INSUFFICIENT CONVERGENCE OF INVERSE MEAN CURVATURE FLOW ON ASYMPTOTICALLY HYPERBOLIC MANIFOLDS. André Neves. Abstract

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1 INSUFFICIENT CONVERGENCE OF INVERSE MEAN CURVATURE FLOW ON ASYMPTOTICALLY HYPERBOLIC MANIFOLDS André Neves Abstract We construct a solution to inverse mean curvature flow on an asymptotically hyperbolic 3-manifold which does not have the convergence properties needed in order to prove a Penrose type inequality. This contrasts sharply with the asymptotically flat case. The main idea consists in combining inverse mean curvature flow with work done by Shi Tam regarding boundary behavior of compact manifolds. Assuming the Penrose inequality holds, we also derive a nontrivial inequality for functions on S Introduction A Penrose inequality for asymptotically flat 3-manifolds was proven independently by Huisken-Ilmanen [8], using inverse mean curvature flow, Hugh Bray [1], using a conformal deformation of the ambient metric. Recently, Hugh Bray Dan Lee [1] extended Bray s approach prove a Penrose inequality for dimensions less than 8. In this paper we investigate an analogous problem for M, g, an asymptotically hyperbolic 3-manifold with scalar curvature R 6. We start by mentioning that these manifolds can arise in General Relativity in two ways: As spacelike hypersurfaces in space-time with a cosmological constant Λ = 3 or as hyperboloidal hypersurface in space-time i.e., second fundamental form h in space-time satisfies h = g with cosmological constant Λ = 0. In both cases, the dominant energy condition translates into R 6. We also recall that the mean curvature of an apparent horizon Σ is given by H = tr Σ h. Hence, in the first case Λ = 3 assuming that h = 0 time-symmetric hypothesis, we have that apparent horizons are minimal surfaces, while in the second case Λ = 0 apparent horizons correspond to H = 2 surfaces. The author was partially supported by NSF grant DMS

2 2 Insufficient convergence of inverse mean curvature flow on... Broadly speaking, the Penrose inequality says that, assuming the dominant energy condition, the presence of outermost apparent horizons implies conjecturally a lower bound on the mass of M. Following the presentation of [2], the lower bound should be given by Mass 16π Σ 1/2 16π 3/2 43 Λ Σ, where Σ is a sphere which is an outermost apparent horizon see next subsection for definitions Λ is the cosmological constant. Xiadong Wang in [16] proposed a definition of mass see also [5] [6] for a less restrictive definition conjectured the following version of the Penrose inequality for Λ = 0. The definition of mass M will be recalled in the next subsection. Conjecture. If Σ 0 is an outermost sphere with HΣ 0 = 2, then Σ0 1/2 M. 16π If equality holds then M, g is isometric to an Anti de Sitter Schwarzschild manifold outside Σ 0. The main purpose of this paper is to show that, contrarily to what was suggested in [16], the inverse mean curvature flow does not have the necessary convergence properties needed to prove this conjecture. As it was pointed out by the referee, our argument does not carry to the case Λ = 3, i.e., where Σ 0 is an outermost minimal surface. At the end of the proof of Theorem 1.2, in Section 5, we suggest a modification in our argument that would hle that case. We should also point out that, even if it is geometrically natural, there is no physical evidence supporting the choice of the mass considered in [16]. An evidence-based choice for the correct mass term in the Penrose inequality was suggested by Chruściel Simon in [4] see also [2] which we now briefly describe. Like before, we refer the reader to the next subsection for the relevant definitions. Assume that there is a smooth function u defined on M so that {u < 0} coincides with the outermost horizon Σ 0, the surfaces Σ t = {u < t} give a smooth solution for inverse mean curvature flow. Set the mass to be the limit of the Hawking masses Then, it is a stard fact that Mass = lim t m H Σ t. Σ 1/2 Mass m H Σ 0 = 16π 3/2 16π 43 Λ Σ. Even if Σ 0 does not admit a smooth solution to inverse mean curvature flow, the work done by Huisken Ilmanen in [8] implies with

3 André Neves 3 no difficulty that if Σ 0 is an outermost horizon, then there is a weak solution Σ t t 0 to inverse mean curvature flow such that m H Σ t is non-decreasing so Mass = lim m H Σ t m H Σ 0 = Σ 0 1/2 t 16π 3/2 16π 4 3 Λ Σ Notation Definitions. Given a complete noncompact Riemannian 3-manifold M, g, we denote its connection by D, the Ricci curvature by Rc, the scalar curvature by R. The induced connection on a surface Σ M is denoted by, the exterior unit normal by ν whenever its defined, the mean curvature by H, the trace free part of the second fundamental form by Å, the surface area by Σ. A sphere Σ M with mean curvature HΣ = 2 is said to be outermost if it is the boundary of a compact set its outside region contains no other spheres with H = 2. We say that Σ is outer minimizing if every compact perturbation lying outside of Σ has bigger surface area. In what follows g 0 denotes the stard metric on S 2. Definition 1.1. A complete noncompact Riemannian 3-manifold M, g is said to be asymptotically hyperbolic if the following are true: i There is a compact set K M such that M \K is diffeomorphic to R 3 minus an open ball. ii With respect to the spherical coordinates induced by the above diffeomorphism, the metric can be written as g = dr 2 + sinh 2 r g 0 + h/3 sinh r + Q where h is a symmetric 2-tensor on S 2 Q + DQ + D 2 Q + D 3 Q C exp 4r for some constant C. For simplicity, the manifolds we consider have only one end. The above definition is stated differently from the one given in [16] see also [6]. Nonetheless, using a simple substitution of variable sinhr/2 t = ln, coshr/2 they can be seen to be equivalent. Note that a given coordinate system on M \ K induces a radial function rx on M \ K. With respect to this coordinate system, we define the inner radius outer radius of a surface Σ M \ K to be r = sup{r B r 0 Σ} r = inf{r Σ B r 0} respectively. Furthermore, we denote the coordinate spheres induced by a coordinate system by { x = r} := {x M \ K rx = r}

4 4 Insufficient convergence of inverse mean curvature flow on... the radial vector by r. We stress that the radial function rx depends on the coordinate system chosen. If γ is an isometry of H 3, the radial function sx induced by this new coordinate system is such that sx rx C for all x M \ K, where C depends only on the distance from γ to the identity. We denote by s s the correspondent quantities defined with respect to this new coordinate system. The mass M of an asymptotically hyperbolic manifold M, g with R 6 is given by M = 1 16π [ S 2 tr g0 hdµ i=1 S 2 tr g0 h x i dµ 0 2 ] 1/2, where x 1, x 2, x 3 are the stard coordinates on S 2 R 3. This quantity is well defined i.e. independent of the coordinate system chosen for M \ K by [16] see also [5] [6] for a less restrictive definition. The Anti de Sitter Schwarzschild metric S 2 [t 0, +, g m is given by g m = dt t 2 m/t + t2 g 0, where we choose t 0 so that the mean curvature of the coordinate sphere Σ 0 = { x = t 0 } is 2. A change of variable see [16, page 294] shows that the metric can be written as g = dr 2 + sinh 2 r + m/3 sinh rg 0 + P, where P is term with order exp 5r. An explicit computation reveals that the scalar curvature equals 6 that M = m 2 = Σ0 1/2. 16π 1.2. Statement of the main results. We start by briefly describing how inverse mean curvature flow could prove the conjecture. Find a family of surfaces Σ t t 0 with initial condition Σ 0 such that dx dt = ν HΣ t. Note that the existence theory for a weak solution developed in [8, Section 3] can be used in the current setting. Moreover, the same arguments in [8, Section 5] show that the quantity called the Hawking mass m H Σ t := Σ t 1/2 16π 3/2 16π H 2 4 dµ t Σ t is monotone nondecreasing along the flow. Therefore, Σ0 1/2 = m H Σ 0 lim 16π m HΣ t. t

5 André Neves 5 The result would follow if one could show that the limit of the Hawking mass is not bigger than M. In the asymptotically flat case, Huisken Ilmanen [8, Section 7] showed this by proving that lim inf t areab rt 0 r t = lim inf = 1, areab rt 0 t r t where r t r t denote the outer radius inner radius of Σ t respectively. In our setting, it is not hard to see that in order for the limit of the Hawking mass to be smaller than M we need to find an isometry γ of H 3 such that, with respect to the induced coordinate system, the following two properties hold: 1 B st 0 lim inf = lim inf t B st 0 s t s t = 0, t where s t s t denote, respectively, the outer radius inner radius of Σ t with respect to the radial function sx induced by γ; 2 If the metric with respect to the coordinates induced by γ is written as then g = ds 2 + sinh 2 s g 0 + h γ /3 sinh s + P, S 2 x i tr g0 h γ dµ 0 = 0 for i = 1, 2, 3, where x i denotes the coordinate functions of the unit sphere in R 3. If these properties do not hold, it is impossible to compare the limit of the Hawking mass with the mass of the manifold. More precisely, we can construct a family of spheres Σ r r 0 an asymptotically hyperbolic metric M, g where m H Σ := lim m H Σ r r is bigger or smaller than the mass M. This comes from the following observation. Pick a 2-tensor h on S 2 such that x i tr g0 hdµ 0 = 0 for i = 1, 2, 3. S 2 Then the mass is given by M = 1 tr g0 hdµ 0 4 S 2, according to Proposition 2.1 e, given any function f in C S 2 we can construct a family of spheres Σ r r 0 such that m H Σ = 1 1/2 exp2fdµ 0 tr g0 h exp fdµ 0. 4 S 2 S 2

6 6 Insufficient convergence of inverse mean curvature flow on... It is simple to recognize that one can choose h positive definite some function f for which m H Σ < M. Moreover, if one chooses f to be zero but x i tr g0 hdµ 0 0 for i = 1, 2, 3. S 2 then we would have in this case m H Σ = 1 tr g0 hdµ 0 > M. 4 S 2 We can now state the main theorem. Theorem 1.2. There is an asymptotically hyperbolic 3-manifold M, g with scalar curvature 6 for which its boundary Σ 0 is an outerminimizing sphere with HΣ 0 = 2 satisfying the following property. There is a smooth solution to inverse mean curvature flow Σ t t 0 with initial condition Σ 0 such that for every coordinate system we have lim inf t s t s t > 0. Remark 1.3. i We note that the sphere Σ 0 might not be outermost, i.e., there could be another H = 2 sphere enclosing Σ 0. If this is the case, the area of this sphere has to be bigger than Σ 0. We point out that the Penrose inequality in the asymptotically flat case also holds with an outer-minimizing minimal sphere instead of an outermost minimal sphere. ii The author does not know whether the manifold constructed constitutes a counterexample to the Penrose inequality. The reason is that it is hard to compute explicitly the mass of the manifold M, g. In the last section of the paper assuming that the manifolds we construct do not violate the Penrose inequality, we deduce a nontrivial inequality for functions on S 2 see Section 5.1. Jointly with Alice Chang, we have verified that such inequality indeed holds. The strategy of the proof is the following. We first construct a solution to inverse mean curvature flow Σ t t 0 on an Anti De Sitter Schwarschild metric g m with mass m/2 such that lim m HΣ t > m/2. t This is done in Sections 2 3, where we prove a long time existence result for inverse mean curvature flow on Anti de Sitter Schwarzschild space. It is important that the estimates in this section do not depend on the area of our initial condition this requires a careful bookkeeping. Unfortunately the initial condition for this solution is not a sphere with H = 2. To fix that, we find a function u such that the metric g = u2 H 2 dt2 + g t

7 André Neves 7 has R = 6 u Σ0 = HΣ 0 /2. Note that, with respect to this metric, Σ 0 has H = 2. This is done in Section 4 where we adapt the work of Shi-Tam [14] Wang-Yau [15] to prove the existence of such function u. A simple computation will show that Σ t t 0 is also a solution to inverse mean curvature flow with respect to the metric g. We are left to show that this solution has the desired asymptotic behavior. Using Proposition 2.1 f we will see that the Gaussian curvature K t of Σ t with respect to the normalized metric ĝ t := 4π Σ t 1 g t does not converge to one when t goes to infinity. Note that it does not matter whether we choose g or g m in this step because the intrinsic geometry is the same. This fact Proposition 2.1 f imply that the metric g is such that for every coordinate system we have lim inf t s t s t > 0. The proof of Theorem 1.2 is done in Section 5. Acknowledgments The author would like to express his thanks to Gang Tian Piotr Chruściel for many useful discussions their interest in this work. The author is also grateful to the referee for the insightful suggestions. 2. Basic properties of graphical surfaces on asymptotically hyperbolic 3-manifolds In this section M, g denotes an asymptotically hyperbolic manifold with some given coordinate system on M \ K. Given a function f on S 2 we consider the surfaces Σq 0 = {q 0 + fθ, θ θ S 2 } M \ K. The function f satisfies hypothesis I if there are constants V, V 0 such that { f V, I 0 f V 0, where 0 denotes the connection with respect to the round metric on S 2. We denote by sq 0 sq 0, respectively, the outer radius inner radius of Σq 0, where sx is the radial function induced by some coordinate system γ. Given any geometric quantity T defined on Σq 0, we use the notation T = Oexp kr when we can find a constant C = Cg, V, V 0 for which T C exp kr.

8 8 Insufficient convergence of inverse mean curvature flow on... The next proposition collects some properties for the surfaces Σq 0 when q 0 is very large. Proposition 2.1. Assume that f satisfies hypothesis I. The following properties hold: a When q 0 goes to infinity, the normalized metrics ĝq 0 := 4π Σq 0 1 g Σq0 converge to 1 ĝ := exp2fdµ 0 exp2fg 0. S 2 b There is a constant C = Cg, V, V 0 such that, for all q 0 1, c Assume that Σq 0 H 2 + Σq 0 Å C + C sup 2 0f S 2 sup 2 0f C Σq 0 H 2 + Σq 0 Å + C; S 2 sup k 0f E for all k = 2,, n 1. S 2 There is a constant C = Cg, E, V, V 0 such that for all q 0 1 Σq 0 n+2 n A 2 C + C sup S 2 n 0 f 2 sup n 0 f 2 C + C Σq 0 n+2 n A 2 ; S 2 d The mean curvature of Σq 0 satisfies e H 2 4 = 4KΣ + 2 Å 2 2tr g 0 h sinh 3 r + Oexp 4r; lim q 0 m HΣq 0 = 1 4 S 2 exp2fdµ 0 1/2 f There is a coordinate system γ for which if only if lim q 0 sq 0 sq 0 = 0 K = lim Kq 0 = 1, q 0 S 2 tr g0 h exp fdµ 0. where Kq 0 is the Gaussian curvature of Σq 0 with respect to ĝq 0.

9 André Neves 9 Note that this proposition also holds, with obvious modifications, if Σq 0 = {q 0 + f q0 θ, θ θ S 2 }, where the functions f q0 converge to a function f on S 2 when q 0 goes to infinity. Proof. Consider tangent vectors to Σq 0 i := f θ i r + θi, i = 1, 2, where θ 1, θ 2 represent coordinates on S 2 which are orthonormal with respect to g 0 at a given point p. The induced metric on Σq 0 is given by g ij = f θ i f θ j + sinh 2 q 0 + fg 0 θi, θj + Oexp r so detgij = sinh 2 q 0 +f detg 0 +O1 = sinh 2 q 0 exp2f detg 0 +O1. This implies that Σq 0 1 lim q 0 4π sinh 2 = exp2f dµ 0 q 0 S 2 the first property follows from the fact that lim sinh q 0 2 g ij = exp2fg 0 θi, θj. q 0 Denoting the connection with respect to the stard hyperbolic metric by D, we have D r r = 0, D θj r = cosh r sinh r θ j, D θi θj = sinh rcosh rδ ij r, D D C exp 3r for some C = Cg. Thus, D i j = 2 f θ j θ i r + f θ j f θ i D r r + f θ j D θi r + f θ i D r θj +D θi θj = cosh r sinh rδ ij r + 2 f θ j θ i r + O1 θj + Oexp r. An easy computation shows that the exterior unit normal is given by 2 ν = 1 + Oexp 2r r + Oexp 2r θ1 + Oexp 2r θ2 thus A ij = 2 cosh r sinh r g ij 2 f + O1. θ j θ i This implies Property b.

10 10 Insufficient convergence of inverse mean curvature flow on... Property c follows from what was done above plus some tedious computations. We now prove Property d. It was shown in [13, Lemma 3.1.] that Rcν, ν+2 = tr g 0 h 2 sinh 3 +Oexp 4r R = 6+Oexp 4r. r Combining this with Gauss equations we obtain that H 2 4 = 4KΣq Å 2 + 4Rν, ν R/2 1 = 4KΣq Å 2 2tr g 0 h sinh 3 r + Oexp 4r. Combining Property a with Property d, it follows from the definition of Hawking mass that lim m Σq 0 1/2 2tr g0 h HΣq 0 = lim q 0 q 0 16π 3/2 Σq 0 sinh 3 r dµ 1 2 Σq 0 3/2 = lim q 0 16π 3/2 Σq 0 sinh 3 tr g0 hdˆµ r = 1 3/2 4 3/2 exp2fdµ 0 2tr g0 h exp 3fdˆµ S 2 S 2 = 1 1/2 exp2fdµ 0 tr g0 h exp fdµ 0. 4 S 2 S 2 Finally, we prove Property e. Given a coordinate system induced by an isometry γ of H 3, we consider the function on Σq 0 given by wx = sx ˆq 0 where Σq 0 = 4π sinh 2 q 0, where sx is the radial function for this coordinate system. For all q 0 sufficiently large, Σq 0 is graphical over the coordinate spheres for this new coordinate system so lim sup q 0 s ν, s Due to [13, Propostion 3.3], we know that s = 4 2 s 2 exp 2s + 2 H exp2q 0 <. + H 21 s, ν + 1 s, ν 2 + Oexp 3s. Therefore, Property d implies that w satisfies the following equation with respect to ĝq 0 3 ˆ w = exp 2w Kq0 + P q 0, where lim P q 0 dˆµ = 0. q 0 Σq 0

11 André Neves 11 Suppose the coordinate system induced by γ is such that lim sq 0 sq 0 = 0. q 0 Then lim w = 0 q 0 so equation 3 implies that Assume for simplicity that lim Kq 0 = 1. q 0 S 2 exp2f dµ 0 = 1 because, according to 1, this implies that lim ˆq 0 q 0 = 0. q 0 If ˆK = 1, then ĝ is a round metric on S 2 hence there is a conformal transformation γ of S 2 for which γ ĝ = g 0. From Property a we know that ĝ = exp2fg 0 so γ g 0 = exp 2f T g 0. This conformal transformation induces an isometry of hyperbolic space which we still denote by γ. The relationship between the radial functions rx sx is determined by sx + f γx r γx C exp rx for some constant C. This implies that for all x in Σq 0 thus wx + ˆq 0 q 0 C exp q 0, lim w = 0. q 0 q.e.d. 3. Long time existence for inverse mean curvature flow on asymptotically hyperbolic 3-manifolds In this section the ambient manifold will be an Anti de Sitter Schwarzschild metric S 2 [s 0, +, g m with mass m > 0. A sphere Σ 0 satisfies hypothesis H if we can find constants Q j j N, ε 0, δ 0 for which H ε 0 Σ 0 H 2 4 Q 0, H ν, r ε 0 ν, r 1 Σ 0 1 Q 1, Å 2 1/4 δ 0 H 2 Σ 0 2 Å 2 Q 2, sup n A 2 Q n+2 Σ 0 n+2 for all n 1, Σ 0 Σ 0 bounds a compact region containing S 2 {s 0 }.

12 12 Insufficient convergence of inverse mean curvature flow on... Recall that r 0 r 0 denotes, respectively, the outer radius the inner radius of Σ 0. Theorem 3.1. Assume that Σ 0 satisfies H. There is a constant r = rq j j N, ε 0, δ 0, r 0 r 0, m such that if r 0 r then the inverse mean curvature flow Σ t with initial condition Σ 0 exists for all time has the following properties: i There is a positive constant C = CQ j j N, ε 0, δ 0, r 0 r 0, m such that the mean curvature of Σ t satisfies H C, for some other constant C = CQ j j N, ε 0, δ 0, r 0 r 0, m, Σ 0 H 2 4 C exp t; ii For every n 0 k 1 there is a constant such that C = CQ j j N, ε 0, δ 0, r 0 r 0, m Σ 0 n+2 k t n A 2 C exp n + 2t Σ 0 n+2 n A 2 C exp n + 2t for n 1; iii The surfaces Σ t can be described as where ˆr t is such that Σ t = {ˆr t + f t θ, θ θ S 2 }, Σ t = 4π sinh 2 ˆr t. Moreover, the functions f t converge to a smooth function f defined on S 2. iv For every n 0 k 1 there is a constant such that C = CQ j j N, ε 0, δ 0, r 0 r 0, m Σ 0 n n f t 2 C exp nt, k t f t C exp t, Σ 0 n k t n f t 2 C exp nt for n 1. We essentially adapt to our setting some of the ideas used in the work of Huisken Ilmanen [9] Claus Gerhardt [7] on smooth solutions to inverse mean curvature flow. We could have been more precise regarding how the constants depend on Q j j N but this version of the theorem suffices for our purposes. The important point is that the estimates do not depend on r 0 only on r 0 r 0.

13 André Neves 13 Proof. During the first part of this proof, given any geometric quantity T defined on Σ t we use the notation T = Oexp kr whenever there is a constant C = Cm such that T C exp kr. Because H > 0 we have short-time existence for the flow. Denoting by Σ m t := { x = rt m } the solution to inverse mean curvature flow with initial condition { x = t 0 } we know that Σ m t = Σ m 0 expt thus we can find a constant K = Km such that t/2 K r m t t 0 t/2 + K. Because two solutions that are initially disjoint must remain disjoint [8, Theorem 2.2], we have that for some constant K = Km 4 t/2 + r 0 K r t r t t/2 + r 0 + K r t r 0. Therefore, we can find K = Km, r 0 r 0 for which K 1 Σ 0 expt exp2r K Σ 0 expt. We now derive the evolution equations that will be needed later on. We use the notation B ij C ij when B ij C ij have the same trace-free part. Set X := φr r β t := exp t/2 X, ν, where the function φ is such that g m = dr 2 + φr 2 g 0 ν is the exterior normal vector to Σ t. Lemma 3.2. The following evolution equations hold. a dβ t dt = β t H 2 + b A 2 H 2 1 3m β t + r sinh 3 r + Oexp 5r dh dt = H H 2 A 2 + Rcν, ν 1 H 2 H 2 H 3 ; βt H 2 ;

14 14 Insufficient convergence of inverse mean curvature flow on... c då dt Å 2 H H H2 H 3 d d Å 2 dt + Å 2Å2 H Å 2 H 2 H2 + 2Rcν, ν + Oexp 3r 2H 2 + Å 1 H + 1 H 2 Oexp 3r; Å 2 H Å 2 H 2 H2 + 2Rcν, ν + Oexp 3r 2H 2 Å 2 2 Å 2 2 Å 2 H H, Å H 2 4 H 3 + Å H 2 + Å Oexp 3r. H Proof. For every vector Y we have that D Y X = φ ry this implies that, using local coordinates y 1, y 2 for Σ t, β t, i = exp t/2a i, X, i = 1, 2 expt/2 β t = i i A i, X + A i, i X Moreover so Therefore dβ t dt = β t H 2 + = H, X + Rcν, X + φ H X, ν A 2. D t ν = H/H 2, D t X = φ t, dβ t dt = φ H, X + H H 2 β t 2. A 2 H β exp t/2 Rcν, X H 2. Note that denoting by e 1, e 2 a g m -orthonormal basis for the coordinates spheres ν, X = 0 ν, r r, X = i ν, e i e i, X

15 André Neves 15 hence, we obtain from [12, Lemma 3.1 iii] that Rcν, X = i ν, e i Rce i, X + ν, r Rc r, X m = 2 sinh 3 r + Oexp 5r ν, e i e i, X i m sinh 3 r + Oexp 5r ν, r r, X = r 2 3m 2 sinh 3 r + Oexp 5r ν, X. The second evolution equation was derived in [8, Section 1]. We now prove the third identity. From [10, Theorem 3.2] it follows that assuming normal coordinates around a point p dåij dt da ij dt A ij i j H H 2 2 ih j H H 3 + ÅikÅkj R νiνj. H Arguing like in the proof of Simons identity for the Laplacian of the second fundamental form A see for instance [10], one can see that Åij i j H + HÅimÅmj + ÅijH 2 /2 Åij Å 2 + HR νiνj R νν Å ij + R kikm Å mj + R kjkm Å im + R kijm Å km + R mjik Å km + D k R νjik + D i R νkjk. Because the metric g m satisfies it follows that R stuv = δ su δ tv δ sv δ tu + Oexp 3r D q R stuv = Oexp 3r R kikm Å mj +R kjkm Å im +R kijm Å km +R mjik Å km = 4Åij+ÅijOexp 3r = 2Rcν, νåij + ÅijOexp 3r, therefore R νiνj = g ij + Oexp 3r, dåij dt Åij H ih j H H 3 Å 2 H 2 H2 + 2Rcν, ν + Oexp 3r 2H 2 + Å ij 1 H + 1 H 2 Oexp 3r.

16 Å2, Å = 0. q.e.d. 16 Insufficient convergence of inverse mean curvature flow on... Using the formula då dt i, j = dåij dt D t i, k Åkj D t j, k Åik we obtain Lemma 3.2 c. The last identity follows from d Å 2 då = 2 dt dt, Å We now argue that we can choose r = rm a positive constant C = Cε 0, r 0 r 0, m such that if r 0 ˆr, then 5 H C ν, r C expr 0 r 0 while the solution exists. Choosing r large enough so that for all r r the term 3m 2 sinh 3 r + Oexp 5r in the equation of Lemma 3.2 a is positive, we obtain that dβ t dt β t H 2 while β t is nonnegative thus β t min β 0 > 0. Note that φr grows like expr so, for some constant C = Cm, β t C expr 0 r, ν. This implies the desired bound for ν, r. Set α t := β t H. Because Rcν, ν = 2 + Oexp 3r, the previous lemma implies that, provided we choose r sufficiently large, dα t dt = α t H 2 2 α t, H H Oexp 3r H 2 α t 2H 2 α t H 2 2 α t, H H H 2 α t 2H 2 Because α t 3min β 0 implies that H 2 3, it follows from the maximum principle that α t min{ 3min β 0, min α 0 } for all t thus we can use the inequalities in 4 in order to obtain the desired bound for the mean curvature.

17 André Neves 17 Lemma 3.3. We can find constants r = rε 0, δ 0, r 0 r 0, m C = Cε 0, δ 0, r 0 r 0, m such that if r 0 r, then Σ 0 H 2 4 C Σ 0 sup H Å 2 + exp r 0 exp t Σ 0 Σ 0 2 Å 2 C while the solution exists. Σ 0 2 sup Å 2 + exp 2r 0 exp 2t Σ 0 Proof. We assume that the bounds in 5 hold. Let α t := Å 2 H 2. From Lemma 3.2 dh 2 = H 2 dt H Å 2 H 2 + H2 + 2Rcν, ν 2H 2 H 2 2 H 2 H 6 thus 6 dα t dt α t H 2 +4α2 t 2α t + α t + α t + H 1 Oexp 3r α t H 2 +Q, where Q := 4 H, Å 2 H 5 H 2 2 Å 2 H 6 2 Å 2 H H, Å H 4 4 H 3. We claim that, given ε > 0, we can find a constant C = Cε 0, ε, r 0 r 0, m so that dα t dt α t H 2 + 4α t α t 12 + ε + C exp 6r 0 3t + Q Qp 4 H 2 H εα t p 14 + ε + C exp 6r 0 3t, whenever p is a critical point of α t. The first inequality follows easily from Cauchy s inequalities combined with properties 4 5. Denote by {v 1, v 2 } an eigenbasis for Å at p assume without loss of generality that Åv 1, v 1 0. Because p is a critical point of α t the following identities hold at p Å 2 = 2 Å 2 H 1 H Å H 1 H = 2 Åv 1, v 1 = 2 Åv 2, v 2. As a result, we obtain that Å 2 = Å 2 H 2 H Åv 1, v 2 2 = α 2 t H Åv 1, v 2 2

18 18 Insufficient convergence of inverse mean curvature flow on... 2 Åv 1, v 2 2 = 2 v2 Av 1, v 1 +Rcν, v v1 Av 2, v 2 +Rcν, v v2 Av 1, v v1 Av 2, v α t H + H Oexp 3r + Oexp 6r = 2 v 2 Åv 1, v 1 + H, v v 1 Åv 2, v 2 + H, v α t H + H Oexp 3r + Oexp 6r = 2 H, v 2 2 αt H, v 1 2 αt α t H + H Oexp 3r + Oexp 6r = αt 2 H 2 + H 2 H H, Å /2 2 H + α t H + H Oexp 3r + Oexp 6r. Moreover, we also have that at the point p thus Qp = 4 H 2 H 4 4 H, Å 2 = 8α t H 2 α t p 1 4 H + α t H + H Oexp 3r H Oexp 6r H 4. The claim follows from Cauchy s inequalities combined with properties 4 5. As a result, there is a constant C 1 = C 1 ε 0, ε, r 0 r 0, m for which if we set β t := α t + C 1 exp 6r 0 3t, then 7 dβ t dt β t H 2 + 4α t α t 12 + ε + Q Qp 4 H 2 H εα t p 14 + ε whenever p is a critical point of β t. Chose ε < δ 0 /4 so that ε 4 δ ε 0

19 André Neves 19 chose r so that C 1 exp 6r δ 0 /4. Thus β 0 1/4 3δ 0 /4 β t x 1/4 δ 0 /2 = α t x 1/4 δ 0 /4. Therefore we can apply the maximum principle to β t conclude that α t sup α 0 + C 1 exp 6r 0 while the solution exists. This implies that α t ε 1/4 δ 0/8 so we obtain from equation 7 that α t sup α 0 + C exp 6r 0 exp 1 + 2δ 0 t for some C = Cε 0, δ 0, r 0 r 0, m. As a result, α 2 t Csup α exp 12r 0 exp 2t 4δ 0 t, αt exp3r Csup α 0 + exp 6r 0 exp 2t δ 0 t for some C = Cε 0, δ 0, r 0 r 0, m. Using this bounds in equation 6 we obtain α t dt α t H C exp 3t/2α t + Csup α 0 + exp 6r 0 exp 2 + δ 0 t + C exp 6r 0 3t + Q for some C = Cε 0, δ 0, r 0 r 0, m hence Å 2 C sup Å 2 + exp 6r 0 exp 2t. Σ 0 The evolution equation for H 2 is given by see [8, Section 1] dh 2 dt = H2 H 2 6 H 2 H 2 2 Å 2 H 2 2Rcν, ν thus, if we set φ t := expth 2 4, we obtain that dφ t dt = φ t H 2 3 φ t, H H 2 2 Å 2 expt expt4 + 2Rcν, ν From the upper bound derived for Å the bounds given in 4 5 we have that dφ t dt φ t H 2 C 3 φ t, H H 2 sup Å 2 + exp 6r 0 Σ 0 exp t C exp t/2 3r 0 where C = Cε 0, δ 0, r 0 r 0, m. The maximum principle implies that H 2 4 C sup H Å 2 + exp 3r 0 exp t. Σ 0

20 20 Insufficient convergence of inverse mean curvature flow on... In order to show the existence of some C = Cε 0, δ 0, r 0 r 0, m for which H C sup H Å 2 + exp 3r 0 exp t Σ 0 it is enough to note that dh 2 dt H2 H H 2 + C exp 3r 0 3t/2. q.e.d. Fix some r for which Lemma 3.3 holds. Note that in this case we have a uniform bound for A 2 so stard estimates can be used to show that the solution Σ t t 0 exists for all time. Nonetheless, we need shaper estimates on all the derivatives of A this will occupy most of the rest of the proof. What we have done so far proves Theorem 3.1 i. The next lemma will be useful in proving Theorem 3.1 iii. Lemma 3.4. There is a constant C = CQ 0, ε 0, δ 0, r 0 r 0, m such that 1 r, ν C sup1 r, ν + exp 3r 0 exp t Σ 0 for all t thus Σ 0 r 2 C exp t for some other constant C = CQ 0, Q 1, ε 0, δ 0, r 0 r 0, m. Proof. We denote by Λ any geometric quantity defined on Σ t for which we can find a constant C = Cε 0, δ 0, r 0 r 0, m such that For every vector Y we have that Therefore Λ C H 2 + Å + exp 2r D Y r = φ r/φr Y Y, r r. d r, ν dt = D t r, ν + r, D t ν = φ 1 φ H φ r, ν 2 φ H + r, H H 2. For every tangent vectors Z W we have r, ν, Z = φ /φ r, ν r, Z + AZ, r Z r, W = φ /φ Z, W φ /φ Z, r W, r r, ν AZ, W,

21 André Neves 21 where r denotes the tangential projection of r. These identities combined with Lemma 3.3 with φ φ = 1 + Oexp 2r imply that div φ φ φ r, ν r 2 = 2 r, ν r φ φ φ r, ν 2 H φ r 2 r, ν φ φ φ r, ν A r, r φ 2 = 2 r, ν + r, ν r 2 φ + φ φ r, ν 2 H + r 2 Λ diva, r = H, r +Rcν, r + φ φ H φ φ A r, r A 2 r, ν As a result we get where φ H 2 Q = 2 φ = H, r + φ φ H r 2 H2 2 r, ν Å 2 r, ν d r, ν dt = r, ν H 2 + Q, + r 2 Λ + Oexp 3r. 2 r, ν r, ν r 2 2 φ φ r, ν 2 H+ r 2 + H2 2 r, ν + Å 2 r, ν + r 2 Λ + Oexp 3r. Setting α t := r, ν 1, we obtain from 5 Lemma 3.3 that Q = 2H 2 H 2 φ 2 4 H φ φ + φ + α t 4 α2 t 2α t 4 + Å 2 r, ν + α t Λ + Oexp 3r α t 4 α2 t 2α t 4 + α t Λ + Oexp 3r α t 1 Λ + Oexp 3r, where the last inequality follows from 0 α t 1. There is C = CQ 0, ε 0, δ 0, r 0 r 0, m

22 22 Insufficient convergence of inverse mean curvature flow on... for which Λ C exp t hence dα t dt α t H C exp tα t + C exp 3t/2 3r 0 for some other C = CQ 0, ε 0, δ 0, r 0 r 0, m. This equation implies the desired result. q.e.d. For the rest of the proof, C will denote any constant with dependence C = CQ j j N, ε 0, δ 0, r 0 r 0, m. Set ˆr t to be such that Σ t = 4π sinh 2 ˆr t we remark that ˆr t t/2 is uniformly bounded. An immediate consequence of the previous lemma is that Σ t can be written as the graph of a function f t over the coordinate sphere { x = ˆr t } with f t C Σ 0 f t 2 C exp t for some constant C. Furthermore, Lemma 3.3 Proposition 2.1 imply the existence of some constant C for which Σ f t 2 C exp 2t. The next lemma is an adaptation of what was done in [7, Section 6]. Given two tensors P S we denote by S T any linear combination of tensors formed by contracting over S T. Lemma 3.5. There is r = rq j j N, ε 0, δ 0, r 0 r 0, m so that if r 0 r the following property holds. For every n 0 there is a constant C such that Σ 0 n n f t 2 C exp nt for all t. Equivalently, for all n 1 there is a constant C for which Σ 0 n+2 n A 2 C exp n + 2t. Proof. We start by showing that it is enough to bound n Å. Lemma 3.6. There exists a constant C for which Σ A C Σ 0 Å + C exp 3t/2 Σ A C Σ Å + C exp 2t. Moreover, if we can find a constant E for which then Σ 0 k+2 k Å 2 E exp k + 2t for all k = 1,... n 1, Σ 0 n+3 n+1 A 2 C 1 Σ 0 n+3 n+1 Å 2 + C 1 exp n + 3t. for some constant C 1 = C 1 E, Q j j N, ε 0, δ 0, r 0 r 0, m.

23 André Neves 23 Proof. On each Σ t consider the 1-form BX = RcX, ν. First we estimate the derivatives of B. In local coordinates x 1, x 2, B can be written as B j = F j r, r, j = 1, 2, where F j r, q 1, q 2 is defined on R 3 D k F j S k exp 3r j = 1, 2 for some constant S k, provided q 1, q 2 lie on a fixed compact set. We denote by P any tensor on Σ t for which P = Oexp 3r by Q any tensor for which Q = Oexp 3r Using this notation we have so we can estimate Q = r P + 2 r P. B = r Q + 2 r Q Σ 0 3 B 2 C exp 3t, Σ 0 4 B 2 C exp 4t for some constant C. Let {v 1, v 2 } be an orthonormal basis for Σ t. We know that for every integer p p Av 1, v 2 = p Åv 1, v 2 p Av 1, v 1 p Av 2, v 2 = p Åv 1, v 1 p Åv 2, v 2. Moreover, Codazzi equations imply that for i j thus p vi Av j, v j = p vj Av 1, v 2 p B i m+1 A C m+1 Å + C m B for every integer m. This implies the desired result when n = 0, 1. To prove the general result we proceed by induction. The inductive hypothesis implies that Σ 0 k k r 2 = Σ 0 k k f t 2 C 1 exp kt for all k = 1,..., n + 1 for some C 1 = C 1 E, Q j j N, ε 0, δ 0, r 0 r 0, m thus, using the expression derived for B, we obtain Σ 0 k+3 k B 2 C 1 exp k + 3t for all k = 1,..., n for some C 1 = C 1 E, Q j j N, ε 0, δ 0, r 0 r 0, m. Hence, the desired result follows. q.e.d.

24 24 Insufficient convergence of inverse mean curvature flow on... In what follows L 1 will denote any tensor that satisfies the following properties. There exists a constant C for which Σ 0 L 1 C exp t, Σ 0 3 L 1 2 C Σ 0 3 A 2 + C exp 3t, if there is a constant E such that then Σ 0 k+2 k A 2 E exp k + 2t for all k = 1,... n 1, Σ 0 n+2 n L 1 2 C 1 Σ 0 n+2 n A 2 + C 1 exp n + 2t for some constant C 1 = C 1 E, Q j j N, ε 0, δ 0, r 0 r 0, m. Likewise, L 0 will denote any tensor with the same properties of L 1 except that we just require L 0 to be uniformly bounded. We can see from Lemma 3.2 that the evolution equation for Å can be written as då dt Å H 2 Å + L 1 Å + M + A A L 0, where the tensor M sts for the term 1 H + 1 H 2 Oexp 3r that appears on Lemma 3.2 c. The relevant property of M is that Σ 0 2 M 2 C exp 3t Σ 0 3 M 2 C exp 3t Σ 0 3 A 2 + C exp 4t for some constant C. If there is a constant E such that for all t then Σ 0 k+2 k A 2 E exp k + 2t for all k = 1,... n 1, Σ 0 n+2 n M 2 C 1 exp 3t Σ 0 n+2 n A 2 + C 1 exp n + 3t for some other constant C 1 = C 1 E, Q j j N, ε 0, δ 0, r 0 r 0, m. This follows from the fact that in local coordinates x 1, x 2 1 M = H + 1 H 2 F r, r, where F r, q 1, q 2 is a matrix-valued function defined on R 3 for which there is a constant S k such that, provided q 1, q 2 lie on a fixed compact set, D k F S k exp 3r. If K denotes the curvature tensor of Σ t, then for any tensor T we know that T = T + K T + K T

25 André Neves 25 d T dt = dt dt T 2 + T Å + T A = dt dt T 2 + T L 1 + T L 0. The last identity comes from the fact that, using normal coordinates, d T dt 1,, n+1 = dt dt 1,, n+1 T 1,, n, D t n+1 Therefore, d T dt = T H 2 T dt 2 + dt T H 2 + T L 1 Proceeding inductively, it can be checked that d n Å dt n Å H 2 n 1 + j=0 n n Å + + T A 1,, n+1. + T L T L 1 + T L 0. n j Å n j L 1 j=0 n 1 j+2 Å n j L 1 + n M + + j=0 j,k,l 0,j+k+l=n thus we can find a constant C for which j Å n j L 0 j+1 A k+1 A l L 0 d n Å 2 n Å 2 dt H 2 2 n+1 Å 2 H 2 n + 2 n Å 2 n n 1 + C j Å n j L 1 n Å + C j+2 Å n j L 1 n Å j=0 + C n M n Å + C j,k,l 0,j+k+l=n j=0 j+1 A k+1 A l L 0 n Å n 1 + C j Å n j L 0 n Å. We now show the desired bound when n = 1. Recall that for some constant C we have see Lemma 3.3 Lemma 3.6 j=0 L 0 + L 1 + A C Å + exp 3t/2 Σ 0 3/2, 2 A C 2 Å + exp 2t Σ 0 2, Σ 0 2 Å 2 C exp 2t.

26 26 Insufficient convergence of inverse mean curvature flow on... In this case, we can find ε > 0 such that d Å 2 dt Å 2 H 2 3 C exp εt Å 2 + C Å 4 + C exp 3 + εt Σ 0 3. Hence, if we set α t := Σ 0 3 Å 2 + exp 3t, then dα t dt α t H 2 3 C exp εtα t + Cαt 2 + C exp 3 + εt for some other constant C. Moreover, from Lemma 3.2 d Lemma 3.3, we can find some positive constant C = CQ j j N, ε 0, δ 0, r 0 r 0, m. so that d Å 2 dt Å 2 H 2 Choose r so that Set + C Σ 0 1 1/2 Å 2 + C Σ Å 2 r 0 r = C Σ 0 1 1/4 ψ t := log α2 t + K Σ Å 2, where the constant K will be chosen later. Note that dψ t dt ψ t H 2 3 C exp ε t + C K/4 Σ 0 3 Å 2 + C exp 3t Σ log α2 t 2 + C exp 3t 4 for some constant C. Choose K such that K > 4C + 4. If p is a maximum of ψ t, then at p log αt 2 2 = K 2 4 Å 2 2 2K 2 Å 2 Å 2 CK 2 Σ 0 2 Å 2. We can now chose r so that for al r 0 r we have Σ 0 3 Å 2 p + log α2 t 2 p 0. 4 The maximum principle implies that for some constant C so for some other constant C. ψ t 3t + C Σ 0 3 Å 2 C exp 3t

27 André Neves 27 For n > 1 we argue by induction. Thus, assume that Σ 0 k+2 k A 2 C exp k + 2t for all k = 1,... n 1 for some constant C. Then, we can find another constant C for which j L j L 1 2 C Σ 0 j 2 exp j + 2t if 1 j n 1, n L n L 1 2 C n Å 2 + C Σ 0 n 2 exp n + 2t, n+1 A 2 C n+1 Å 2 + C Σ 0 n 3 exp n + 3t, n A 2 C n Å 2 + C Σ 0 n 2 exp n + 2t, n M 2 C exp 3t n Å 2 + C Σ 0 n 2 exp n + 3t. Looking at the evolution equation of n Å 2, we see that we can find ε > 0 a constant C such that d n Å 2 dt n Å 2 H 2 n + 2 C exp εt n Å 2 + C Σ 0 n 2 exp n εt the maximum principle implies the desired result. q.e.d. In what follows, C continues to denote any constant with dependence C = CQ j j N, ε 0, δ 0, r 0 r 0, m. One immediate consequence of this lemma is that if we denote by 0 the connection determined by g 0 the round metric on S 2, then for every n 0 n 0 f t C for some constant C. Moreover, dˆr t dt = sinh ˆr t 2 cosh ˆr t thus, combining Lemma 3.3 with Lemma 3.4, we have df t dt = 1 H sinh ˆr t + r, ν 1H 1 2 cosh ˆr t C H 2 + exp 2r 0 2t + r, ν 1 C sup H Å 2 + r, ν 1 + exp 2r 0 Σ 0 exp 2t, for some other constant C. As a result, we get that the functions f t converge to a smooth function f on S 2 so this proves Theorem 3.1 iii. We will now argue that for all integers k 1 n 0 there is a constant C such that Σ 0 n k t n f t 2 C exp nt for n 1, k t f t C exp t,

28 28 Insufficient convergence of inverse mean curvature flow on... Σ 0 n+2 t k n A 2 C exp n + 2t. This estimates finish the proof of the theorem. We start with the case k = 1. Using normal coordinates, we have that t f t, i = t i f t f t, D t i = i r, ν H 1 r, ν H, i H 2 A f t, i H 1 = φ φ r, ν f t, i H 1 r, ν H, i H 2 this implies that Σ 0 t f t 2 C exp t. The same type of computations shows that for every n 1 we can find C such that Σ 0 n t n f t 2 C exp nt This implies that, for each n 1, Σ 0 n+2 t n A 2 C exp n + 2t for some constant C. Having this estimates one can then show that, for each n 1, 2 t f t C exp t Σ 0 n 2 t n f t 2 C exp nt. Repeating this process gives the desired estimates. 4. A modified Shi-Tam flow q.e.d. In this section Σ 0 denotes a sphere satisfying hypothesis H Σ t t 0 is a solution to inverse mean curvature flow for which Theorem 3.1 holds. Consider the manifold N := t 0 Σ t where the metric g m can be written as g m = dt2 H 2 + g t. The metric ḡ is defined to be ḡ := u2 H 2 dt2 + g t, where function u satisfies 12. Lemma 4.1. The metric ḡ has Rḡ = 6.

29 André Neves 29 Proof. The mean curvature the exterior normal vector of Σ t computed with respect to ḡ equal HΣ t = HΣ t /u ν = ν/u respectively. Thus ν H = ν H this implies that Σ t t 0 is indeed a solution to inverse mean curvature flow for the new metric with HΣ 0 = 2. We now check that the scalar curvature of ḡ is 6. According to formula 1.10 of [14], given metrics h 0 := dt 2 + g t h 1 = v 2 dt 2 + g t, the scalar curvature R 0 of h 0 R 1 of h 1 are related by 8 H 0 v t = v2 t v v v3 R t 1 2 ur0 + u3 2 R1, where H 0 denotes the mean curvature of Σ t with respect to h 0. Let g 0 be the metric dt 2 + g t. Because the scalar curvature of g m is 6, we obtain from combining 8 setting v = H 1 both with Gauss equations with dh 1 dt = H 1 H 2 + A 2 + Rcν, ν H 3 that the scalar curvature of g 0 is given by Rg 0 = R t 1 A 2 H 2. Consider the function v := u/h. Using 8 with h 0 = g 0 h 1 = ḡ, the condition that Rḡ = 6 is equivalent to v t = v2 t v v v3 R t 1 2 vrg 0 3v 3. The evolution equation for u follows from the above equation, Gauss equations, the evolution equation for H 1. q.e.d. Using the identification of Σ t with S 2 via Σ t = {ˆr t + f t θ, θ θ S 2 }, the function u t can be identified with a function on S 2 which we still denote by u t. Recall that the normalized metrics ĝ t defined on Lemma 5.1 converge to a smooth metric on S 2. The main purpose of this section is to prove

30 30 Insufficient convergence of inverse mean curvature flow on... Theorem 4.2. Assume that Σ 0 satisfies HΣ 0 > 0 that on Σ t we have R t + 6 2H t H 1 > 0 for all t. Equation 12 admits a smooth solution u with initial condition u Σ0 = HΣ 0 /2 satisfying the following properties. i If we denote by u t the restriction of u to Σ t, then the functions w t := 2 exp3t/2 Σ 0 u t 1/4π converge smoothly to a function w defined on S 2. ii For every integer n k we can find such that Λ = ΛQ j j N, ε 0, δ 0, r 0, r 0, m n w t 2 Λ exp nt k t n w t Λ exp n + 2t; iii The metric ḡ is asymptotically hyperbolic. More precisely, we can find a coordinate system s, θ a symmetric 2-tensor Q such that ḡ = ds 2 + sinh 2 sg 0 + m + Σ 0 /4π 1/2 exp3f w 3 sinh s Q + DQ + D 2 Q + D 3 Q Λ exp 4r for some Λ = ΛQ j j N, ε 0, δ 0, r 0, r 0, m. g 0 + Q Except for property iii, this theorem was essentially proven in [15, Theorem 2.1] when the deformation vector of the foliation Σ t t 0 equals the unit normal vector. In light of Theorem 3.1 the same techniques apply with no modification see also [14]. Nonetheless, we need to make sure that some estimates are independent of r 0 so we sketch its proof. During the proof Λ will denote any constant with dependence Proof. Set Λ = ΛQ j j N, ε 0, δ 0, r 0, r 0, m. Rt + 6 2H t H 1 h + t = sup Σ t 2H 2 h t = h + t if inf u 0 1 or, in case inf u 0 > 1, Moreover, define W + = 1 h t = inf Σ t Rt + 6 2H t H 1 2H sup u 0, W = 1 inf u 0, Σ 0 Σ 0

31 André Neves 31 γ ± t = 1 W ± exp t 0 2h ± sds 1/2. From [14, Lemma 2.2] see also [15, Section 2.2] we have that comparison with the ODE dγ dt = h± tγ γ 3, implies 9 γ t u t γ + t while the solution exists. Moreover, we know from from Theorem 3.1 that h ± t 3/4 Λ exp t w 0 Λ. for some constant Λ. Therefore, the inequalities in 9 imply that, while the solution exists, w t Λ for some other constant Λ. Performing the change of variable s = 4π Σ t 1 = 4π Σ 0 1 exp t, the evolution equation for w t becomes see also [15, Theorem 2.1] 10 dw s ds = u2 H 2 t w s + 2u 2 H 1 ĝ t wt, + w s 4π 1 Σ t H 1 3 t H 1 + uu H R t + 6 2H 2, where the operators t are computed with respect to the normalized metric ĝ t the range os s is 4π Σ 0 1 s < 0. In order to use the stard theory for quasilienar parabolic equations, we need to make some remarks regarding the last term on the right-h side of equation 10. Direct computation shows that Thus the term uu + 1 = π 1 Σ t Σ0 16π s3/2 w s Σ 0 16π s3 w 2 s. 3 t H 1 + uu H R t + 6 2H 2 can be decomposed as 3 Σ 0 164π s2 ws 2 Σ0 9 sws + uu + 1F t, 164π where F t = 4π 1 t H 1 Σ t H + 6H2 4 4R t 8H 2.

32 32 Insufficient convergence of inverse mean curvature flow on... Therefore, we obtain from Theorem 3.1 that the term in 11 is bounded by some constant Λ. Stard theory for quasilinear parabolic equations [11, Section VI, Theorem 6.33] gives a uniform C 0,α -bound in space-time for w s, i.e., for all θ, θ S 2 4π Σ 0 1 s, s < 0 w s θ w s θ distθ, θ 2α + w sθ w s θ s s α Λ for some constant Λ. The term in 11 has a uniform C 0,α -bound so stard Schauder estimates imply that w s 2 w s are uniformly C 0,α -bounded in space-time. Bootstrapping implies the existence of a solution w s for all s with n w s + s n w s Λ for every integer n. Rewriting the equation for w t in terms of the variable t differentiating it with respect to time we obtain that, for every integer n k, n w t 2 Λ exp nt k t n w t 2 Λ exp n + 2t. As a result, w t converges smoothly to a smooth function w defined on S 2. Finally, we show that the metric ḡ satisfies the definition of asymptotic hyperbolicity given in the Introduction. The manifold N defined in the beginning of this section is diffeomorphic to S 2 [0, + thus, besides polar coordinates r, θ, admits also coordinates t, θ where r = f t + ˆr t. In what follows we will use these coordinate systems, Theorem 3.1, the previous estimates for the function u without further mention. Let h := w exp3f Σ 0 1/2 4π 1/2 denote by Q any 2-tensor that satisfies Then Due to the fact that Q + DQ + D 2 Q + D 3 Q = Oexp 4r. ḡ = g m + u2 1 H 2 dt 2 = g m + u 1 dt 2 + Q 2 = g m + 2u 1dr 2 + Q. Σ t = 4π sinh 2 r f t,

33 André Neves 33 we get that ḡ = g m + 2u 1 exp3t/2 Σ 0 3/2 4π 3/2 sinh 3 dr 2 + Q r f t = g m + 8h exp3r dr2 + Q = 1 + 4h exp 3r 2 dr 2 + sinh 2 r + m/3 sinh rg 0 + Q. Thus, if we set we obtain that s := r 4/3h exp 3r, ḡ = ds 2 + sinh 2 s + h + m/3 sinh sg 0 + Q this implies that ḡ is asymptotic hyperbolic if one uses the coordinate system s, θ. q.e.d. 5. Proof of the main theorem We now prove the main theorem. Proof of Theorem 1.2. Consider the ambient manifold to be Anti de Sitter Schwarzschild S 2 [t 0, +, g m with positive mass. Set f to be a smooth function on S 2 with S 2 exp2fdµ 0 = 1 that is invariant under reflection with respect to the three coordinate planes consider Σr 0 = {r 0 + fθ, θ θ S 2 } S 2 [t 0, +. According to Proposition 2.1 e we know that lim m HΣr 0 = m exp fdµ 0 > m r 0 2 S 2 2, where the last inequality is a consequence of Hölder s inequality. Choose r 0 sufficiently large such that Σr 0 satisfies hypothesis H of Section 3 m H Σr 0 > m 2. This is possible because, due to Proposition 2.1, we know that lim H = 2 lim r 0 r 0 Å 2 = 0. Therefore, we can apply Theorem 3.1 conclude the existence of a smooth solution Σ t t 0 to inverse mean curvature flow where, by monotonicity of Hawking mass, m 2 < m HΣr 0 lim m H Σ t. t Denote the induced metric on Σ t by g t. The above inequality implies

34 34 Insufficient convergence of inverse mean curvature flow on... Lemma 5.1. The Gaussian curvature K t of Σ t with respect to the normalized metric ĝ t := 4π Σ t 1 g t does not converge to one when t goes to infinity. Proof. From Theorem 3.1 iii we know that Σ t = {ˆr t + f t θ, θ θ S 2 }, where ˆr t is such that Σ t = 4π sinh 2 ˆr t the functions f t converge to a smooth function f defined on S 2. Moreover, Proposition 2.1 more precisely, identity 1 implies that the metric ĝ t converges to ĝ = exp2f g 0. Note that the ambient metric is preserved by reflections with respect to the coordinate planes thus the metric ĝ also shares these symmetries. Suppose that K t converges to one. Then ĝ is a constant scalar curvature metric which is symmetric under reflection on the coordinate planes so f must be identically zero. If this were true, it would follow from Proposition 2.1 e that this is impossible. lim t m HΣ t = m 2 Outside Σr 0, i.e., on the region q.e.d. N := t 0 Σ t, the metric g m can be written as g m = dt2 H 2 + g t. We want to find a new asymptotically hyperbolic metric ḡ with Rḡ = 6 such that, with respect to this new metric, the mean curvature of Σ 0 is 2, Σ t t 0 is a solution to inverse mean curvature flow, the induced metric on Σ t by ḡ coincides with g t. This would finish the proof for the following two reasons. First, because Σ t t 0 is a smooth solution to inverse mean curvature flow for ḡ, N is foliated by a family of spheres with positive mean curvature thus Σr 0 is outer-minimizing. Second, the intrinsic geometry of Σ t is maintained so we know from Lemma 5.1 that the Gaussian curvature of Σ t with respect to the normalized metric does not converge to one. Proposition 2.1 f implies that no matter the coordinate system we choose we will always have lim inf t s t s t > 0.

35 André Neves 35 The construction of the metric ḡ is inspired by the work of Shi Tam [14]. Consider smooth positive functions u defined on N such that u Σ0 := HΣ 0 /2 12 2H 2 u t = 2u2 t u + 4u 2 H u, H 1 + u u 3 R t + 6 2H t H 1, where the Laplacian gradient term are computed with respect to the metric g t R t is the scalar curvature of Σ t. Having such a function u, the new metric is defined to be ḡ := u2 H 2 dt2 + g t it has scalar curvature 6 by Lemma 4.1. Note that the intrinsic geometry of Σ t is preserved the mean curvature exterior normal vector of Σ t computed with respect to ḡ equal HΣ t = HΣ t /u ν = ν/u respectively. Thus ν H = ν H this implies that Σ t t 0 is indeed a solution to inverse mean curvature flow for the new metric with HΣ 0 = 2. We are only left to check that equation 12 has a solution. Note that, provided we choose r 0 sufficiently large, Proposition 2.1 Theorem 3.1 imply that R t + 6 2H t H 1 > 0 for all t. It is important to remark that this estimate holds because the constants on Theorem 3.1 do not depend on r 0 but only on r 0 r 0. Therefore, Theorem 4.2 implies that equation 12 admits a solution that the metric ḡ is asymptotically hyperbolic. q.e.d. Remark 5.2. To extend our argument to the case Λ = 3, i..e, the case where the initial condition for the flow is a minimal surface, the obvious modification would be to consider a family of metrics having g ε := u2 H 2 dt2 + g t R = 6 u Σ0 := HΣ 0 /ε. In this case, the sphere Σ 0 would have mean curvature H = ε with respect to g ε Σ t 0 would still be a solution to inverse mean curvature with respect to g ε. One would then need to show that the metrics g ε converge when ε goes to zero. We remark that if Σ 0 is a coordinate

36 36 Insufficient convergence of inverse mean curvature flow on... sphere, then each g ε is an Anti De Sitter Schwarschild they indeed converge when ε goes to zero A nontrivial consequence of the Penrose inequality. Assuming that the Penrose inequality holds as conjectured by Xiadong Wang, we will argue that for every smooth function f defined on S 2 with exp2fdµ 0 = 1 S 2 we have K f exp3fdµ 0 K f exp3f x i dµ 0 1, S 2 S 2 i=1 where K f denotes the Gaussian curvature of exp2fg 0. A simple computation shows that an equality is attained if exp2fg 0 has constant scalar curvature. Moreover, if we denote If = K f exp3fdµ 0 K f exp3f x i dµ 0 S 2 S 2 consider T to be a conformal transformation of S 2 such that i=1 T g 0 = exp2ug 0, then If = If T + u. One could check this directly or note, as we shall see next, that If can be obtained as the limit of masses of sequence of metrics so If = If T +u because they are the limit of masses for the same sequence of metrics but written with different coordinate systems. In what follows we use the same notation as in the proof of Theorem 1.2. Set f to be a smooth function on S 2 with S 2 exp2fdµ 0 = 1 consider Σr 0 = {r 0 + fθ, θ θ S 2 } S 2 [t 0, +. Denote by Mr 0 the mass of the metric ḡ constructed in the proof of Theorem 1.2. It is not hard to see that, by choosing r 0 sufficiently large, we can have f defined in Theorem 3.1 iii w defined in Theorem 4.2 i respectively, as close to f w 0 as we want. Moreover, from Proposition 2.1 d, we have that 2w 0 = Σ 0 H 2/4π = KΣr 0 + Oexp r 0, where the Gaussian curvature is computed with respect to the normalized metric ĝr 0 := 4π Σr 0 1 g Σr0. Therefore, denoting the mass two tensor of ḡ by h, we have that 16π 1/2 h Σr0 1/2