AN INVERSE CURVATURE FLOW IN A SPACETIME WITH A FUTURE SINGULARITY

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1 AN INVERSE CURVATURE FLOW IN A SPACETIME WITH A FUTURE SINGULARITY HEIKO KRÖNER Abstract. We consider an inverse curvature flow (ICF) (0.1) ẋ = F 1 ν in a Lorentzian manifold N with a certain future singularity, which is still quite general. Here, F denotes a curvature function of class (K ), which is homogeneous of degree one, e.g. the n-th root of the Gaussian curvature, and ν the past directed normal. We prove longtime existence of the ICF and that the leaves of the ICF provide a foliation of the future of the initial flow hypersurface. Contents 1. Introduction and main result 2 2. Notations and definitions 4 3. Lower order estimates 6 4. Estimates for general spacelike graphs in N 9 5. C 2 -estimates 12 References 17 Date: November 26, Universität Tübingen, Mathematisches Institut, Auf der Morgenstelle 10, Tübingen, Germany address: kroener@na.uni-tuebingen.de This work was supported by the DFG and the Heidelberger Graduiertenakademie. 1

2 2 HEIKO KRÖNER 1. Introduction and main result In [6] it is shown that the inverse mean curvature flow in cosmological spacetimes satisfying some structural assumptions exists for all times and that the leaves of the flow provide a foliation of the future of the initial flow hypersurface. There is a result of the author [11], where it is shown that the structural assumptions in [6] can be weakened. Our aim is to extend the above mentioned result in [6] to a class of inverse curvature flows, for which the normal speed is given by the inverses of fairly general functions of the principal curvatures, i.e. we prove longtime existence for these flows and that the corresponding flow hypersurfaces provide a foliation of the future of the initial flow hypersurface. Concerning the necessary structure of the ambient space N we let us inspire by [7], where the inverse mean curvature flow is considered in Lorentzian manifolds with a special future singularity, so-called ARW spaces, which are defined by several technical assumptions. For a definition of ARW spaces we refer to [10, Chapter 7] and especially concerning their relationship to the Einstein equations to [10, Section 7.9]. For our purposes we assume a spacetime as in Definition 1.1, which differs from ARW spaces mainly in the fact, that we don t need two assumptions concerning the asymptotical behaviour of higher order derivatives of the metric. There is a paper in preparation by the author, in which it is shown that for our inverse curvature flows we get in ARW spaces after rescaling an analogous asymptotical behaviour as in [7] for the rescaled inverse mean curvature flow, especially the rescaled flows define a natural diffeomorphism through the singularity into a mirrored spacetime. The present paper is based on parts of the author s PhD thesis [12] and the author would like to thank Claus Gerhardt under whose supervision the thesis was carried out. Definition 1.1. (Definition of the ambient space N) Let N = N n+1 be a globally hyperbolic spacetime within which a future end N + of N can be written as a product [a, 0) S 0, where S 0 is a compact Riemannian space and there exists a future directed time function τ = x 0 so that the metric in N + can be written as (1.1) (ḡ αβ ) = e 2 ψ{ (dx 0 ) 2 + σ ij (x 0, x)dx i dx j }, where S 0 corresponds to (1.2) x 0 = a. We assume that ψ is of the form (1.3) ψ(x 0, x) = f(x 0 ) + ψ(x 0, x) and that there exists a positive constant c 0 and a smooth Riemannian metric σ ij on S 0 such that for τ 0 (1.4) e ψ(τ,x) c 0 σ ij (τ, x) σ ij (x) f(τ) f (τ). We assume that the derivatives of arbitrary order with respect to space and time of e 2f ğ αβ converge for x 0 0 uniformly to the corresponding derivatives of the following metric (1.5) (dx 0 ) 2 + σ ij (x)dx i dx j.

3 AN INVERSE CURVATURE FLOW IN A SPACETIME WITH A FUTURE SINGULARITY 3 Furthermore there exists γ > 0 such that (1.6) lim f e γf > 0 lim (f + γ f 2 ) exists. τ 0 τ 0 If γ < 1 we assume that (1.7) σ ij has non-negative sectional curvature. Remark 1.2. The limit (1.8) lim τ 0 γf τ 1 τ 2 exists. Proof. See [10, Lemma 7.3.4]. We can now state our main theorem, cf. also Section 2 for notations. Theorem 1.3. Let N be as in Definition 1.1. Let F C (Γ + ) C 0 ( Γ + ) be a curvature function of class (K ), cf. Definition (2.3), in the positive cone Γ + R n, which is in addition positiv homogeneous of degree one and normalized such that (1.9) F (1,..., 1) = n. Let M 0 be a smooth, closed, spacelike hypersurface in N which can be written as a graph over S 0 for which we furthermore assume that it is convex and that it satisfies (1.10) ɛ < inf M 0 x 0 < 0, where (1.11) ɛ = ɛ(n, ğ αβ ) > 0. Then the inverse curvature flow (ICF) given by the equation (1.12) ẋ = 1 F ν with initial surface x(0) = M 0 exists for all times and the flow hypersurfaces M(t) provide a foliation of the future I + (M 0 ) of M 0. Here, ν denotes the past directed normal. The flow hypersurfaces M(t) can be expressed as graphs over S 0 (1.13) M(t) = graph u(t, ), and there exist constants c 1, c 2 > 0 with (1.14) a < c 1 e γ n t u c 2 e γ n t < 0. Moreover there is a constant c > 0 such that (1.15) F ce ( γ+1) n t for all t 0.

4 4 HEIKO KRÖNER 2. Notations and definitions In this section, where we want to introduce some general notations, we let N be as in Definition 1.1; now the special structure of ψ and σ ij, cf. (1.3) et seqq., is not necessarily needed. Let M N be a connected and spacelike hypersurface with differentiable normal ν (which is then timelike). Geometric quantities in N are denoted by (ḡ αβ ), ( Rαβγδ ) etc. and those in M by (g ij ), (R ijkl ) etc.. Greek indices range from 0 to n, Latin indices from 1 to n; summation convention is used. Coordinates in N and M are denoted by (x α ) and ( ξ i) respectively. Covariant derivatives are written as indices, only in case of possibly confusion we precede them by a semicolon, i.e. for a function u the gradient is (u α ) and (u αβ ) the hessian, but for the covariant derivative of the Riemannian curvature tensor we write R αβγδ;ɛ. In local coordinates, (x α ) in N and ( ξ i) in M, the following four important equations hold; the Gauss formular (2.1) x α ij = h ij ν α. In this implicit definition (h ij ) is the second fundamental form of M with respect to ν. Here and in the following a covariant derivative is always a full tensor, i.e. (2.2) x α ij = x α,ij Γ k ijx α k + Γ α βγx β i xγ j and the comma denotes ordinary partial derivatives. The second equation is the Weingarten equation (2.3) ν α i = h k i x α k, where ν α i is a full tensor. The third equation is the Codazzi equation (2.4) h ij;k h ik;j = R αβγδ ν α x β i xγ j xδ k and the fourth is the Gauß equation (2.5) R ijkl = {h ik h jl h il h jk } + R αβγδ x α i x β j xγ k xδ l. As an example for the covariant derivative of a full tensor we give (2.6) Rαβγδ;i = R αβγδ;ɛ x ɛ i, where this identity follows by applying the chain rule from the definition of the covariant derivative of a full tensor; it can be generalized obviously to other quantities. Let (x α ) be a future directed coordinate system in N, then the contravariant vector (ξ α ) = (1, 0,..., 0) is future directed; as well its covariant version (ξ α ) = e 2ψ ( 1, 0,..., 0). Now we want to express normal, metric and second fundamemtal form for spacelike hypersurfaces, which can be written as graphs over the Cauchyhypersurface. Let M = graph u S0 be a spacelike hypersurface in N, i.e. (2.7) M = {( x 0, x ) : x 0 = u(x), x S 0 }, then the induced metric is given by (2.8) g ij = e 2 ψ { u i u j + σ ij },

5 AN INVERSE CURVATURE FLOW IN A SPACETIME WITH A FUTURE SINGULARITY 5 where σ ij is evaluated at (u, x) and the inverse ( g ij) = (g ij ) 1 is given by (2.9) g ij 2 ψ = e {σ ij + ui u j }, where ( σ ij) = (σ ij ) 1 and (2.10) u i = σ ij u j v 2 v 2 = 1 σ ij u i u j 1 Du 2, v > 0. We define ṽ = v 1. From (2.8) we conclude that graph u is spacelike if and only if Du < 1. The covariant version of the normal of a graph is (2.11) (ν α ) = ±v 1 e ψ (1, u i ) and the contravariant version (2.12) (ν α ) = v 1 e ψ ( 1, u i). We have Remark 2.1. Let M be a spacelike graph in a future directed coordinate system, then (2.13) (ν α ) = v 1 e ψ ( 1, u i) is the contravariant future directed normal and (2.14) (ν α ) = v 1 e ψ ( 1, u i) the past directed. In the following we choose ν always as the past directed normal. Let us consider the component α = 0 in (2.1), so we have due to (2.14) that (2.15) e ψv 1 h ij = u ij Γ 0 00u i u j Γ 0 0ju i Γ 0 0iu j Γ 0 ij, where u ij are covariant derivatives with respect to M. Choosing u const, we deduce (2.16) e ψ hij = Γ 0 ij, where h ij is the second fundamental form of the hypersurface { x 0 = const }. An easy calculation shows (2.17) e ψ hij = 1 2 σ ij ψσ ij, where the dot indicates differentiation with respect to x 0. Now we define the classes (K) and (K ), which are special classes of curvature functions; for a more detailed treatment of these classes we refer to [10, Section 2.2]. For a curvature function F (i.e. symmetric in its variables) in the positive cone Γ + R n we define (2.18) F (h ij ) = F (κ i ), where the κ i are the eigenvalues of an arbitrary symmetric tensor (h ij ), whose eigenvalues are in Γ +.

6 6 HEIKO KRÖNER Definition 2.2. A symmetric curvature function F C 2,α (Γ + ) C 0 ( Γ + ), positively homogeneous of degree d 0 > 0, is said to be of class (K), if (2.19) F i = F κ i > 0 in Γ +, (2.20) F Γ+ = 0, and (2.21) F ij,kl η ij η kl F 1 (F ij η ij ) 2 F ik hjl η ij η kl η S, where F is evaluated at an arbitrary symmetric tensor (h ij ), whose eigenvalues are in Γ + and S denotes the set of symmetric tensors. Here, F i is a partial derivative of first order with respect to κ i and F ij,kl are second partial derivatives with respect to (h ij ). Furthermore ( h ij ) is the inverse of (h ij ). In Theorem 1.3 the κ i in (2.18) are the eigenvalues of the second fundamental form (h ij ) with respect to the metric (g ij ), i.e. the principal curvatures of the flow hypersurfaces. Definition 2.3. A curvature function F (K) is said to be of class (K ), if there exists 0 < ɛ 0 = ɛ 0 (F ) such that (2.22) ɛ 0 F H F ij h ik h k j, for any symmetric (h ij ) with all eigenvalues in Γ +, where F is evaluated at (h ij ). H represents the mean curvature, i.e. the trace of (h ij ). In the following a + sign attached to the symbol of a metric of the ambient space refers to the corresponding Riemannian background metric, if attached to an induced metric, it refers to the induced metric relative to the corresponding Riemannian background metric. Let us consider as an example the metrics ğ αβ and g ij introduced as above, then (2.23) +ḡαβ = e 2 ψ{(dx 0 ) 2 + σ ij (x 0, x)dx i dx j }, + gij = + ḡ αβ x α i x β j. 3. Lower order estimates Let M τ = {x 0 = τ} denote the coordinate slices. Then (3.1) M τ = det σ ij (τ, x) dx 0, τ 0. S 0 e n ψ(τ,x) And for the second fundamental form h ij of the M τ we have (3.2) hi j = e ψ( 1 2 σik σ kj + ψδ i j ), hence there exists τ 0 such that M τ is convex for all τ τ 0. From Lemma 4.2 (iii) and the homogeneity of F we conclude that for m N there is a < τ 0 < 0 and c m > 0 such that for all τ 0 τ < 0 we have (3.3) ϕ 1 (τ) := nf + c m τ m e ψf Mτ = e ψf ( h i j) = F ( 1 2 σik σ kj ψδ i j ) nf c m τ m =: ϕ 2 (τ). We will show that the flow does not run into the future singularity within finite time.

7 AN INVERSE CURVATURE FLOW IN A SPACETIME WITH A FUTURE SINGULARITY 7 Lemma 3.1. There exists a time function x 0 = x 0 (x 0 ), so that the F -curvature F of the slices { x 0 = const} satisfies (3.4) e ψ F 1. e ψ is the conformal factor in the representation of the metric with respect to the coordinates ( x 0, x i ), i.e. (3.5) d s = e 2 ψ{ (d x 0 ) 2 + σ ij ( x 0, x)dx i dx j }. Furthermore there holds (3.6) x 0 ({τ 0 x 0 < 0}) = [0, ) and the future singularity corresponds to x 0 =. Proof. Define x 0 by (3.7) τ x 0 = ϕ 2 (s)ds = nf(τ) c m τ 0 m + 1 τ m+1 +nf(τ 0 )+ m + 1 τ 0 m+1, τ 0, where ϕ 2 is chosen as in (3.3). For the conformal factor in (3.5) we have (3.8) e 2 ψ 2 ψ x0 x 0 = e x 0 x and therefore = ψϕ 2 e2 0 2 (3.9) e ψ F = e ψ F ϕ The evolution problem (1.12) is a parabolic problem, hence a solution of the ICF exists on a maximal time interval [0, T ), 0 < T. Lemma 3.2. We assume the situation of Theorem 1.3, 0 < T T finite and that for 0 t T the flow hyperfurfaces M(t) of the ICF can be expressed as a graph of a function u over S 0 (3.10) M(t) = {(x 0, x) : x 0 = u(t, x), x S 0 }. Then the flow stays in a precompact set Ω T for 0 t < T. Proof. We choose for x 0 the special time function in Lemma 3.1 with (3.11) e ψ F 1 for the coordinate slices {x 0 = const}. Let (3.12) M(t) = graph u(t, ) be the flow hypersurfaces in this coordinate system and (3.13) ϕ(t) = sup S 0 u(t, ) = u(t, x t ) with suitable x t S 0. It is well-known that ϕ is Lipschitz continuous and that for a.e. 0 t < T (3.14) ϕ(t) = t u(t, x t). From (2.15) we deduce in x t the relation (3.15) h ij h ij, c m

8 8 HEIKO KRÖNER hence (3.16) F F. We look at the component α = 0 in (1.12) and get (3.17) u = ṽ F e ψ, where (3.18) u = u t + u iẋ i is a total derivative. This yields (3.19) so that we have in x t (3.20) With (3.14) we conclude u t = e ψv 1 F, u t = 1 e ψf 1 e ψ 1. F (3.21) ϕ ϕ(0) + t 0 t < T, which proves the lemma, since the future singularity corresponds to x 0 =. Lemma 3.3. We assume in the situation of Theorem 1.3 that the flow exists for all times, i.e. T =, and that the flow hypersurfaces can be written as graphs over S 0 then the flow runs into the future singularity, i.e. (3.22) s [a,0) t0>0 t t0 inf M(t) x0 s. Proof. Similar arguments as in the proofs of Lemmata 3.1 and 3.2 using the functions ϕ 1 and (3.23) ϕ(t) = inf S 0 u(t, ) prove the lemma. We have actually a precise rate of convergence with respect to time. Corollary 3.4. In the situation of Lemma 3.3, especially the time function x 0 is assumed as in Theorem 1.3, there are c 1, c 2 > 0 such that (3.24) c 1 e γ n t x 0 M(t) c 2 e γ n t for all t 0. Proof. A careful view on the proofs of the previous lemmas shows that there are c 1, c 2 > 0 such that for large t (3.25) f 1 ( c 1 t n ) x0 M(t) f 1 ( c 2 t n ). Inserting in the monotone decreasing function e γf yields (3.26) e γ( c1 t n ) e γf(x0) M(t) e γ( c2 t n ). The claim follows in view of (1.8) and (1.6).

9 AN INVERSE CURVATURE FLOW IN A SPACETIME WITH A FUTURE SINGULARITY 9 4. Estimates for general spacelike graphs in N Lemma 4.1. Let N be as in Theorem 1.3. Let Ω N be precompact and M Ω a convex spacelike graph over S 0, then the quantity ṽ is bounded by a constant, which only depends on Ω. Proof. We use the convexity of M together with [10, Theorem ]. In the following we generalize Lemma 4.1 by omitting the assumption that M Ω, Ω precompact. We will get a bound for ṽ, which is related to the data of the ambient space. We introduce a special convention for our notation, which is only used in this section. The metric of N, which is defined in (1.1), will be denoted by ğ αβ, deduced quantities like the induced metric of hypersurfaces, the second fundamental form, a unit normal or the curvature tensor by ğ ij, h ij, ν, R αβγδ. We can equip N with the conformal metric (4.1) ḡ αβ = e 2 ψğ αβ = (dx 0 ) 2 + σ ij (x 0, x)dx i dx j, and deduced quantities like the induced metric of hypersurfaces, the second fundamental form, a unit normal or the curvature tensor will be denoted by g ij, h ij, ν, R αβγδ. The second fundamental form with respect to ḡ αβ of a coordinate slice {x 0 = const} is denoted by h ij. The second fundamental forms h j i and hj i are related by (4.2) e ψ hj i = hj i + ψ α ν α δ j i = hj i ṽf δ j i + ψ αν α = def ȟ j i, cf. [10, Proposition ]. When we insert h j i into F we will denote the result in accordance with our convention as F. Due to a lack of convexity it would not make any sense to insert h j i into the curvature function F, so that we stipulate that the symbol F will stand for (4.3) F = e ψ F = F (h j i ṽf δ j i + ψ αν α ). We prove a decay property of certain tensors. Lemma 4.2. In the following all covariant derivatives are taken with respect to the metric ḡ αβ and will denote the norm with respect to (4.4) +ḡαβ = (dx 0 ) 2 + σ ij (x 0, x)dx i dx j. (i) Let ϕ C ([a, 0)), a < 0, and assume (4.5) lim τ 0 ϕ (k) (τ) = 0 k N, then for every k N there exists a c k > 0 such that (4.6) ϕ(τ) c k τ k. (ii) Let T be a tensor such that for all k N (4.7) D k T (x 0, x) 0 as x 0 0 uniformly in x then (4.8) k N ck >0 x S0 T (x 0, x) c k x 0 k.

10 10 HEIKO KRÖNER (iii) Let (η α ) = ( 1, 0,..., 0) be a covariant vector field, then for its first covariant derivative T = (η αβ ) the relation (4.8) holds, analogously for η αβγ, Dψ, R αβγδ η α, or more generally for any tensor that would vanish identically, if it would have been formed with respect to the product metric (4.9) (dx 0 ) 2 + σ ij dx i dx j. Proof. See [10, Lemma 7.4.4]. Lemma 4.3. Let N be as in Theorem 1.3. Let ɛ > 0 be arbitrary, then there exists δ = δ((n, ğ αβ ), ɛ) > 0 such that for every closed, spacelike, convex hypersurface M in the end N + δ = {x0 > δ} holds (4.10) ṽ ɛ f 1 γ. Proof. Let p > γ 1 and define (4.11) w = ṽ{e f + u p } and look at a point, where w attains its maximum, and infer (4.12) where 0 = w i = ṽ i {e f + u p } + ṽ{e f f p u p 1 }u i = { h ik u k + ṽ 1 hik u k }{e f + u p } + ṽ{e f f p u p 1 }u i = { h ik u k e ψ ṽf u i + ɛṽu i }{e f + u p } + ṽ{e f f p u p 1 }u i, (4.13) ɛ c m u m m N. Multiplying by u i and assuming Du 0 we get the inequality (4.14) if δ > 0 small, since 0 ( f + ɛ){e f + u p } + e f f p u p 1 = f u p + ɛ{e f + u p } p u p 1 < 0, (4.15) f u γ 1 + cu 2, c.f. (1.8). This is a contradiction, hence Du = 0. Since (4.16) ϕ(τ) = e f(τ) + τ p, a τ < 0, is monotone decreasing we conclude (4.17) ṽ ef(umin) + u min p e f(u) + u p (e f(umin) + u min p )e f(u), where u min = inf u. Choosing δ appropiately small finishes the proof. Corollary 4.4. Let δ > 0 be small and N + δ and M be as in Lemma 4.3, then (4.18) F ij Rαβγδ ν α x β i νγ x δ j cδf ij g ij, if the limit metric σ ij has non-negative sectional curvature.

11 AN INVERSE CURVATURE FLOW IN A SPACETIME WITH A FUTURE SINGULARITY 11 Proof. We define (4.19) Rαβγδ (0, ) = lim τ 0 Rαβγδ (τ, ) and have (4.20) F ij Rαβγδ ν α x β i νγ x δ j =F ij ( R αβγδ (0, ) + R αβγδ (u, ) R αβγδ (0, ))ν α x β i νγ x δ j F ij ( R αβγδ (u, ) R αβγδ (0, ))ν α x β i νγ x δ j F ij ν α x β i νγ x δ j R αβγδ (u, ) R αβγδ (0, ) c m u m F ij g ij, for arbitrary m N and suitable c m. Note that we used for the last inequality that (4.21) Rαβγδ (x 0, ) R αβγδ (0, ) satisfies (4.7). We want to formulate the relation of the curvature tensors for conformal metrics. Lemma 4.5. The curvature tensors of the metrics ğ αβ, ḡ αβ are related by (4.22) e 2 ψ Rαβγδ = R αβγδ ḡ αγ ψβδ ḡ βδ ψαγ + ḡ αδ ψβγ + ḡ βγ ψαδ + ḡ αγ ψβ ψδ + ḡ βδ ψα ψγ ḡ αδ ψβ ψγ ḡ βγ ψα ψδ + {ḡ αδ ḡ βγ ḡ αγ ḡ βδ } D ψ 2, where covariant derivatives and are taken with respect to ḡ αβ. Proof. Straightforward calculation. Now we are able to prove the following lemma which is necessary for the C 2 - estimates in the next section. Lemma 4.6. Let c > 0, then there is δ = δ(n, ğ αβ ) > 0 such that for N + δ as in Lemma 4.3 holds ij (4.23) F Rαβγδ ν α x β i νγ x δ j c f 2 2 e ψ. Here F ij is evaluated at h j i. Proof. In view of the homogeneity of F we have (4.24) F i j = F i j, hence (4.25) F ij = e 2 ψ F ij. We have due to Lemma 4.5 (4.26) We have (4.27) e 2 ψ F ij Rαβγδ ν α x β i νγ x δ j =F ij Rαβγδ ν α x β i νγ x δ j + F ij x β i xδ j ψ βδ F ij g ij ψαγ ν α ν γ F ij x β i xδ j ψ β ψδ + F ij g ij ψα ψγ ν α ν γ + F ij g ij D ψ 2. + gij 2σ ij 2ṽ 2 g ij. and M

12 12 HEIKO KRÖNER Now we estimate each summand in (4.26) separately with the help of the Riemmanian background metric + ḡ αβ, namely (4.28) F ij Rαβγδ ν α x β i νγ x δ j cṽ 2 (F ij F ī j + g iī + gj j) 1 2 cṽ 2 F ij σ ij cṽ 4 F ij g ij, (4.29) F ij x β i xδ j ψ βδ = F ij u i u j f + F ij x β i xδ jψ βδ F ij u i u j f cṽ 2 F ij g ij, (4.30) F ij g ij ψαγ ν α ν γ = ṽ 2 F ij g ij f F ij g ij ψ αγ ν α ν γ ṽ 2 F ij g ij f cṽ 2 F ij g ij, F ij x β i xδ ψ j β ψδ = F ij u i u j (ψ 0 + f ) 2 F ij ψ i ψ j 2F ij u j ψ i (ψ 0 + f ) F ij u i u j (ψ 0 + f ) 2 c(1 + f Du )F ij σ ij Dψ (4.31) F ij u i u j (ψ 0 + f ) 2 cṽ 2 (1 + f Du )F ij g ij Dψ F ij u i u j (ψ 0 + f ) 2 c f ṽ 2 F ij g ij, where Dψ 2 = σ ij ψ i ψ j, (4.32) F ij g ij ψα ψγ ν α ν γ ṽ 2 (ψ 0 + f ) 2 F ij g ij cṽ 2 f F ij g ij, (4.33) F ij g ij D ψ 2 = (f + ψ 0 ) 2 F ij g ij + σ ij ψ i ψ j F ij g ij (f + ψ 0 ) 2 F ij g ij cf ij g ij. Thus we conclude (using u i u j (ṽ 2 1)g ij ) (4.34) e 2 ψ F ij Rαβγδ ν α x β i νγ x δ j cṽ 4 F ij g ij + F ij u i u j f ṽ 2 f F ij g ij cṽ 2 f F ij g ij + (ψ 0 + f ) 2 F ij (ṽ 2 g ij u i u j g ij ) cṽ 4 F ij g ij ṽ 2 f F ij g ij c f ṽ 2 F ij g ij. Now, the claim follows with Lemma 4.3 in case γ 1. Let us now consider the case γ < 1. Due to assumption the limit metric σ ij has non-negative sectional curvature. Now we use Corollary 4.4 to bound the first summand of the right side of (4.26) from below by the term cf ij g ij, one easily checks that this term replaces the summand with ṽ 4 in (4.34) completing the proof. 5. C 2 -estimates In this section we consider N to be equipped only with the metric in (1.1) and apply standard notation to this case, i.e. the notation conventions in Section 2 are used. Lemma 5.1. The following evolution equation holds ( ) d 1 (5.1) 1 ( ) 1 dt F F 2 F ij = 1 F F 3 F ij h ik h k j 1 F 3 F ij Rαβγδ ν α x β i νγ x δ j. Proof. cf. [10, Lemma 2.3.4]. ij

13 AN INVERSE CURVATURE FLOW IN A SPACETIME WITH A FUTURE SINGULARITY 13 Lemma 5.2. In the situation of Theorem 1.3 we assume that the flow exists on a maximal time interval [0, T ), 0 < T, and that the flow hypersurfaces can be written as graphs over S 0. Furthermore let ɛ in (1.11) be so, that (4.23) holds for all flow hypersurfaces. Then (5.2) F inf M 0 F for 0 t < T. If T = we have for a constant c > 0 (5.3) F ce ( γ+1) n t for all times. Proof. We define (5.4) ϕ(t) = inf M(t) F and infer from Lemma 5.1 d (5.5) dt F F 2 F ij F ij = 1 F F ij h ik h k j + 1 F F ij Rαβγδ ν α x β i νγ x δ j 2 F 3 F ij F i F j, hence using Lemma 4.6, (2.22) and the well-known relation F H, cf. [10, Lemma ], we deduce (5.6) ϕ(t) c f 2 ϕ e 2f + ɛ 0 ϕ, especially ϕ(t) 0 for a.e. 0 < t < T. If T = we use (3.24), (1.8) and (1.6) to conclude from (5.6) that there are positive constants c 3, c 4 such that 2( γ+1) c4+ (5.7) ϕ(t) c 3 e n t ϕ 1 + ɛ 0 ϕ, hence (5.8) ϕ c 5 e ( γ+1) n t, c 5 > 0. Remark 5.3. Due to [10, Lemma 1.8.3], and the remark at the beginning of Section 3, especially inequality (3.2), for every relative compact subset Ω of N lying sufficiently far in the future of N, i.e. inf Ω x 0 close to 0, there exists a strictly convex function χ C 2 ( Ω), this means (5.9) χ αβ c 0 ḡ αβ with a constant c 0 > 0. Lemma 5.4. The following evolution equation holds (5.10) χ 1 F 2 F ij χ ij = 2 F χ αν α 1 F 2 F ij χ αβ x α i x β j Proof. Direct calculation. Lemma 5.5. The following evolution equation holds (5.11) (log F ) 1 F 2 F ij (log F ) ij = 1 F 2 F ij h ik h k j + 1 F 2 F ij Rαβγδ ν α x β i νγ x δ j 1 F 4 F ij F i F j

14 14 HEIKO KRÖNER Proof. Use Lemma 5.1. Lemma 5.6. The following evolution equation holds (5.12) ṽ 1 F 2 F ij ṽ ij = 1 F 2 F ij h ik h k j ṽ 2 F η αβν α ν β 2 F 2 F ij h k j x α i x β k η αβ where (η α ) = e ψ( 1, 0,..., 0). Proof. cf. [10, Lemma 2.4.4]. 1 F 2 F ij η αβγ x β i xγ j να 1 F 2 F ij Rαβγδ ν α x β i xγ k xδ jη ɛ x ɛ l g kl, Lemma 5.7. Let Ω N be precompact and assume that the flow stays in Ω for 0 t T < T, then the F -curvature of the flow hypersurfaces is bounded from above, (5.13) 0 < F < c(ω). Proof. Consider the function (5.14) w = log F + λṽ + µχ, where λ, µ > 0 will be chosen later appropiately. Assume (5.15) w(t 0, x 0 ) = sup sup w with 0 < t 0 T, then we have in (t 0, x 0 ) (5.16) [0,T ] M(t) 0 ẇ 1 F 2 F ij w ij = 1 F 2 F ij h ik h k j + 1 F 2 F ij Rαβγδ ν α x β i νγ x δ j 1 F 4 F ij F i F j λ F 2 F ij h ik h k j ṽ 2λ F η αβν α ν β 2λ F 2 F ij h k j x α i x β k η αβ λ F 2 F ij η αβγ x β i xγ j να λ F 2 F ij Rαβγδ ν α x β i xγ k xδ jη ɛ x ɛ l g kl 2µ F χ αν α µ F 2 F ij χ αβ x α i x β j ɛ 0 ( λ cλ 1)ṽ + 2 F 2 F ij g ij + c(µ + λ) 1 F c µ 0 F 2 F ij g ij. Now we choose λ > 2 arbitrary and µ >> 1 large and we deduce that F is a priori bounded from above in (t 0, x 0 ) from which we conclude the Lemma. Let Ω N be precompact and assume that the flow stays in Ω for 0 t T < T, then there exist as we have just proved constants 0 < c 1 (Ω) < c 2 (Ω) such that (5.17) c 1 (Ω) < F < c 2 (Ω). It remains to prove that there also holds an estimate for the principal curvatures from above (5.18) κ i c 3 (Ω), yielding (5.19) 0 < c 4 (Ω) κ i c 3 (Ω) due to the convexity of the flow hypersurfaces and (5.17).

15 AN INVERSE CURVATURE FLOW IN A SPACETIME WITH A FUTURE SINGULARITY 15 Lemma 5.8. The mixed tensor h j i (5.20) satisfies the parabolic equation ḣ j i 1 F 2 F kl h j i;kl = F 2 F kl h rk h r l h j i + 1 F h rih rj + 1 F hk i h j k Proof. cf. [10, Lemma 2.4.1]. + 1 F 2 F kl,rs h kl;i h ;j rs 2 F 3 F if j + 2 F 2 F kl Rαβγδ x α mx β i xγ k xδ rh m l g rj 1 F 2 F kl Rαβγδ x α mx β k xγ r x δ l h m i g rj 1 F 2 F kl Rαβγδ x α mx β k xγ i xδ l h mj 1 F 2 F kl Rαβγδ ν α x β k νγ x δ l h j i + 2 F R αβγδ ν α x β i νγ x δ mg mj + 1 F 2 F kl Rαβγδ;ɛ { ν α x β k xγ l xδ i x ɛ mg mj + ν α x β i xγ k xδ mx ɛ l g mj}. Lemma 5.9. Let Ω N be precompact and assume that the flow stays in Ω for 0 t < T, then there exists c 3 (Ω) such that (5.21) κ i c 3 (Ω). Proof. The proof is more or less word by word as in [4, Lemma 5.1], for reasons of completeness we present it here. Let ϕ and w be defined respectively by (5.22) ϕ = sup{h ij η i η j : η = 1}, w = log ϕ + λṽ + µχ, where λ, µ are large positive parameters to be specified later. We claim that w is bounded for a suitable choice of λ, µ. Let 0 < T < T, and x 0 = x 0 (t 0 ), with 0 < t 0 T, be a point in M(t 0 ) such that (5.23) sup w < sup{ sup w : 0 < t T } = w(x 0 ). M 0 M(t) We then introduce a Riemannian normal coordinate system (ξ i ) at x 0 M(t 0 ) such that at x 0 = x(t 0, ξ 0 ) we have (5.24) g ij = δ ij and ϕ = h n n. Let η = ( η i ) be the contravariant vector field defined by (5.25) η = (0,..., 0, 1), and set (5.26) ϕ = h ij η i η j g ij η i η j. ϕ is well defined in a neighbourhood of (t 0, ξ 0 ). Now, define w by replacing ϕ by ϕ in (5.22); then w assumes its maximum at (t 0, ξ 0 ). Moreover, at (t 0, ξ 0 ) we have (5.27) ϕ = ḣn n, and the spatial derivatives do also coincide; in short, at (t 0, ξ 0 ) ϕ satisfies the same differential equation (5.20) as h n n. For the sake of greater clarity, let us therefore treat h n n like a scalar and pretend that w is defined by (5.28) w = log h n n + λṽ + µχ.

16 16 HEIKO KRÖNER W.l.o.g. we assume that µ, λ and h n n are larger than 1. At (t 0, ξ 0 ) we have ẇ 0 and in view of the maximum principle, we deduce from (5.20), (5.12), (5.10) and (5.17) (5.29) 0 ch n n + cλf ij g ij λ 2 ɛ 0ṽ H F + µc c µ 0 F 2 F ij g ij + 1 F 2 F ij (log h n n) i (log h n n) j 2 h n nf 3 F n F n + 1 h n nf 2 F kl,rs h kl;n h rs;i g ni. Because of [10, Lemma 2.2.6] we have (5.30) F kl,rs h kl;n h rs;n F 1 (F ij h ij;n ) 2 1 h n F ij h in;n h jn;n so that we can estimate the last two summands of (5.29) from above by (5.31) 1 1 (h n n) 2 F 2 F ij (h n n;i + R i )(h n n;j + R j ); here (5.32) Ri = R αβγδ ν α x β nx γ i xδ n = h in;n h nn;i denotes the correction term which comes from the Codazzi equation when changing the indices from h in;n to h nn;i. Thus the terms in (5.29) containing derivatives of h n n are estimated from above by 1 (5.33) 2 (h n n) 2 F 2 F ij h n R 1 n;i j = 2 h n nf 2 F ij (log h n n) i Rj. Moreover Dw vanishes at ξ 0, i.e., (5.34) (log h n n) i = λṽ i µχ i Hence we conclude from (5.29) that = λη αβ x β i να λη α x α k h k i µχ α x α i. 0 ch n n + cλf ij g ij λ 2 ɛ 0ṽ H F + µc + µ c h (5.35) n F ij µ g ij c 0 F 2 F ij g ij c 1 h n n + c 2 λf ij g ij λc 3 h n n + µc 4 + µ c 5 h n F ij g ij c 0 µf ij g ij, where c i, i = 0,..., 5, are positive constants and the value of c 0 changed. We note that we used the estimate (5.36) F ij Rj η α x α k h k i cf, which can be immediately proved. Now suppose h n n to be so large that (5.37) c 5 h n < 1 2 c 0, and choose λ, µ such that λ (5.38) 2 c 1 3 > c 1 and 4 c 0µ > c 2 λ yielding that estimating the right side of (5.35) yields (5.39) 0 λ 2 c 3h n n c 0 4 µf ij g ij + µc 4,

17 AN INVERSE CURVATURE FLOW IN A SPACETIME WITH A FUTURE SINGULARITY 17 hence h n n is a priori bounded at (t 0, ξ 0 ). Remark Now all necessary a priori estimates are proved so that we can deduce existence of the flow for all times. This is standard theory and can be found more detailed in [10, Section 6.6]. Furthermore the flow provides a foliation of the future of M 0 as can be seen literally as in [10, Section 6.7]. References [1] Bernal, A. N. und Sánchez, M.: On Smooth Cauchy Hypersurfaces and Geroch s Splitting Theorem, Commun. Math. Phys., 243 (2003), [2] Bernal, A. N. und Sánchez, M.: Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes, Commun. Math. Phys., 257 (2005), [3] Gerhardt, C.: Closed Weingarten hypersurfaces in Riemannian manifolds, J. Diff. Geom. 43 (1996), [4] Gerhardt, C.: Hypersurfaces of prescribed curvature in Lorentzian manifolds, Indiana Univ. Math. J. 49 (2000), , arxiv:math.dg/ [5] Gerhardt, C.: Hypersurfaces of prescribed mean curvature in Lorentzian manifolds, Math. Z. 235 (2000), [6] Gerhardt, C.: The inverse mean curvature flow in cosmological spacetimes, Adv. Theor. Math. Phys. 12, (2008), arxiv:math.dg/ [7] Gerhardt, C.: The inverse mean curvature flow in ARW spaces - transition from big crunch to big bang, 2004, arxiv:math.dg/ , 39 pages. [8] Gerhardt, C.: Analysis II, International Series in Analysis, International Press, Somerville, MA, 2006, 395 pages. [9] Gerhardt, C.: On the CMC foliation of future ends of spacetime, Pacific J. Math. 226 (2006), [10] Gerhardt, C.: Curvature problems, Series in Geometry and Topology, International Press, Somerville, MA, 2006, 323 pages. [11] Kröner, H.: A note on inverse mean curvature flow in cosmological spacetimes, 2012, arxiv: [12] Kröner, H.: The inverse F -curvature flow in ARW spaces, PhD thesis (2012), Heidelberg, (preprint version on arxiv: ). [13] Krylov, N. V.: Nonlinear elliptic and parabolic equations of the second order, Mathematics and its Applications (Soviet Series), vol. 7, D. Reidel Publishing Co., Dordrecht, [14] Li, G., Salavessa, M. C.: Power Mean Curvature Flow in Lorentzian Manifolds, 2006, arxiv:math/ v1. [15] O Neill: Semi-Riemannian geometry. With applications to relativity., Pure and Applied Mathematics, 103. New York-London etc.: Academic Press. XIII, 1983.