ENTIRE SPACELIKE HYPERSURFACES OF CONSTANT GAUSS CURVATURE IN MINKOWSKI SPACE. 1. Introduction.. (1 Du 2 ) n+2

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1 ENTIRE SPACELIKE HYPERSURFACES OF CONSTANT GAUSS CURVATURE IN MINKOWSKI SPACE PIERRE BAYARD AND OLIVER C. SCHNÜRER Abstract. We prove existence an stability of smooth entire strictly convex spacelike hypersurfaces of prescribe Gauß curvature in Minkowski space. The proof is base on barrier constructions an local a priori estimates. 1. Introuction In Minkowski space L n+1, the Gauß curvature of graph u, u : R n R, is given by et D u K[u] =. 1 Du n+ We consier that equation for strictly convex strictly spacelike functions u see Section for efinitions. The Gauß map sens every point of the hypersurface graph u to its future irecte unit normal which is a point in hyperbolic space { x L n+1 : x, x = 1, x n+1 > 0 }. In this paper we stuy convex spacelike hypersurfaces of constant Gauß curvature with prescribe Gauß map image. We solve the corresponing fully nonlinear elliptic partial ifferential equation on a sequence of growing balls an pass to a limit. In aition, we stuy logarithmic Gauß curvature flow for convex spacelike hypersurfaces with given Gauß map image. These solutions converge to solutions of the equation of constant Gauß curvature. Thus the solutions to the elliptic equation are ynamically stable. The construction of hypersurfaces of prescribe Gauß curvature an of solutions to logarithmic Gauß curvature flow uses barriers. For these barriers, we have the following existence result. For etails, we refer to Theorem 4.3. Theorem 1.1. There exist spacelike viscosity sub- an supersolutions u an u to the equation of prescribe constant positive Gauß curvature which are at infinity close to V F, where V F x := sup x λ an F is the closure of λ F Date: December 006. Key wors an phrases. Gauß curvature, Minkowski space, stability, regularity, spacelike, Gauß map image. 1

2 PIERRE BAYARD AND OLIVER C. SCHNÜRER some open non-empty subset of the ieal bounary S n 1 of hyperbolic space with F C 1,1. Moreover, the subsolution u is convex. Our main results are the existence results for solutions to the equation of prescribe Gauß curvature an for solutions to logarithmic Gauß curvature flow. The result for the equation of prescribe Gauß curvature is Theorem 1.. Let u u be barriers as above with K[u] > 1 > K[u] in the viscosity sense, close to V F at infinity. Then there exists a unique smooth strictly convex strictly spacelike function u : R n R with u u u which solves the equation of prescribe Gauß curvature one K[u] = et D u 1 Du n+ = 1 in R n, 1.1 i.e. graph u L n+1 is a strictly convex strictly spacelike hypersurface of Gauß curvature one. Moreover, the image of the Gauß map is the hyperbolic space convex hull of F. For logarithmic Gauß curvature flow, we have Theorem 1.3. Let u an u be barriers as in Theorem 1.. Let u 0 : R n R be a smooth strictly convex strictly spacelike function with u u 0 u such that log K[u 0 ] is uniformly boune. Then there exists a unique strictly convex strictly spacelike function u : C R n 0, C 0 R n [0, solving u = 1 Du log K[u] in R n 0,, u, 0 = u 0 in R n, 1. u u, t u in R n for all t 0 such that log K[u, t] is uniformly boune for all t 0. Moreover, as t, the functions u, t converge exponentially to the solution in Theorem 1.. The barriers for Theorem 1.1/4.3 are obtaine as follows. We apply a Lorentz transformation to a semitrough, a hypersurface of constant Gauß curvature, which has Gauß map image equal to half the hyperbolic space. Then we take suprema an infima over such semitroughs an obtain barriers u an u. It is essential for the following to control the behavior of these barriers near infinity uring this construction. In orer to prove Theorem 1. an Theorem 1.3, we solve 1.1 an an equation similar to 1. between the barriers u an u on balls with aitional Dirichlet bounary conitions impose. We o such on a sequence of growing balls. For these auxiliary solutions, we prove a priori estimates for the first an secon erivatives which are local in space an uniform in time. For proving C 1 -estimates it is necessary to have barriers which are close to

3 HYPERSURFACES OF CONSTANT GAUSS CURVATURE IN MINKOWSKI SPACE 3 each other at spatial infinity. Then C -estimates require only control of the C 0 -behavior of u. These a priori estimates allow to extract subsequences converging to the esire solutions. Many of our techniques exten to Eucliean space an to the situation where f = fx, ν, t. Theorem 1.3 shows that the solutions foun in Theorem 1. are ynamically stable. In the flow equation, the logarithm is useful to preserve convexity. The flow equation u = log K is a less geometric alternative for which analogous results can be prove by methos similar to the ones use here. In our case, the stability issue follows irectly from the evolution equation of the normal velocity. In orer to guarantee convergence to the elliptic solution, we have to impose that solutions are C 0 -close at infinity to the elliptic solution ũ. Otherwise, u might converge to ũ + c instea. Thus it is not too restrictive to start between these barriers. Of course, the barriers are also crucial for proving local C 1 a priori estimates. Note that 1. escribes hypersurfaces moving with normal velocity equal to the logarithm of the Gauß curvature K, tx = log Kν. By scaling graph u, we can use Theorem 1. to fin convex spacelike hypersurfaces of constant Gauß curvature f 0 > 0. Replacing the flow equation 1. in Theorem 1.3 by u = 1 Du log K[u] log f Du et D u log log f 1 Du n+ 0 an using barriers with K[u] > f 0 > K[u], lim u, t converges to the t hypersurface of Gauß curvature f 0 mentione above. Let us quote some results concerning hypersurfaces of prescribe mean an Gauß curvature in Minkowski space. In [0], Anrejs Treibergs classifies all the entire spacelike hypersurfaces of constant mean curvature in L n+1 by their bounary values at infinity, an in [5], Hyeong In Choi an Anrejs Treibergs escribe the Gauß maps of the entire spacelike constant mean curvature hypersurfaces in L n+1 ; they prove the following: for any close set in the ieal bounary at infinity of the hyperbolic space which has more than two points, there exists an entire spacelike hypersurface with constant mean curvature whose Gauß map image is the hyperbolic space convex hull of the set. The Dirichlet problem for the prescribe Gauß curvature equation in Minkowski space is solve on convex omains by Philippe Delanoë in [7]. Bo

4 4 PIERRE BAYARD AND OLIVER C. SCHNÜRER Guan solve the problem in [14] uner the weaker assumption of the existence of a lower barrier. In [17], An-Min Li prove the existence of entire convex spacelike hypersurfaces of prescribe positive Gauß curvature which stay at a boune istance of a light-cone. In [15], Bo Guan, Huai-Yu Jian, an Richar Schoen prove the following: for every close set in the ieal bounary at infinity of the hyperbolic space which is not containe in any hyperplane, there exists a Lipschitz hypersurface whose graph solves the prescribe constant Gauß curvature equation in a weak sense an whose Gauß map image is the convex hull of the set. Moreover, they stuy a Minkowski type problem on half of hyperbolic space. Our results are similar to a result by Bo Guan, Huai-Yu Jian, an Richar Schoen [15, Theorem 3.5]. Our a priori estimates, especially the local C 1 - estimates, o not egenerate when we solve auxiliary problems on a sequence of growing balls. Thus the limit of the solutions to our auxiliary problems is smooth an strictly spacelike. Theorem 1.3 allows to eform entire hypersurfaces by a fully nonlinear flow equation. Before, such has been one for mean curvature flow by Klaus Ecker an Gerhar Huisken in Eucliean space [9] an by Klaus Ecker in Minkowski space [8]. There is a mean curvature flow approach by Mark Aarons [1] to fin entire spacelike hypersurfaces of constant mean curvature as classifie by Anrejs Treibergs in [0]. Stability of non-compact solutions to geometric flow equations was stuie by the secon author an Albert Chau [4] for Kähler-Ricci flow an for mean curvature flow with Julie Clutterbuck an Felix Schulze [6]. Ricci flow of non-compact manifols has been consiere by Wan-Xiong Shi [19]. He solves the initial value problem for metrics of boune curvature. Similarly, we assume initially boune Gauß curvature in Theorem 1.3. Note that for Kähler manifols, Ricci flow can be rewritten as u = log et u i j for the Kähler potential. The rest of the paper is organize as follows: In Section we introuce some terminology. We stuy the Gauß map an construct barriers in Sections 3 an 4. Local C 1 - an C -estimates are erive in Sections 5 an 6. We obtain the existence an convergence results mentione above in Section 7. We mention non-compact comparison principles in A. In Appenix B, we solve auxiliary problems on balls an prove a local normal velocity boun in Appenix C that allows to relax the uniform initial normal velocity boun. A technical lemma in Section D finishes the paper. Acknowlegement: The first author was supporte by SFB 647 uring his visit in Berlin in December 006.

5 HYPERSURFACES OF CONSTANT GAUSS CURVATURE IN MINKOWSKI SPACE 5. Definitions an Notation We say that a function u : Ω R is strictly convex, if its Hessian D u = u ij has positive eigenvalues. We say that such a function is uniformly strictly convex, if the eigenvalues of D u on Ω are uniformly boune below by a positive constant. A function u : Ω [0, is sai to be uniformly strictly convex, if u, t is uniformly strictly convex for each t an, in the uniformly strictly convex case, if the positive lower boun on the eigenvalues of the Hessian is inepenent of t. A function u : Ω R is calle strictly spacelike, if graph u is strictly spacelike, i.e. if Du < 1. Such a function is uniformly strictly spacelike, if sup Ω Du < 1. We say that a Lipschitz function u is strictly spacelike in a set Ω, if there exists some ϑ > 0 such that ux uy 1 ϑ x y for all x, y Ω. Similar to the efinition of convexity, we say that u : Ω [0, is uniformly strictly spacelike, if u, t has this property for any t. A function f is calle uniformly positive, if it is boune below by a uniform positive constant everywhere on its omain of efinition..1. Notation. We say that a function u solving a parabolic equation is in C, if u, t is in C for every t. The space C,1 enotes those functions, where in aition all first erivatives are continuous. We use Greek inices running from 1 to n + 1 for tensors in n + 1- imensional Minkowski space. Latin inices refer to quantities on spacelike hypersurfaces an run from 1 to n. The Einstein summation convention is use to sum over pairs of upper an lower inices. We raise an lower inices of tensors with the respective metrics or its inverses. An exception is the Latin subscript t, we efine f t = f u t. We set u = t. We use L n+1 to enote n + 1-imensional Minkowski space with its metric g αβ = iag1,..., 1, 1. We agree to always use coorinate systems in Minkowski space such that the metric has this form. Therefore the Coazzi equations imply that the first covariant erivative of the secon funamental form is completely symmetric. We use X = Xx, t to enote the embeing vector of a manifol M t into L n+1 an tx = Ẋ for its total time erivative. It is convenient to ientify M t an its embeing in L n+1. An embeing inuces a metric g ij. We will consier strictly spacelike hypersurfaces M t L n+1. If M t is locally represente as graph u, u : Ω R, Ω R n, being strictly spacelike is equivalent to Du < 1, where Du enotes the Eucliean norm of the graient of u. Let us use u i to enote partial erivatives of u. Using the Kronecker elta δ, we have u i δ ij u j u i u i = Du. The inuce

6 6 PIERRE BAYARD AND OLIVER C. SCHNÜRER metric g ij of graph u an its inverse g ij are given by respectively. g ij = δ ij u i u j an g ij = δ ij + ui u j 1 Du, We choose ν α to be the future irecte unit normal vector to M t. If M t is locally represente as graph u, we get ν α Du, 1 =, 1 Du ν α g αβ ν β = Du, 1 1 Du. The embeing also inuces a secon funamental form h ij. In the graphical setting it is given in terms of partial erivatives by h ij = u ij / 1 Du. We enote its inverse by h ij. We write inices, precee by semi-colons, e. g. h ij; k, to inicate covariant ifferentiation with respect to the inuce metric. For erivatives in L n+1, we use expressions like f α. Setting X;ij α := Xα,ij Γk ij Xα k, where a colon inicates partial erivatives, the Gauß formula is an the Weingarten equation is X α ;ij = h ij ν α ν α ;i = h k i X α k h ilg lk X α k. The eigenvalues of h ij with respect to g ij are the principal curvatures of the hypersurface an are enote by λ 1,..., λ n. A hypersurface is calle strictly convex, if all principal curvatures are strictly positive. The Gauß curvature is the prouct of the principal curvatures K = λ 1 λ n = et h ij et g ij. For graph u, it is given by K[u] := et u ij 1 Du n+, so the evolution equation t Xα = Ẋα = log K log fν α F ˆf ν α can be rewritten for graphs as et D u u = 1 Du log 1 Du n+ log fx, u, t an is therefore parabolic precisely when u is strictly convex an strictly spacelike.

7 HYPERSURFACES OF CONSTANT GAUSS CURVATURE IN MINKOWSKI SPACE 7 Let us also efine the mean curvature H = h ij g ij = λ λ n an the square norm of the secon funamental form A = h ij h kl g ik g jl = λ λ n. It is often convenient to choose coorinate systems such that the metric tensor equals the Kronecker elta, g ij = δ ij, an h ij is iagonal, h ij = iagλ 1,..., λ n. Note that in such a coorinate system F ij := F h ij,g ij h ij = h ij is iagonal. For tensors A an B, A ij B ij means that A ij B ij is positive efinite. Finally, we use c to enote universal, estimate constants. In orer to compute evolution equations, we use the Gauß equation an the Ricci ientity for the secon funamental form R ijkl = h ik h jl + h il h jk,.1 h ik; lj =h ik; jl + h a k R ailj + h a i R aklj.... Evolution Equations. Recall, see e. g. [10, 11], that for a hypersurface moving accoring to t Xα = log K log fx, t ν α F h ij, g ij ˆfX, t ν α, we have t g ij = F ˆf h ij,.3 t h ij = F ˆf ; + F ˆf h k i h kj,.4 ij t hj i = F ˆf j F ˆf h k i h j k,.5 ;i t να =g ij F ˆf ; i Xα ; j,.6 F t ˆf =F ij F ˆf ;ij F ˆf F ij h k i h kj.7 ˆf α ν α F ˆf ˆf t =F ij F ˆf ;ij F ˆf H ˆf α ν α F ˆf ˆf t. We efine η α = 0,..., 0, 1 an ṽ = η α ν α. The evolution equation for ṽ is given by tṽ F ij ṽ ;ij = ṽh η α X α ;i g ij X β ;j ˆf β.

8 8 PIERRE BAYARD AND OLIVER C. SCHNÜRER 3. The Gauß Map of an Entire Spacelike Hypersurface of Constant Gauß Curvature We recall here some results from [5, 15, 0] concerning the Gauß map of an entire spacelike hypersurface of constant Gauß curvature in L n+1. Following [0, Section 6], the blow own V u : R n R of a convex spacelike function u is efine by urx V u x = lim. r r Denoting by Q the set of the convex homogeneous of egree one functions whose graient has norm one whenever efine, the following hols see [0, Theorem 1] for the prescribe constant mean curvature equation: Lemma 3.1. For every amissible solution u of the prescribe constant Gauß curvature equation 1.1 the blow own V u belongs to Q. Proof. V u is clearly convex homogeneous of egree one. To prove that its graient has norm one whenever efine, we just observe that the barrier construction of Treibergs for the prescribe constant mean curvature equation can be use for the prescribe constant Gauß curvature equation as well see [0] page 5, step 1 in the proof of Theorem 1. It is prove in [5, Lemma 4.3] that the set Q is in one-to-one corresponence with the set of close subsets of S n 1 : Lemma 3.. [5, 0] If F is a close non-empty subset of S n 1, V F x = sup x λ 3.1 λ F belongs to Q; the map F V F is one-to-one, an its inverse is the map w Q F = {x S n 1 R n wx = 1}. In particular, the blow own of a convex solution u of 1.1 is etermine by the set of its lightlike irections L u = {x S n 1 R n V u x = 1}. As in [5], let us ientify the unit ball B 1 0 R n with the Klein moel of the hyperbolic geometry {x, 1 L n+1, x < 1}; the Gauß map of the graph of an entire spacelike function u in the natural chart x x, ux is then simply the function R n B 1 0, x Dux see [5, Lemma 4.5]. We also ientify S n 1 with the ieal bounary at infinity of the Klein moel B 1 0. The following lemma hols: Lemma 3.3. The image of the Gauß map of the graph of an amissible solution u of 1.1 is the convex hull in B 1 0 of the set L u.

9 HYPERSURFACES OF CONSTANT GAUSS CURVATURE IN MINKOWSKI SPACE 9 Proof. The proofs of Lemma 4.4 an Lemma 4.6 of [0] given there for the prescribe constant mean curvature equation exten to the prescribe constant Gauß curvature equation without moification. See also [15]. 4. The Construction of the Barriers In this section we escribe known examples of entire hypersurfaces of constant Gauß curvature, the semitroughs, constructe for n = by Jun-ichi Hano an Katsumi Nomizu in [16] an for n 3 by Bo Guan, Huai-Yu Jian an Richar Schoen in [15]. We prove some of their properties that we finally use to construct the barriers The Semitroughs. Let us first recall the properties of the stanar semitrough constructe in [16] n = an in [15] n 3 see also [5] for the construction of the semitroughs with constant mean curvature: it is the graph M of a function u of the form u x 1, x,..., x n = f x 1 + x, 4.1 where x = x,..., x n, an where f satisfies the following prescribe Gauß curvature equation: f f n 1 1 f n+ = 1, 4. i.e., graph u has Gauß curvature equal to one. Integrating equation 4., we get 1 f n f n c, 4.3 where c = 1 f 0 n f0 n. Choosing a 0, an b 0, 1 such that 1 b n a n = 1, it is prove in [15] that equation 4.3 has a unique solution f : R R such that f0 = a, f 0 = b. The function f has the following properties see [15, Section ]: i f > 0, 1 > f > 0, f > 0; ii lim ft = 0, lim f t = 0; t t iii 1 + t ft = λ < 0. lim t + Consiering ft + λ instea of ft, we may suppose instea of iii that iii lim 1 + t ft = 0. t +

10 10 PIERRE BAYARD AND OLIVER C. SCHNÜRER We thus get an entire function u given by 4.1 whose graph has constant Gauß curvature 1, such that an DuR n = { x B 1 0 : x 1 > 0 }, lim x ux V B + x = 0, where B + is the close ball in S n 1 efine by B + = { x S n 1 : x 1 0 }. Definition 4.1. The stanar semitrough is the entire convex spacelike function u whose graph has constant Gauß curvature 1 an is asymptotic to V B +, where B + = { x S n 1 : x 1 0 }. By the comparison principle, Lemma A.1, such a function is unique. It is given by 4.1, where f solves 4. an satisfies i, ii, an iii. In the Klein moel, the image of the Gauß map of its graph is B 1 0 { x 1 > 0 }. Let us enote by S the natural istance on S n 1 : for x, y S n 1, we have S x, y = arccosx y [0, π], 4.4 where the ot stans for the canonical scalar prouct in R n. A ball in S n 1 is a ball in the metric space S n 1, S, i.e. a set B = { x S n 1 : S x, x 0 < δ }, where x 0 is some point of S n 1 the center of the ball an where δ is a positive constant; δ is the raius of B, also enote by δb. B will enote the closure of B in S n 1, an B c the complement of B in S n 1. We note that B c is a ball of raius π δb. Applying Lorentz transformations an homotheties we thus get from the existence of u the existence of the so-calle semitroughs. This is one implicitly for the mean curvature case in [5, 0].: for every ball B of S n 1 there exists an entire function z B on R n whose graph is a hypersurface with constant Gauß curvature k > 0 in L n+1 which is asymptotic to V B. The image of the Gauß map of such a hypersurface is the convex hull of B in B 1 0. We also observe that, by the comparison principle Lemma A.1, the entire convex hypersurface with given Gauß curvature k asymptotic to V B is unique. The next lemma gathers the properties of the semitroughs that we will use to construct the barriers see Section 4. below. Lemma 4.. Let B be a close ball of S n 1 such that π δ 0 δb δ 0 for some δ 0 > 0 an let z B be the semitrough with Gauß curvature k which is asymptotic to V B. The following hols:

11 HYPERSURFACES OF CONSTANT GAUSS CURVATURE IN MINKOWSKI SPACE 11 i Let h k x = an k n + x. Then h k z B > V B 4.5 z B x V B x 0 as x 4.6 uniformly in B such that π δ 0 δb δ 0. ii For all compact sets K R n there exists δ = δk, δ 0, k > 0 such that: for all x K, z B x V B x + δ. 4.7 iii For all compact sets K R n there exists ϑ K = ϑ K K, δ 0, k 0, 1] such that: for all x, y K, z B x z B y 1 ϑ K x y. 4.8 iv For i = 1,, let z i,b enote the semitroughs with Gauß curvature k i which are asymptotic to V B. If k 1 > k, then z,b > z 1,B. More precisely, for all compact sets K R n there exists δ > 0 such that, for all x K, z,b x z 1,B x + δ. 4.9 The constant δ epens on K, δ 0, k 1, k, an k 1 k. Proof. We first prove i. We may suppose that k = 1. For the stanar semitrough u c.f. Definition 4.1 above we have h 1 u > V B +. Let ψ be a Lorentz transformation which maps B + to B recall that the Lorentz transformations act as the conformal maps on the bounary S n 1 of the Klein moel, which is ientifie with the projective lightcone in L n+1. The function ψ maps the graphs of V B +, u an h 1 to the graphs of V B, z B an h 1 respectively. This implies the inequalities in 4.5. We then focus on the stuy of the limit. Let us first choose coorinates such that the ball B is centere aroun 1, 0,..., 0 an such that the semitrough z B is the image of the stanar semitrough u uner the Lorentz transformation x 1 coshϕ sinhϕ x 1 x n+1 = sinhϕ coshϕ x n+1, x i = x i for i n The raius δ B of B is given by δ B = arccostanhϕ, an the conition π δ 0 δb δ 0 reas ϕ 0 ϕ ϕ 0, 4.11 where ϕ 0 = tanh 1 cosδ 0 is a finite number. Now the rest of the claim follows from Lemma D.1.

12 1 PIERRE BAYARD AND OLIVER C. SCHNÜRER For the proof of ii-iv, we observe that the properties 4.7, 4.8 an 4.9 clearly hol on the compact set K, for each close ball B. Keeping the notations from above, they are uniform in ϕ [ ϕ 0, ϕ 0 ] by continuity an compactness ϕ 0 is finite. 4.. The Barriers. The aim of this section is to construct two barriers for the prescribe Gauß curvature equation with given asymptotics. We obtain the following existence theorem. Theorem 4.3. Let S n 1 enote the ieal bounary at infinity of hyperbolic space an F S n 1 a non-empty close subset. Assume that F is of the following form: if n 3, F is the closure of some open subset of S n 1 with C 1,1 -bounary; if n =, F is a finite union of non-trivial intervals on the unit circle. Define V F : R n R by V F x := sup x λ. λ F Let k 1 > k be two positive constants. Then there exist two functions u, u : R n R, such that u is a convex subsolution to the equation of prescribe Gauß curvature k 1 an u is a supersolution to the equation of prescribe Gauß curvature k. More precisely, u is the supremum of functions u : R n R which are strictly convex, strictly spacelike, an fulfill the equation K[u] = k 1. Similarly, u is the infimum of strictly convex strictly spacelike functions u such that K[u] = k. The barriers u u have the following properties: i ux V F x + ux V F x 0 as x. 4.1 ii For every compact subset K R n, there exists a constant ϑ > 0 such that for every x, y K ux uy 1 ϑ x y, 4.13 ux uy 1 ϑ x y iii For every compact subset K R n, there exists a constant δ > 0 such that for every x K V F x + δ ux ux δ By construction, u is the supremum of subsolutions of the form z 1,B. So it is a subsolution in the viscosity sense. When we wish to compare a solution with u, we can apply the comparison principle for the functions z 1,B an show that u z 1,B for each B consiere. Then we obtain that u u. The situation for u is similar. The following lemma states that the barriers obtaine in Theorem 4.3 guarantee that our solutions u have the esire asymptotics at infinity.

13 HYPERSURFACES OF CONSTANT GAUSS CURVATURE IN MINKOWSKI SPACE 13 Lemma 4.4. Let F, u, an u be as in Theorem 4.3. Let u : R n R be a smooth convex function with u u u. Then the Gauß map image of the hypersurface graph u is the hyperbolic space convex hull of F, i.e. the convex hull of F in hyperbolic space. Proof. Since u u u with u, u asymptotic to V F, we have ux V F x 0 as x. Thus V u = V F an F is the set of lightlike irections of u. Lemma 3.3 implies the result. The en of this section is evote to the proof of Theorem 4.3. We first state preliminary lemmas. Uner the hypotheses of Theorem 4.3, we obtain the following technical properties which essentially rely on the C 1,1 regularity of F : Lemma 4.5. Let S be the natural istance on the sphere S n 1. For F as above, there exists δ 0 > 0 such that the following hols: i F an F c are the union of close balls of S n 1 of raius δ 0 ; ii For every x F c, there exists a close ball B with raius boune below by δ 0 which contains x an is containe in F c such that S x, B c = S x, F. Proof. If n = these properties are evient an we focus on the case n 3. Since the bounary of F is compact an C 1,1, F an F c are a union of close balls of raius δ 0 for some δ 0 > 0 sufficiently small. Moreover, we may choose δ 0 smaller such that the following tubular neighborhoo property hols: for every x F c, if S x, F δ 0 then there exists a unique y F with S x, F = S x, y, an, enoting by N the exterior normal of F at y, x may be written x = cosδy +sinδn for some δ [0, δ 0 ]. We first assume that x F c is such that S x, F δ 0 : if B is the close ball tangent to F at y, of raius δ 0 an exterior to F, then B F c an S x, B c = S x, F. If now x is such that S x, F δ 0, we take the close ball of raius S x, F an center x. Let k 1 > k be two positive constants. We suppose that the set F in S n 1 satisfies the hypotheses of Theorem 4.3 an we fix δ 0 > 0 as in Lemma 4.5. Let us efine ux = sup z 1,B x an ux = inf B F B F δb δ 0 δb π δ 0 z,b x, 4.16 where z 1,B resp. z,b is the semitrough asymptotic to V B whose curvature is k 1 resp. k. To stuy the properties of u an u, we will nee the following escriptions of V F :

14 14 PIERRE BAYARD AND OLIVER C. SCHNÜRER Lemma 4.6. For every close subset F of S n 1 an for all x S n 1, V F x = cos S x, F, 4.17 where S enotes the natural istance on the sphere S n 1. In particular, if F an F are two close subsets of S n 1 an if x S n 1 is such that S x, F = S x, F, then V F x = V F x. Thus if y 0 F is such that S x, F = S x, y 0, we have V F x = x y 0. Proof. Let y 0 F be such that S x, F = S x, y 0. In view of 4.4, to prove the lemma, we have to prove that V F x = x y 0. By efinition, V F x = sup λ F x λ. Since y 0 F, we have V F x x y 0. For every λ F, S x, λ S x, y 0. Thus x λ x y 0 an V F x x y 0. Lemma 4.7. Uner the hypotheses on F mae above, we have: for all x R n, V F x = inf V B F B x = sup V B x B F δb π δ 0 δb δ 0 Proof. We may suppose that x S n 1. Since V F1 V F if F 1 F, we obviously have: sup V B x V F x B F δb δ 0 inf V B F B x. δb π δ 0 We first prove that V F x inf { V B x : B F, δ B π δ 0 } : if x F, the result is obvious since V F x = V B x = 1 for every ball B containing F. We thus assume that x / F, an we consier a ball B of S n 1 with raius boune below by δ 0 which contains x an is containe in F c, such that S x, B c = S x, F Lemma 4.5. Thus, from Lemma 4.6, we have V F x = V B cx, an, since F B c with δb c π δ 0, the esire inequality. We now prove that V F sup { V B x : B F, δ B } δ 0 : if x F, VF x = 1. By the efinition of δ 0, there exists a ball B F of raius δ 0 which contains x. Since V B x = 1 we obtain the result. If x / F, we consier y 0 F such that S x, F = S x, y 0 an a ball B F of raius δ 0 which contains y 0. We have S x, B = S x, F. Thus, from Lemma 4.6, we have V F x = V B x, an thus the last inequality follows. Proof of Theorem 4.3. For proving ii, we observe that the inequalities 4.13 an 4.14 are irect consequences of Lemma 4. iii, since the close balls involve in the efinition of u an u have raius between δ 0 an π δ 0. The secon inequality in 4.15 is a irect consequence of Lemma 4. iv, since z,b z,b if B F B to prove this last claim, note that V B V B

15 HYPERSURFACES OF CONSTANT GAUSS CURVATURE IN MINKOWSKI SPACE 15 an apply the comparison principle, Lemma A.1. The first inequality in 4.15 follows irectly from Lemma 4. ii an Lemma 4.7. We now prove i. We first note that 4.15 implies that V F < u < u on R n. To prove 4.1, it is thus sufficient to prove that ux V F x 0 as x x. If x F, ux V F x = ux x h k x x where h k x = k n + x. Thus ux V F x tens to zero when x tens to, with x x x F. If x / F, we consier a ball B in S n 1 with raius boune below by δ 0, containing x x an containe in F c such that S x/ x, B c = S x/ x, F, given by Lemma 4.5. From Lemma 4.6 we get V F x = V B cx. Since F B c an δb c π δ 0, we thus obtain 0 ux V F x z B cx V B cx. Since π δ 0 δb c δ 0, Lemma 4. i gives the result. 5. Local C 1 -Estimates We have the following estimate which is inepenent of the ifferential equation. Lemma 5.1. Let Ω R n be a boune open set. Let u, u, ψ : Ω R be strictly spacelike. Assume that near Ω, we have ψ > u. Everywhere in Ω we assume that u u. Consier the set, where u > ψ. For every x in that set, we get the following graient estimate for u Dux ux ψx sup u ψ. {u>ψ} 1 Dψ This lemma allows to get uniform graient estimates on the set, where u ψ is estimate from below by a positive constant. Note that this a priori estimate is extene in [3]. Proof of Lemma 5.1. Consier u ψ 1 Du in the set {u > ψ}. It vanishes at the bounary. Therefore it has an interior positive maximum. We euce that also ϕ efine by has an interior maximum. ϕ := 1 log 1 Du + logu ψ

16 16 PIERRE BAYARD AND OLIVER C. SCHNÜRER At that local maximum of w, we get We euce there that 0 = ϕ i = uk u ki 1 Du + u i ψ i u ψ. 0 = ϕ i u i = uk u ki u i 1 Du + Du Dψ, Du. u ψ As u is convex, we may rop the first term an obtain there Du Du, Dψ Du Dψ. It follows everywhere on {u > ψ} that u ψ 1 Du sup {u>ψ} We arrive at the estimate claime above. u ψ 1 Du sup {u>ψ} u ψ 1 Dψ. We now construct the function ψ. Let us fix λ 0, 1. We consier x u λ x := λu, λ where u is the lower barrier constructe in Section 4.. Since V F is a homogeneous function of egree one, we have V F < u λ < u. Let K R n be a compact set, an δ > 0 a constant such that u u λ δ on K. We set ψ = u λ + δ. We have ψ u δ on K, an ψ u δ as x, where u is the upper barrier constructe in Section 4., since u is also asymptotic to V F. The latter implies that ψ > u near the bounary of some boune open set Ω R n which contains K. Smoothing ψ by a convolution, Lemma 5.1 gives the C 1 estimate on K for every spacelike convex function u between u an u. 6. Local C -Estimates Similarly to [13, Chapter 17.7], we obtain local C -estimates. Theorem 6.1. Let Ω R n be a boune omain, l : R n R be an affine linear function. Suppose that u : Ω [0, is strictly convex. Assume that u is smooth, spacelike, u + 1 is uniformly boune, an lx ux, t < 0 1 Du for x, t Ω [0,. If u is uniformly boune an u solves et D u u = 1 Du log 1 Du n+ log f

17 HYPERSURFACES OF CONSTANT GAUSS CURVATURE IN MINKOWSKI SPACE 17 in Ω [0,, where f is a positive constant, then β lx ux, t x, t exp 1 Du 1 Du max ξ =1 u ij ξ i ξ j δ ij u i u j ξ i ξ j, where inices of u enote partial erivatives an β is sufficiently large, is uniformly boune in the set {x, t Ω [0, : lx > ux, t} in terms of its sup at t = 0 an the bouns assume above. Invariantly, this quantity is rewritten as l η α X α e βṽ h ij ξ i ξ j max ξ =1 g ij ξ i ξ j, where l is extene trivially to R n+1 an η α = 0,..., 0, 1. Remark 6.. If we consier solutions between barriers, we can control the set, where l u > 1. So Theorem 6.1 implies bouns on the principal curvatures an thus spatial C - an C,1 -bouns. If we consier a sequence of such functions, we thus get uniform C,1 -bouns on K [0, for compact sets K. We can weaken the regularity assumption on our initial ata u t=0. If this function can be approximate in C 0 by C -functions for which Theorem B.4 implies the existence of a solution with boune log K, we can also obtain spatial C -bouns on the sequence as long as t is uniformly boune below by a positive constant. In orer to prove this, one applies arguments as in the proof of Theorem 6.1 to the function l η α X α e βṽ max ξ =1 h ij ξ i ξ j g ij ξ i ξ j Proof of Theorem 6.1. Assume that the function l η α X α e βṽ h ij ξ i ξ j max ξ =1 g ij ξ i ξ j attains a new positive maximum at some positive time an coorinates are chosen such that ξ = e 1 gives this maximal value there. Define t. w := log l η α X α + βṽ + log h 11 g 11. We will assume for the rest of the proof that l η α X α > 0 at the point consiere. Following [10], it suffices to apply the maximum principle to the function w if we want to boun the expression above. We have the evolution equation t l η αx α F ij l η α X α ;ij = l α η α ν α F ˆf n.

18 18 PIERRE BAYARD AND OLIVER C. SCHNÜRER Combining.1,., an.4, we get t h ij F kl h ij;kl = Hh ij + In the maximum consiere, we get F ˆf + n 0 t w = l αẋα η α Ẋ α l η α X α + β ṽ + 1 h 11 ḣ 11 1 g 11 ġ 11, 0 =w ;i = l αx α ;i η αx α ;i l η α X α + βṽ ;i + 1 h 11 h 11;i, h k i h kj h kr hls h kl;i h rs;j. 0 w ;ij = l αx;ij α η αx;ij α l η α X α + βṽ ;ij + 1 h 11;ij 1 h 11 h h 11;i h 11;j 11 l α X;i α η αx;i α l β X β ;j η βx β ;j l η α X α, 1 0 l η α X α t l η αx α F ij l η α X α ;ij + 1 h 11 t h 11 F ij h 11;ij + 1 h F ij h 11;i h 11;j 11 + F ij l α X α ;i η αx α ;i l β X β ;j η βx β ;j l η α X α 1 g 11 ġ β tṽ F ij ṽ ;ij Let us use C 1 -bouns an normal velocity bouns. Let us also assume that h 11 1 at an interior maximum an β 1. Then we conclue that c 0 l η α X α β ṽh 1 h hkr hls h kl;1 h rs;1 + 1 h ij 11 h h 11;i h 11;j 11 l α X α ij ;i + h η αx;i α l β X β ;j η βx β ;j l η α X α. We have ṽ =η α ν α, ṽ ;i =η α ν α ;i = η α h k i X α ;k. We may assume that we have chosen coorinates such that h ij is iagonal an g ij = δ ij. We now want to consier the term that requires the most complicate estimates. Using the extremal conition 0 = l αx α ;i η αx α ;i l η α X α + βṽ ;i + 1 h 11 h 11;i,

19 HYPERSURFACES OF CONSTANT GAUSS CURVATURE IN MINKOWSKI SPACE 19 we get l α X α h ij ;i η αx;i α l β X β ;j η βx β ;j l η α X α lα X;1 α η αx;1 α l β X β ;1 η βx;1 β = h 11 l η α X α l α X α h ii ;i η αx;i α l β X β ;i η βx β ;i l η α X α + i>1 c h 11 l η α X α + h ii βṽ ;i + 1 h 11;i h i>1 11 c = h 11 l η α X α + h ii β ṽ;i + i>1 i>1 + 1 h ii h h 11;i h 11;i i>1 11 c = h 11 l η α X α + h ii β ṽ;i h ii β ṽ;i i>1 i>1 h ii l α X;i α βṽ η αx;i α ;i l η i>1 α X α + i>1 c h 11 l η α X α i>1 h ii βṽ ;i l α X α ;i η αx α ;i l η α X α + i>1 1 h 11 1 h 11 h ii βṽ ;i 1 h 11 h 11;i h ii h 11;i h 11;i h ii h 11;i h 11;i. We get β h ii l η α X α ṽ ;i lα X;i α η α X;i α i>1 β = h ii l η α X α h ii η α X;i α = i>1 l β X β ;i η βx β ;i β l η α X α l β X β ;i η βx β ;i X;i α η α βc l η α X α. i>1

20 0 PIERRE BAYARD AND OLIVER C. SCHNÜRER So we obtain l α X α h ij ;i η αx;i α l β X β ;j η βx β ;j l η α X α In a local maximum of w, we euce that 0 cβ l η α X α β ṽh c h 11 l η α X α + + i>1 1 h 11 h ii h 11;i h 11;i. 1 h hkr hls h kl;1 h rs;1 + 1 h ij 11 h h 11;i h 11;j 11 c + h 11 l η α X α + 1 h ii h h 11;i h 11;i. i>1 11 βc l η α X α Accoring to the Coazzi equations, h ij;k is symmetric in all inices. Thus 1 h ii hjj h ij;1 + 1 h ii h 11 h h 11;i + 1 h ii i,j 11 h h 11;i i i>1 11 = 1 h rr hss h rs;1 0. h 11 Therefore we get r,s>1 0 cβ l η α X α β ṽh + c h 11 l η α X α. We use ṽ 1 an h 11 H. For fixe β 1, we get h 11 l η α X α c cβ + h 11 l η α X α. So we obtain an interior C -boun. 7. Existence of Entire Solutions an Convergence Consier a sequence of functions u R as in Theorem B.4 solving u R = 1 Du R log et D u R log f 1 Du R n+ in B R [0,. In orer to apply Theorem B.4, we may shift the barriers so that u < u 0 < u. Our uniform bouns on the normal velocity of graph u R, the local spatial bouns in C an higher orer erivative estimates ue to Krylov, Safonov, an Schauer for positive times imply that a subsequence converges in C R n 0, C 1,α;0,α/ R n [0, for every 0 < α < 1

21 HYPERSURFACES OF CONSTANT GAUSS CURVATURE IN MINKOWSKI SPACE 1 to a solution u C R n 0, C 1,1;0,1 R n [0, of the initial value problem u = 1 Du log et D u 1 Du n+ log f 0 in R n 0,, u, 0 = u 0 in R n. Accoring to B.4, the normal velocity F ˆf converges exponentially to zero. Therefore u, t converges exponentially fast to a smooth strictly convex, strictly spacelike solution ũ : R n R of et D ũ 1 Dũ n+ = f 0 in R n with u ũ u. As u u converges to zero at infinity, the maximum principle, Lemma A.1, implies that ũ is the only solution like that. This finishes the proof of Theorem 1.3. Similarly, for proving Theorem 1., we first apply Theorem B.1 to construct a sequence of smooth strictly convex strictly spacelike functions ϕ R such that u ϕ R u an a sequence u R of solutions to B.1 with Ω = B R 0. The functions ϕ R can be obtaine from mollifications of u. Applying Krylov- Safonov estimates an Schauer theory, we get higher erivative estimates. Therefore, we fin a subsequence of u R that converges in C R n to the solution u of 1.1 with u u u. Thus we have prove Theorem 1.. Appenix A. Comparison Principles We state a comparison principle for the Gauß curvature operator on spacelike functions efine on R n. Lemma A.1. Let u an v be two strictly spacelike functions belonging to C R n. Let us assume that v is strictly convex, the Gauß curvatures satisfy K[u] K[v] on R n an lim inf x ux vx 0. Then u v on Rn. We omit the proof, which is very close to the proof of the following comparison principle for the parabolic operator u P [u] = + log K[u] 1 Du acting on spacelike functions efine on R n 0,. Lemma A.. Let u an v be two strictly spacelike functions belonging to C,1 R n 0, C 0 R n [0,. Let us assume that v., t is strictly convex for all t, that P [u] P [v] on R n 0,, u., 0 v., 0 on R n,

22 PIERRE BAYARD AND OLIVER C. SCHNÜRER an lim inf inf ux, t vx, t 0 for every t 0 > 0. x t [0,t 0 ] R n [0,. Then u v on Proof. If there exists x 0, t 0 R n 0, such that ux 0, t 0 < vx 0, t 0, let δ > 0 be such that δ < vx 0, t 0 ux 0, t 0, an set u δ = u + δ. Since lim inf inf u δx, t vx, t δ for all t 0 > 0, we see that Ω δ,t0 = x t [0,t 0 ] {x, t R n 0, t 0 u δ x, t < vx, t} is a non-empty boune open set. We have v u δ on the parabolic bounary of Ω δ,t0 an P [u δ ] = P [u] P [v] in Ω δ,t0. Since v is convex, the stanar maximum principle, see also Remark B.3, implies that v u δ on Ω δ,t0, which is impossible. Appenix B. Existence of Solutions on Balls In orer to prove existence of graphical solutions to the equation of prescribe Gauß curvature or to logarithmic Gauß curvature flow, we construct solutions u R with Dirichlet bounary conitions on balls B R 0 an let R. Then we use local a priori estimates to show that a subsequence of the u R converges to an entire solution as R. In this appenix, we escribe how to obtain auxiliary solutions on balls. B.1. Elliptic Dirichlet Problem. In the elliptic case, an existence theorem in Minkowski space is known [7] for convex omains. We will assume in Theorem B.1 an Theorem B.4 that the properties of being positive, convex, an spacelike are all strict an uniform. Assume also that the ata are smooth with uniform a priori estimates. Theorem B.1. Let Ω R n be a boune convex omain, f : Ω R be positive, an ϕ : Ω R be convex an spacelike. Then there exists a smooth convex spacelike solution u : Ω R to the Dirichlet problem et D u = fx in Ω, 1 Du n+ u = ϕ on Ω. B.1 Proof. See [7]. Remark B.. Let us note that in the proof of Theorem B.1 in [7], Philippe Delanoë constructs spacelike convex barriers of given positive constant Gauß curvature that touch graph ϕ Ω from below at a given bounary point an lie below ϕ. We will use these barriers also in the parabolic setting.

23 HYPERSURFACES OF CONSTANT GAUSS CURVATURE IN MINKOWSKI SPACE 3 Remark B.3. If there exist upper an lower barriers, u an u, which are strictly spacelike, strictly convex, of class C, an fulfill et D u 1 Du n+ u ϕ u fx then a solution u to B.1 fulfills u u u. et D u 1 Du n+ in Ω, on Ω, It even suffices to require that the ifferential inequality for u hols at those points, where u is spacelike an convex. This follows from the observation, see [1], that for a strictly convex strictly spacelike function, graph u can touch graph u from below only in points, where u is strictly spacelike an strictly convex. B.. Parabolic Dirichlet Problem. We want to consier an initial value problem of the form u = 1 Du log et D u ˆf 1 Du n+ 0 in Ω [0,, B. u = u 0 on Ω [0,, u = u 0 on Ω {0}. In orer to get a smooth solution, we have to moify f 0 such that compatibility conitions are fulfille along Ω {0}. This is one in the following theorem. Note, however, that the evolution equation is unchange outsie of a neighborhoo of Ω. Note further that f 0 is moifie in such a way that the normal velocity F ˆf of the evolving hypersurfaces graph u, t stays uniformly boune along a sequence of balls B R = Ω for which we consier such problems. Theorem B.4. Let Ω := B R 0, R. Let u 0 : Ω R be convex an spacelike. Assume that u < u 0 < u with u an u as in Remark B.3. Exten u 0 by setting u 0 x, t := u 0 x. Let f 0 be a positive constant. Choose a smooth function η : Ω R [0, 1] such that η = 0 on B R 1 0 R graph u graph u an η = 1 near graph u 0 Ω. Choose also a smooth function ζ : [0, [0, 1], inepenent of the other ata, such that ζt = 1 near t = 0 an ζt = 0 for t 1. Define f by log fx, u, t ˆf := ηx, u ζ t ε log et D u 0 1 Du 0 n+ log f 0 +log f 0. If ε > 0 is fixe sufficiently small, then there exists a uniformly strictly convex, uniformly strictly spacelike solution u C Ω [0, to the

24 4 PIERRE BAYARD AND OLIVER C. SCHNÜRER initial value problem et D u u = 1 Du log ˆfx, u, t in Ω [0,, 1 Du n+ u = u 0 on Ω [0,, u = u 0 on Ω {0}, B.3 u such that the normal velocity is uniformly boune in terms of f 1 Du 0 an K[u 0 ] := et D u 0. 1 Du 0 n+ Proof. Short Time Existence: At the bounary, compatibility conitions of any orer are fulfille, so for a short time interval, we get a smooth solution. We will assume for the a priori estimates that a smooth solution exists for all positive times. C 0 -Estimates: The functions u an u are barriers an imply that u u u. The C 0 -bouns follow. C 1 -Estimates: As in [7], it suffices to prove graient estimates at the bounary Ω. Note that the absolute value of the moifie function f is controlle inepenent of R. So we can use the lower barrier constructe in [7]. The maximal hypersurface foun by Robert Bartnik an Leon Simon in [] serves as an upper barrier for all bounary points simultaneously. These barriers stay above or below the solution u uring the evolution. They are strictly spacelike an coincie at a given bounary point with the solution. Moreover, the tangential graients of the barriers an of u, t coincie at that bounary point, so we get Du 1 c everywhere along the bounary Ω for some estimate positive constant c. Convexity implies interior C 1 -bouns. Note that a positive lower boun on c can be chosen so that it oes not epen on ε. Velocity Estimates: Uner the evolution equation F Ẋ = ˆfX, t ν, which is equivalent to the flow equation in B.3, the normal velocity fulfills F t ˆf F ij F ˆf ;ij = F ˆf H ˆf α ν α F ˆf ˆf t. Along the bounary, F ˆf vanishes. Define V t := sup F ˆf. graph u,t Accoring to the maximum principle, we get in the sense of ifference quotients t V t C V t + c 1 ε,

25 HYPERSURFACES OF CONSTANT GAUSS CURVATURE IN MINKOWSKI SPACE 5 where c 1 epens only on f 0 an K[u 0 ], but C may grow in R as ux ux + ux ux 0 as x. We may assume that C 1. The function V t = V 0 + c 1 e Ct c 1 εc εc solves this ifferential inequality with equality. If we pick ε = 1 C, we get sup V t cv 0, c 1. 0 t ε For the rest of the proof, we fix that value of ε. For t ε, we get F t ˆf F ij F ˆf ;ij = F ˆf H. As the evolving hypersurfaces are convex, we have H 0 an the maximum principle implies V t V ε for all t ε. This implies in particular a lower boun on F. Thus solutions stay convex. Accoring to the geometric-arithmetic means inequality, there exists a positive lower boun on H, epening only on cv 0, c 1. We apply the maximum principle once again to the evolution equation for the normal velocity an get F ˆf c e c 3 t B.4 with c, c 3 > 0 an c + 1 c 3 boune above in terms of f 0 an K[u 0 ]. C -Estimates at the Bounary: The following C -estimates epen on R. Tangential-Tangential Derivatives: These estimates follow irectly from ifferentiating the bounary conition twice. Tangential-Normal Derivatives: These a priori estimates follow from a stanar barrier construction using Lw =ẇ vu ij w ij + 1 v F ˆf u i w i n v ui w i, T u =u r + u n ω r, ϑ = µ, Θ =Aϑ + B x x 0 ± T u u 0 + l, the ifferential inequality LΘ 0 in Ω δ = Ω B δ x 0 an Θ 0 on Ω δ with equality at x 0 Ω. This bouns the normal erivatives of T u u 0 an thus u τν. Details can be foun in [18] an many other papers. Normal-Normal Derivatives: Sketch: This boun follows from techniques as in [1] an a similar barrier construction as above.

26 6 PIERRE BAYARD AND OLIVER C. SCHNÜRER Interior C -estimates: Consier the test function w := log H + βṽ for some β 1. For a family of convex spacelike hypersurfaces moving with normal velocity F ˆf, we get the following evolution equations t H F ij H ;ij =n A H F ˆf A H ˆf α ν α g ij hkr hls h kl;i h rs;j ˆf αβ X α ;i X β ;j gij, tṽ F ij ṽ ;ij = ṽh η α X;i α g ij X β ˆf ;j β, t w F ij w ;ij = 1 H t H F ij H ;ij + 1 H F ij H ;i H ;j + β F ˆf A = n H tṽ F ij ṽ ;ij H ˆf α ν α 1 H ˆf αβ X α ;i X β ;j gij 1 H gij hkr hls h kl;i h rs;j + 1 H h ij g kl g rs h kl;i h rs;j βṽh βη α X α ;i g ij X β ;j ˆf β. As the terms on the right-han sie involving erivatives of the secon funamental form are non-positive, we get in a point with H 1 ue to our C 1 a priori estimates t w F ij w ;ij c 1 + H βṽh c. The maximum principle implies for β 1 fixe sufficiently large a global boun on H. As u is strictly convex, u is boune in C,1. The estimates obtaine so far guarantee that u is uniformly strictly convex an spacelike. Long Time Existence: The estimates of Krylov, Safonov, an Schauer imply bouns on higher erivatives of u. Thus a solution exists for all times justifying our assumption above an the theorem follows. Remark B.5. There is a non-compact maximum principle by Klaus Ecker an Gerhar Huisken [9]. We can t apply this as the coefficients in our equation grow at infinity. There is also a non-compact maximum principle that allows for coefficients growing at infinity [1]. We were not able to unerstan the proof. That s why we use the compact maximum principle for our auxiliary problems instea.

27 HYPERSURFACES OF CONSTANT GAUSS CURVATURE IN MINKOWSKI SPACE 7 Appenix C. Velocity Bouns The estimates obtaine here allow to consier initial ata with log K[u 0 ] uniformly boune below. Theorem C.1. Let Ω R be a boune omain, ˆf R, an l : R R be a spacelike affine linear function. Assume that u C Ω [0, is a solution of u = 1 Du log et D u 1 Du n+ ˆf in Ω [0,, 1 such that l u < 0 on Ω [0, an u + is uniformly boune. 1 Du Then l u u is boune above in the set {x, t Ω [0, : lx > ux, t} in terms of an upper boun on K[u, 0] on Ω an in terms of the boun assume above. Proof. We first suppose that the imension is arbitrary. We will restrict it when necessary. Consier the test function w = log l η α X α + logf ˆf + log ṽ, where the notation is as above. We have t w F ij 1 w ;ij = l η α X α t l η αx α F ij l η α X α ;ij + 1 F ˆf t F F ij F ;ij + 1 ṽ tṽ F ij ṽ ;ij 1 + l η α X α F ij l η α X α ;i l η β X β ;j 1 + F ˆf F ij F ;i F ;j + 1 ṽ F ij ṽ ;i ṽ ;j. C.1 At a point where w attains a new maximum, we get 0 t w F ij w ;ij C. an 0 = w ;i = l η αx α ;i l η α X α + F ;i ṽ;i + F ˆf ṽ. C.3

28 8 PIERRE BAYARD AND OLIVER C. SCHNÜRER Using C.1, C., an C.3, together with the evolution equations, see Section, we get 1 0 l η α X α l α η α ν α F ˆf n H H + l η α X α F ij l η α X α ;i l η β X β C.4 ;j + ṽ F ij ṽ ;i ṽ ;j + l η α X α ṽ F ij l η α X α ;i ṽ ;j. We may assume that F ˆf n an H 1 at the maximum of w. From the graient estimate an ṽ ;i = η α h k i Xα ;k, we have l η α X α ṽ F ij l η α X α ;i ṽ ;j c l η α X α. The first term in C.4 is non-positive: since l an u are spacelike, l α η α ν α 1 = l i u i Du i n C.5 Dropping this term in C.4, using the inequality ṽ F ij ṽ ;i ṽ ;j H H ṽ, an C.5, we thus get 0 H ṽ + l η α X α F ij l η α X α ;i l η β X β + c ;j l η α X α. We thus obtain H c l η α X α n i=1 1 c 1 + λ i l η α X α. Now we nee to suppose that n =. Multiplying by l η α X α K where K is the Gauß curvature, we obtain l η α X α KH c H + c 1 l η α X α K. C.6 We first assume that 1 l η αx α KH c 1 l η α X α K. C.7 Inequality C.6 then reas Thus l η α X α l η α X α K c. C.8 log K ˆf l η α X α logc logl η α X α ˆf. The right han sie term is boune above an so is w at its maximum.

29 HYPERSURFACES OF CONSTANT GAUSS CURVATURE IN MINKOWSKI SPACE 9 We now suppose that C.7 oes not hol. This implies that l η α X α H c 1. Applying the geometric-arithmetic means inequality K 1 H gives an inequality as C.8. We then conclue as above. Appenix D. Closeness at Infinity Lemma D.1. Let u, v : R n R be spacelike functions. For ϕ R, let cosh ϕ 0 sinh ϕ A ϕ := 0 I 0 sinh ϕ 0 cosh ϕ be a Lorentz transformation. Then there exist spacelike function u ϕ, v ϕ : R n R such that Let ϕ 0 > 0. If then graph u ϕ = A ϕ graph u an graph v ϕ = A ϕ graph v. lim x lim ux vx = 0, x sup u ϕ x v ϕ x = 0. ϕ ϕ 0 Proof. It is known that there exists a function u ϕ such that A ϕ graph u = graph u ϕ in the original coorinate system as a Lorentz transformation preserves the property of being spacelike. In this proof, we use Eucliean space, equippe with the stanar norm. We claim first that the Hausorff istance at infinity between graph u ϕ an graph v ϕ is zero: lim R sup a, graph v ϕ + b, graph u ϕ = 0. a graph u ϕ \B R 0 b graph v ϕ \B R 0 Fix ε > 0. Consier R 0 an a graph u ϕ \ B R 0. Then there exists ã graph u such that A ϕ ã = a. We have ã R cϕ 0. As ux vx 0 for x, there exists b graph v such that ã b ε provie that R is chosen sufficiently large. Define b := A b ϕ graph v ϕ. By the linearity of A ϕ, there exists L = Lϕ 0 such that A ϕ p A ϕ q Lϕ 0 p q for all p, q R n+1, ϕ ϕ 0. Thus L ε a b a, graph v ϕ. Since v ϕ x y v ϕ x+y v ϕ x+ y for y R n, we see that for x R n such that u ϕ x v ϕ x ζ > 0, we get x, u ϕ x, graph v ϕ ζ. Thus the lemma follows.