Convex Spacelike Hypersurfaces of Constant Curvature in de Sitter Space

We show that for a very general and natural class of curvature functions (for example the curvature quotients $(\sigma_n/\sigma_l)^{\frac{1}{n-l}}$) the problem of finding a complete spacelike strictly convex hypersurface in de Sitter space satisfying $f(\kappa) = \sigma \in (1,\infty)$ with a prescribed compact future asymptotic boundary $\Gamma$ at infinity has at least one smooth solution (if l = 1 or l = 2 there is uniqueness). This is the exact analogue of the asymptotic plateau problem in Hyperbolic space and is in fact a precise dual problem. By using this duality we obtain for free the existence of strictly convex solutions to the asymptotic Plateau problem for $\sigma_l = \sigma$; $1\leq l


Introduction
Let R n+1,1 be the (n + 2) dimensional Minkowski space, that is, the real vector space R n+2 endowed with the Lorentz metric for all u, v ∈ R n+2 . The one sheeted hyperboloid consisting of all unit spacelike vectors and equipped with the induced metric is called de Sitter space. It is a geodesically complete simply connected Lorentzian manifold with constant curvature one . De Sitter space corresponds to a vacuum solution of the Einstein equations with a positive cosmological constant.
Choose a non-zero null vector a ∈ R n+1,1 in the past half of the null cone with vertex at the origin, i.e. a, a = 0, a, e > 0 where e = (0, . . . , 0, 1). Then the open region of de Sitter space defined by Research of the first author was supported in part NSF grant DMS 0904009. 1 is called the steady state space. Since the steady state space is only half the de Sitter space, it is incomplete. Its boundary as a subset of dS n+1 is the null hypersurface which represents the past infinity of H. The spacelike hypersurfaces are umbilic hypersurfaces of de Sitter space with constant mean curvature one with respect to the past oriented unit normal N τ (p) = −p + 1 τ a and foliate the steady state space. The limit boundary L ∞ represents a spacelike future infinity for timelike and null lines of de Sitter space [8].
In this paper we are interested in finding complete spacelike (i.e. the induced metric is Riemannian) strictly locally convex immersions ψ : Σ n → H n+1 with constant curvature and with prescribed (compact) future asymptotic boundary Γ . That is we want to find Σ satisfying where κ[Σ] = (κ 1 , . . . , κ n ) denote the positive principal curvatures of Σ (in the induced de Sitter metric with respect to the future oriented unit normal N) and σ > 1 is a constant. This is the exact analogue of the problem considered by Guan and Spruck [7] in hyperbolic space and as we shall make precise later, the two problems are essentially dual equivalent problems. Our study was motivated by the beautiful paper of Montiel [9] who treated the case of mean curvature.
As in our earlier work [10,12,4,5,6,7], we prefer to use the half space model because we find it has great advantages. Following Montiel [9] we can define a half space model for the steady state space in the following way. Define the map φ : R n+1,1 \ {p ∈ R n+1,1 | p, a = 0} → R n × R = R n+1 given by (1.7) φ(p) = 1 p, a (p − p, a b − p, b a, 1) where b ∈ R n+2 is a null vector such that < a, b >= 1( b is in the future directed null cone and R n stands for the orthogonal complement of the Lorentz plane spanned by a and b). Then the image of H by the map φ lies in the half space R n+1 which is called the half space model for H n+1 . It is important to note that the isometry φ reverses the time orientation.
Thus ∂ ∞ H n+1 is naturally identified with R n = R n × {0} ⊂ R n+1 and (1.6) may be understood in the Euclidean sense. For convenience we say Σ has compact asymptotic boundary if ∂Σ ⊂ ∂ ∞ H n+1 is compact with respect the Euclidean metric in R n .
The curvature function f (λ) in (1.5) is assumed to satisfy the fundamental structure conditions in the convex cone (1.10) K := K + n := λ ∈ R n : each component λ i > 0 : (1.11) f is symmetric, In addition, we shall assume that f is normalized (1.17) f is homogeneous of degree one and satisfies the following more technical assumption for some fixed ε 0 > 0 and δ 0 > 0, where B δ 0 (1) is the ball of radius δ 0 centered at The assumption (1.14) is closely related to the well-known fact [11], [3] , [13] that the Gauss map n of a spacelike locally strictly convex hypersurface Σ n in de Sitter space is an embedding into hyperbolic space H n+1 which inverts principal curvatures. We shall formulate a precise global version of this correspondence (see Theorem 2.2 and Corollary 2.3 in Section 2 ) which will be important for our deliberations. For the moment note that if f = (σ n /σ l ) 1 n−l , 0 ≤ l < n, defined in K where σ l is the normalized l-th elementary symmetric polynomial (σ 0 = 1), then f * (λ) = (σ n−l (λ)) 1 n−l . Also one easily computes that Since f is symmetric, by (1.13), (1.16) and (1.17) we have Using (1.14), we see that In this paper all hypersurfaces in H n+1 we consider are assumed to be connected and orientable. If Σ is a complete spacelike hypersurface in H n+1 with compact asymptotic boundary at infinity, then the normal vector field of Σ is chosen to be the one pointing to the unique unbounded region in R n+1 + \ Σ, and the (both de Sitter and Minkowski) principal curvatures of Σ are calculated with respect to this normal vector field.
Because Σ is strictly locally convex and strictly spacelike, we are forced to take Γ = ∂Ω where Ω ⊂ R n is a smooth domain and seek Σ as the graph of a "spacelike" function u(x) over Ω, i.e.
We will compute the first and second fundamental forms g ij , h ij with respect to the induced de Sitter metric as well asg ij ,h ij the corresponding forms in the induced Minkowski metric viewing Σ as a graph in the Minkowski space R n,1 with unit normal ν. We use where w = 1 − |∇u| 2 . The first fundamental form g ij is then given by For computing the second fundamental form we use The principal curvature κ i of Σ in de Sitter space are the roots of the characteristic equation Therefore, By ( We seek solutions of equation (1.33) satisfying the spacelike condition (1.11) and (1.28). Following the literature we call such solutions admissible. By [2] condition (1.28) implies that equation (1.33) is elliptic for admissible solutions. Our goal is to show that the Dirichlet problem (1.33)-(1.34) admits smooth admissible solutions for all σ > 1 which is optimal.
Our main result of the paper may be stated as follows.
where Ω is a bounded smooth domain in R n . Suppose that σ > 1 and that f satisfies conditions (1.11)-(1.18) with K = K + n . Then there exists a complete locally strictly convex spacelike hypersurface Σ in H n+1 satisfying (1.5)-(1.6) with uniformly bounded principal curvatures As a concrete application we have existence for the canonical curvature functions.
where Ω is a bounded smooth domain in R n . Then there exists a complete locally strictly convex spacelike hypersurface Σ in H n+1 satisfying (σ n /σ l ) 1 n−l = σ > 1, 0 ≤ l < n with ∂Σ = Γ and uniformly bounded principal curvatures As we mentioned earlier, in Section 2 we prove strong duality theorems (see Theorem 2.2 and Corollary 2.3 ) which allows us to transfer our existence results for H n+1 in [7] to H n+1 and conversely. In particular we have where Ω is a bounded smooth domain in R n . Then there exists a complete locally strictly convex spacelike hypersurface Σ in H n+1 satisfying (σ l ) 1 l = σ > 1, 1 ≤ l ≤ n with ∂Σ = Γ and uniformly bounded principal curvatures Moreover, Σ is the graph of an admissible solution u ∈ C ∞ (Ω)∩C 1 (Ω) of the Dirichlet problem (1.33)-(1.34). Furthermore, u 2 ∈ C ∞ (Ω) ∩ C 1,1 (Ω) and Further, if l = 1 or l = 2 (mean curvature and normalized scalar curvature) we have uniqueness among convex solutions and even among all solutions convex or not if Ω is simple.
The uniqueness part of Corollary 1.3 follows from the uniqueness Theorem 1.6 of [7] and a continuity and deformation argument like that used in [12]. Note that Montiel [9] proved existence for H = σ > 1 assuming ∂Ω is mean convex. Our result shows that for arbitrary Ω there is always a locally strictly convex solution. If Ω is simple and mean convex the solutions constructed by Montiel must agree with the ones we construct.
Transferring by duality the results of Corollary 1.2 to H n+1 gives the mildly surprising It is therefore natural to approximate the boundary condition (1.34) by When ǫ is sufficiently small, we shall show that the Dirichlet problem (1.33),(1.43) is solvable for all σ > 1.
Let Ω be a bounded smooth domain in R n and σ > 1. Suppose f satisfies conditions (1.11)-(1.18) with K = K + n . Then for any ǫ > 0 sufficiently small, there exists an admissible solution u ǫ ∈ C ∞ (Ω) of the Dirichlet problem (1.33),(1.43) . Moreover, u ǫ satisfies the a priori estimates where C is independent of ǫ.
In the proof of Theorem 1.5 we mostly follow the method of [5], [7] except that we will appeal to the duality results of Section 2 to use the global maximal principle of [7] to control the principal curvatures and prove the first inequality in (1.36). Because there is no a priori uniqueness (i.e G u ≥ 0 need not hold) it is not sufficient to derive estimates just for solutions of (1.33) with constant right hand side σ, one must also consider perturbations. However to avoid undue length and tedious repetition, we will prove the estimates for solutions of constant curvature σ as the necessary modifications are straightforward (see [5]). By Theorem 1.5, the hyperbolic principal curvatures of the admissible solution u ǫ of the Dirichlet problem (1.33),(1.43) are uniformly bounded above independent of ǫ.
Since f (κ[u ǫ ]) = σ and f = 0 on ∂K + n , the hyperbolic principal curvatures admit a uniform positive lower bound independent of ǫ and therefore (1.33) is uniformly elliptic on compact subsets of Ω for the solution u ǫ . By the interior estimates of Evans and Krylov, we obtain uniform C 2,α estimates for any compact subdomain of Ω. The proof of Theorem 1.1 is now routine.
An outline of the contents of the paper are as follows. Section 2 contains the important duality results, Theorem 2.2 and Corollary 2.3. Section 3 contains preliminary formulas and computations that are used in Section 4 to prove the asymptotic angle result Theorem 4.2. In Section 5 we use the linearized operator to bound the principal curvatures of a solution on the boundary. Here is where the condition (1.18) comes into play. Finally in Section 6 we use duality to establish global curvature bounds and complete the proof of Theorem 1.5. The use of duality to prove this global estimate is unusual but seems to be necessary since F (A) is a concave function of A but G is a convex function of {u ij }.

The Gauss map and Legendre transform
Let ψ : Σ n → H n+1 be a strictly locally convex spacelike immersion with prescribed (compact) boundary Γ either in the timeslice L τ or in L ∞ = ∂ ∞ H. We are constructing such Σ n as the graph of an admissible function u : where Ω is a smooth bounded domain in R n . We know that the Gauss map n : Σ n → H n+1 takes values in hyperbolic space. Using the map φ defined in (1.7) that was used to identify the de Sitter and upper halfspace models of the steady state space H n+1 , Montiel [9] showed that if we use the upper halfspace representation for both H and H n+1 , then the Gauss map n corresponds to the map We now identify the map L in terms of a hodograph and associated Legendre transform. Define the map y = ∇p(x) : Ω ⊂ R n → R n by . Note that p is strictly convex in the Euclidean sense by (1.28) and hence the map y is globally one to one. Therefore v(y) := u(x) 1 − |∇u(x)| 2 is well defined in Ω * := y(Ω). The associated Legendre transform is the function q(y) defined in Ω * by p(x) + q(y) = x · y or q(y) = −p(x) + x · ∇p(x).
Theorem 2.2. Let L be defined by (2.1) and let y be defined by (2.2). Then the image of S by L is the locally strictly convex graph (with respect to the induced hyperbolic metric) Here κ i > 0, i = 1, . . . , n are the principal curvatures of S with respect to the induced de Sitter metric. Moreover the inverse map L * : S * → S defined by L * ((y, v(y)) = (y + v(y)∇v(y), v(y) 1 + |∇v(y)| 2 y ∈ Ω * is the dual Legendre transform and hodograph map x = ∇q(y).

Formulas on hypersurfaces
In this section we will derive some basic identities on a hypersurface by comparing the induced metric in steady state space H n+1 ⊂ dS n+1 and Minkowski space.
Let Σ be a hypersurface in H n+1 . We shall use g and ∇ to denote the induced metric and Levi-Civita connection on Σ, respectively. As Σ is also a submanifold of R n,1 , we shall usually distinguish a geometric quantity with respect to the Minkowski metric by adding a 'tilde' over the corresponding quantity. For instance,g denotes the induced metric on Σ from R n,1 , and∇ is its Levi-Civita connection.
Let x be the position vector of Σ in R n,1 and set where e= (0, · · · , 0, 1) is the unit vector in the positive x n+1 direction in R n+1 , and '·' denotes the Euclidean inner product in R n+1 . We refer u as the height function of Σ.
Throughout the paper we assume Σ is orientable and let n be a (global) unit normal vector field to Σ with respect to the de Sitter metric. This also determines a unit normal ν to Σ with respect to the Minkowski metric by the relation ν = n u .
We denote ν n+1 = e · ν. Let (z 1 , . . . , z n ) be local coordinates and The de Sitter and Minkowski metrics of Σ are given by while the second fundamental forms are where D andD denote the Levi-Civita connection of H n+1 and R n,1 , respectively, and , D , , M denote the corresponding inner product.

The asymptotic angle maximum principle and gradient estimates
In this section we show that the upward unit normal of a solution tends to a fixed asymptotic angle on approach to the asymptotic boundary and that this holds approximately for the solutions of the approximate problem.. This implies a global (spacelike) gradient bound on solutions.
Our estimates are all based on the use of special barriers (see section 3 of [9]). These correspond to horospheres for the dual problem in hyperbolic space and our argument follows that of section 3 of [6]. Let be a ball of radius r centered at c in Minkowski space, where c ∈ R n+1 . Moreover, let Q + (r, c) denote the region above the upper hyperboloid and Q − (r, c) denote the region below the lower hyperboloid. If we choose a = (a ′ , −rσ), then S + (r, a) = ∂Q + (r, a) ∩ H n+1 is an umbilical hypersurface in H n+1 with constant curvature σ with respect to its upward normal vector. For convenience we sometimes call S + (r, a) an upper hyperboloid of constant curvature σ in H n+1 . Similarly, when we choose b = (b ′ , rσ), then S − (r, b) = ∂Q − (r, b) ∩ H n+1 is the lower hyperboloid of constant curvature σ with respect to its upward normal vector. These hyperboloids serve as useful barriers. Now let Σ be a hypersurface in H n+1 with ∂Σ ⊂ P (ε) := {x n+1 = ε} so Σ separates {x n+1 ≥ ε} into an inside (bounded) region and outside (unbounded) one. Let Ω be the region in R n × {0} such that its vertical lift Ω ε to P (ε) is bounded by ∂Σ (and R n \ Ω is connected and unbounded). (It is allowable that Ω have several connected components.) Suppose κ[Σ] ∈ K + n and f (κ) = σ ∈ (1, ∞) with respect to its outer normal.
Proof. For (i) let c = min x∈Σ x n+1 and suppose 0 < c < ε. Then the horizontal plane P (c) satisfies f (κ) = 1 with respect to the upward normal, lies below Σ, and has an interior contact point. Then f (κ[Σ]) ≤ 1 at this point, which leads to a contradiction (notice that in the Euclidean case we have the reverse inequality). For (ii), (iii), (iv) we consider the family {h s } s∈R of isometries of H n+1 consisting of Euclidean homotheties. We perform homothetic dilations from (a ′ , 0) and (b ′ , 0) respectively, and then use the maximum principle. For (ii), choose s 0 big enough so that h s 0 (Q − (r, b)) contains Σ and then decrease s. Since the curvature of Σ and S − (r, b) are calculated with respect to their outward normals and both hypersurfaces satisfy f (κ) = σ, there cannot be a first contact.
For (iii) and (iv) we shrink Q + (r, a) and Q − (r, b) until they are respectively inside and outside Σ. When we expand Q − (r, b) there cannot be a first contact as above. Now decrease s to a certain value s 1 ∈ R such that h s 1 (Q + (r, a)) is disjoint from Σ (outside of). Then we increase s 1 and suppose there is a first interior contact. The outward normal to Σ at this contact point is the upward normal to S + (r, a). Since the curvatures of S + (r, a) are calculated with respect to the upward normal and S + (r, a) satisfies f (κ) = σ, we have a contradiction of the maximum principle.
In order to prove (4.3), we first assume r 2 < ∞. Fix a point x 0 ∈ ∂Ω and let e 1 be the outward pointing unit normal to ∂Ω at x 0 . Let S + (R 2 , a), S − (R 1 , b) be the upper and lower hyperboloid with centers a = (x 0 + r 2 e 1 , −R 2 σ), b = (x 0 − r 1 e 1 , R 1 σ) and radii R 2 , R 1 respectively satisfying is an n-ball of radius r 1 internally tangent to ∂Ω ε at x 0 while Q + (R 2 , a) ∩ P (ǫ) is an n-ball of radius r 2 externally tangent to ∂Ω ε at x 0 . By Lemma 4.1 (iii) and (iv), Q ± ∩ Σ = ∅. Hence Moreover, by a simple calculation we have   |∇u ǫ | ≤ C < 1 in Ω where C is independent of ǫ.

5.
The linearized operator and boundary estimates for second derivatives.
In this section we establish boundary estimates for second derivatives of admissible solutions.
Theorem 5.1. Suppose that f satisfies conditions (1.11)-(1.18) with K = K + n . If ε is sufficiently small, then where C is independent of ε.
Define the linearized operator of G at u (recall (1.32)) Note that After some straightforward but tedious calculations we derive It follows that where C 0 denotes a controlled constant independent of ε.
Since γ sj u s = u j /w, Then from (5.4) and (5.7) we obtain (5.9) In the following we denote by C 1 , C 2 , . . . controlled constants independent of ε.
The following lemma is proven in [6]; it applies to our situation since horizontal rotations are isometries for H n+1 .  L(x i u j − x j u i ) = 0, Lu i = 0, 1 ≤ i, j ≤ n.
Proof of Theorem 5.1. Consider an arbitrary point on ∂Ω, which we may assume to be the origin of R n and choose the coordinates so that the positive x n axis is the interior normal to ∂Ω at the origin. There exists a uniform constant r > 0 such that ∂Ω ∩ B r (0) can be represented as a graph Since u = ε on ∂Ω, we see that u(x ′ , ρ(x ′ )) = ε and u αβ (0) = −u n ρ αβ , α, β < n.
6. Completion of the proof of Theorem 1.5 As we emphasized in the introduction, we will derive a global curvature estimate for solutions of the Dirichlet problem (1.33),(1.43) . In Theorem 5.1 of the previous section we have shown that the principal curvatures satisfy 0 < κ i ≤ C, i = 1, . . . , n on Γ = ∂Ω, hence lie in a compact set E of the cone K. Since f (κ) = σ and f (κ) → 0 uniformly on E when any κ i → 0, it follows that (6.1) 1 C < κ i ≤ C on Γ .
Hence by the global maximum principal for principal curvatures proved in Theorem 4.1 of [7], Once more using duality to return to the graph S, we obtain the desired global estimate The proof of Theorem 1.1 follows by letting ε → 0 as mentioned in the introduction.