ(r) i ϕ = 1 λ i r and v = 1 + ϕ 2 g S n = 1 + r 2 g S n. Since r = 0 at the maximum point, λ 2 we have v = 1. A simple calculation gives i j ϕ = λ λ 2 i r j r + 1 λ i j r. Using r = 0 and 2 r 0, we can see that [ i j ϕ] 0 at the maximum point. rom Proposition 3.2.2, we have h i j = 1 λv (λ δj i (σ ik ϕi ϕ k v )ϕ kj). 2 39

47 ϕ j = 0 and [ i j ϕ] 0. Using ϕ j = 0, [ i j ϕ] 0 and [σ ik ] > 0 at the maximum point, we get [h i j] λ λ δi j and = (h i j) ( λ λ δi j) = λ λ (δi j) = λ n k. λ k+1 Using v = 1 and λ n k λ k+1 at the maximum point, we have d dt r(t) = v (k + 1)λ( r(t)) (n k)λ ( r(t)), hence d (k + 1)λ( r(t)) λ( r(t)). dt n k rom this, the first statement follows. The second statement follows similarly. Using this lemma, we can derive an estimate of r. Lemma We have r(t) k + 1 n k t M 1 and e (n+1)r O(e (k+1)(n+1) n k t ) Proof. When r(, t) attains its minimum, we may derive d (k + 1)λ(r(t)) r(t) dt (n k)λ (r(t)), On the other side, we know R 1 e k+1 n k t λ from previous lemma and λ λ = λ 1 + λ2 mλ 1 n λ 1 + λ 2 = λ R1 2 e 2(k+1)t n k (k+1)t 1 e n k. R 1 Thus d dt r(t) (k + 1) (n k) (k+1)t e n k (1 ) R 1 and r(t) r(t) k+1 t M for M is a positive real number. n k 40

48 Then e (n+1)r O(e (k+1)(n+1) n k t+(n+1)m ) = O(e (k+1)(n+1) n k t ) Next we will show is uniformly bounded above and below. irst, we prove an upper bound for. Lemma We have (n k) k Le k+1 n k t where L a positive number. Proof. rom (3.4.8), we know that the evolution equation of is t = L 2 ij i j 2 3 ij h k i h kj Since i i is bounded above from (4.2.1), then we have t = L 2 ij i j 2 3 ij h k i h kj + i i + i i O(e (n+1)r ) + i i + O(e (n+1)r ). (4.2.1) Multiplying by 2 to both sides and using kl h km h m l k+1 2 and i n k i n k from Lemma 4.2.1, we get 2 t L ij i j 2 k n k 2 + 2(n k) + L 1 e (n+1)(k+1)t n k Let 2 max(t) = max x Σt 2 (x, t). By maximum principle, we get the ODE d dt max 2 2 k + 1 n k max 2 + 2(n k) + L 1 e (n+1)(k+1)t n k. 41

49 Multiplying the integration factor e 2(k+1)t n k, we have d dt 2(k+1)t (e n k 2 max ) (2(n k) + L 1 e (n+1)(k+1)t n k )e 2(k+1)t n k. It follows that we can find two constants L 2 and L 3 depending on n, k and max (0) such that e 2(k+1)t n k 2 max (n k)2 k + 1 e 2(k+1)t n k + L 2 e (n 1)(k+1)t n k + L 3 and 2 max (n k)2 k L 2e (n+1)(k+1)t n k + L 3 e 2(k+1)t n k. Since n 2, we can choose L 4 = L 2 + L 3 such that 2 max (n k)2 k L 4e 2(k+1)t n k and (n k) k L 4 e (k+1) n k t. We have shown that has an upper bound, now we show that has a lower bound. Recall that associated with r, we define ϕ(θ, t) := Φ(r(θ, t)), where Φ(r) is a positive function satisfying d Φ(r) = 1. Then ϕ satisfies dr λ(r) ϕ t = d dr Φ(r) r t = v λ = 1. (4.2.2) 42

50 Here we denote = ( λ v hi j) = ([b i j]) and [b i j] = λ v [hi j] = λ 1 v λv (λ δj i (σ ip ϕi ϕ p v )ϕ pj) = λ δj i σ ip ϕ pj 2 v 2 where σ ip = σ ip ϕi ϕ p v 2 and v = 1 + ϕ 2 g S n Lemma Ce (k+1)t n k. Proof. We let ϕ = ϕ t = 1 and differentiate ϕ with respect to t. We may consider b i j = b i j(ϕ kl, ϕ m, ϕ), then we may use chain rule to get ϕ t = 1 ( bi j ϕ 2 b i kl + bi j ϕ k + bi j j ϕ kl ϕ k ϕ ϕ) A straightforward calculation gives b i j = σik δj l ϕ kl v 2 and bi j ϕ = bi j r r Φ = λ δj i 1 v 2 Φ (r) = λλ δj i (4.2.3) v 2 and previous equation can be rewritten as ϕ t = 1 ( σik δj l 2 b i j v 2 = σik v 2 2 b i j σik v 2 2 b i j ϕ kl + bi j ϕ k + λλ δj i ϕ k v 2 i i ϕ kj 1 ϕ k λλ 2 ϕ k v 2 2 ϕ kj 1 2 ϕ k ϕ k ϕ ϕ) (n k)λλ ϕ. (k + 1)v2 2 (4.2.4) In the last step, we have used i i n k. k+1 Recall that λ (r) = 1 + λ 2 mλ 1 n. So we have λ (r) = 2λλ + m(n 1)λ n λ 2 = 2λλ + m(n 1)λ n λ 1 + λ 2 mλ 1 n 2λ (r) = λ m(n 1)λ n > 0. (4.2.5) 43

51 Since ( σ ik ) > 0, b i j > 0 and λ (r) > 0, we can conclude that max ϕ is decreasing and ϕ is bounded from above from (4.2.4) and maximum principle. So 1 positive upper bound and so = (b i j) c = ϕ has a and = (h i j) = v (bi j) λ c R 2 e (k+1)t n k. In the last step. we have used v 1 and λ R 2 e k+1 n k t. Based on previous estimates, we will improve the estimate of and show that has a uniform positive and lower bound. Before we improve the estimate of, we first prove the next two lemmas that will be used later. Lemma We have ϕ gs n = O(e t n k ), r g = O(e t n k ) and v = 1 + O(e t n k ). Proof. Let ψ = 1 ϕ 2 2 g S n. Then ψ = 1 ϕ 2 t 2 t g S n = m ϕ m ( ϕ) = m ϕ t m ( 1 ). rom and consider b i j = b i j(ϕ kl, ϕ k, ϕ), we have m ( 1 ) = 1 2 = 1 2 ( bi j b i m ϕ kl + bi j m ϕ k + bi j j ϕ kl ϕ k ϕ m ϕ) ( σik δj l b i j v 2 m ϕ kl + bi j m ϕ k + λλ δj i ϕ k v 2 m ϕ)). Thus we get: t ψ = 1 ( σik δj l 2 b i j v 2 m ϕ m ϕ kl bi j ϕ m ϕ m ϕ k δi jλλ k v 2 m ϕ m ϕ) = 1 σ ik δj l ( 2 b i j v 2 m ϕ m ϕ kl ψ k 2 i i λλ ψ) ϕ k v 2 44

52 On the other hand, we may derive: ψ kl = ϕ mkl ϕ m + ϕ mk ϕ m l = ϕ klm ϕ m + (δ p mσ kl δ p l σ km)ϕ p ϕ m + ϕ mk ϕ m l = ϕ klm ϕ m + σ kl ϕ 2 g S n ϕ k ϕ l + ϕ km ϕ m l, then the evolution equation can be rewritten as: t ψ = σ ik δ l ( j b i j v 2 ψ kl (σ kl ϕ 2 2 g S n ϕ k ϕ l ) ϕ km ϕ m l ) 1 ψ k 2 i 2 ϕ k v 2 2 i λλ ψ Since following ODE: σik v 2 (σ 2 kl ϕ gs n ϕ k ϕ l ) and ϕ km ϕ m l are both positive definite, we get the Using i i n k, = λ, λ k+1 v λ from Lemma 4.2.4, we have d dt ψ max 2 i v 2 2 i λλ ψ max. = m(n 1)λ n 1 1 and 2 (n k)2 k+1 +L2 e 2(k+1) n k t 2 i i λλ v 2 2 2(n k)λλ (k + 1)λ 2 2(n k) 2 (k + 1)( (n k)2 + L k+1 2 e 2(k+1) n k t ) = 1 n k + L2 (k+1) e 2(k+1) n k t 2 2 Note that 1 n k + L2 (k+1) e 2(k+1) n k t 2 2 = 2 n k L2 (k+1) (n k) 2(k+1) e n k t 2 n k (1 L2 (k + 1) 2(k+1) (n k) e n k t ) It implies that where D = 2L2 (k+1) (n k) 2. d dt ψ 2 max ( n k De 45 (k+1) n k t )ψ max

53 Then by maximum principle, we conclude that d dt ψ 2 max ( n k De (k+1) n k t )ψ max Solving the ODE, we get ψ max = O(e 2 n k t ) and so ψ = O(e 2 n k t ). Hence it is straightforward to see that ϕ gs n = O(e t n k ). Using d dr ϕ = 1 λ(r), we have r 2 g = λ 2 ϕ 2 g. Recall that g ij = λ 2 (σ ij ϕi ϕ j v 2 ) from (3.2.2). We may derive that r 2 g = λ 2 ϕ 2 g = (σ ij ϕi ϕ j v 2 )ϕ iϕ j = ϕ 2 g S n ϕ 4 g Sn 1 + ϕ 2 g S n = ϕ 2 g S n 1 + ϕ 2 g S n = O(e 2t n k ) and so r g = O(e t n k ). Now we can see that v = 1 + ϕ 2 g S n = 1 + O(e t n k ). Lemma Let X be a tangent vector in M. Then X r = λ λ (X r, X r ) and X (λ r ) = λ X. In particular, we have t (λ r ) = λ ν. Proof. Given a tangent vector X in M. We have X = X r r + X i i. Then X r = X r r+x i i r = X r r r + X i i r = X i Γ k i0 k = X i λ λ δk i k = λ λ Xi i. We have used the fact that i r = Γ k i0 k, Γ k i0 = λ λ δk i, Γ k 00 = Γ 0 00 = Γ 0 0i = 0 from Also, we can see that X i i = X X r r, so we have X r = λ λ (X Xr r ) = λ λ (X r, X r ). 46

54 Now X (λ r ) = X(λ) r + λ X ( r ) = X r λ (r) r + λ (X r, X r ) = λ X. The equation t (λ r ) = λ ν follows from the definition t = 1 ν. Lemma Let ω = ν, λ r. Then ω satisfies the following evolution equation ω t = Lω + ij h ik h k j ω + λo(e (n+1)r ). (4.2.6) Proof. Let ω = ν, λ r. Using d dt ν = ( 1 ) = 2 and t (λ r ) = λ ν, we have ω t = t ν, λ r = ν t, λ r + ν, t (λ r) = λ, r + λ 2. Similarly, we have j ω = j ν, λ r + ν, j (λ r ) = h k j k, λ r and i j ω = ( i h k j ) k, λ r h ik h k j ν, λ r + h k j k, λ i = ( k h ij ) k, λ r h ik h k j ν, λ r + h k j k, λ i + R oijk k, λ r = ( k h ij ) k, λ r h ik h k j ν, λ r + h k j k, λ i + λo(e (n+1)r ). So now we can see that: Lω = ij i j ω = ij (( k h ij ) k, λ r h ik h k j ν, λ r + h k j k, λ i + λo(e (n+1)r )) =, λ r ij h ik h k j ν, λ r + λ + λo(e (n+1)r ) inally, we derive the evolution equation of ω in AdS-Schwartzchild space: ω t = Lω + ij h ik h k j ω + λo(e (n+1)r )