Starshaped compact hypersurfaces with prescribed m-th mean curvature in hyperbolic space

We study the existence of starshaped compact hypersurfaces with prescribed m-th mean curvature in hyperbolic space.


Introduction
Let S n be the unit sphere in the Euclidean space R n+1 , and let e be the standard metric on S n induced from R n+1 . Suppose that (u, ρ) are the spherical coordinates in R n+1 , where u ∈ S n , ρ ∈ [0, ∞). By choosing the smooth function ϕ(ρ) := sinh 2 ρ on [0, ∞) we can define a Riemannian metric h on the set {(u, ρ) : u ∈ S n , 0 ≤ ρ < ∞} as follows This gives the space form R n+1 (−1) which is the hyperbolic space H n+1 with sectional curvature −1. For a smooth hypersurface M in R n+1 (−1), we denote by λ 1 , · · · , λ n its principal curvatures with respect to the metric g := h| M . Then, for each 1 ≤ k ≤ n, the k-th mean curvature of M is defined as Let ψ(u, ρ), u ∈ S n , ρ ∈ (0, ∞), be a given positive smooth function satisfying suitable conditions. We are interested in the existence of a smooth hypersurface M embedded in R n+1 (−1) as a graph over S n so that its k-th mean curvature is given by ψ. We refer the readers to [7] and [5] for the introductory material and the history of this problem.
Then there exists a positive smooth k-admissible function z on S n such that the closed , and its k-th mean curvature is given by ψ: H k (λ 1 (z(u)), · · · , λ n (z(u))) = ψ(u, z(u)) ∀ u ∈ S n .
In the Euclidean space (R n+1 (0)), such results were obtained in the case k = 0 by Bakelman and Kantor [3], [4] and by Treibergs and Wei [19], in the case k = n by Oliker [15], and for general k by Caffarelli, Nirenberg and Spruck [7]. In the elliptic space (R n+1 (+1)), such result is the combination of the work of Barbosa, Lira and Oliker [5] and that of Li and Oliker [14]. The k = n case in Theorem 1 was established by Oliker [16]. Our proof of Theorem 1 uses the C 0 and C 1 a priori estimates obtained in [5] and the arguments in [14] which is based on the degree theory for fully nonlinear elliptic operator of second order developed in [12]. The main work for us to prove Theorem 1 is to give the C 2 a priori estimates. In establishing the C 2 estimates, we make use Lemma 2, a quantitative version of a theorem of Davis [9] which, to our knowledge, was given in [1]. The theorem in [9] says that a rotationally invariant function on symmetric matrices is concave if and only if it is concave on the diagonal matrices, while Lemma 2 allows the use of this term in making C 2 a priori estimates. The use of such a concave term in C 2 estimates for solutions of the Monge-Ampère equation has been extensive, see e.g. Calabi [8] and Pogorelov [17]. The use of Lemma 2 in C 2 estimates for solutions of more general equations can be found in [1], [2], [11], [18], [20] and [21].
Acknowledgement. The work of the second author is partially supported by NSF grant DMS-0401118.

Some fundamental formulae
Let us define a function f on Γ k by where λ := (λ 1 , · · · , λ n ) ∈ Γ k . It is well known that f is smooth, positive, concave, and strictly increasing with respect to each variable, see e.g. [6]. Now our problem is equivalent to finding a smooth positive k-admissible function z on S n so that Suppose now M is the graph of a smooth positive k-admissible function z on S n . Let us recall the formulae given in [5] for the components of g = (g ij ) and B = (b ij ) on M under a local coordinate. Let θ 1 , · · · , θ n be a smooth local coordinate of S n , which of course gives a local coordinate of M. If we denote by {e ij } the components of e under this local coordinate, and set z i = ∂z ∂θ i and z ij = ∂ 2 z ∂θ i ∂θ j , then and where (g ij ) = (g ij ) −1 , (e ij ) = (e ij ) −1 , and ∇ ′ denotes the Levi-Civita connection on S n . Moreover, for the second fundamental form we have We also need the following well-known fundamental equations for a hypersurface M in R n+1 (−1): Gauss equation: Ricci equation: where R ijkl denotes the Riemannian curvature tensor of M, and ∇ i and ∇ i ∇ j the covariant differentiations in the metric g with respect to some local coordinates on M.
As the preparation for deriving the C 2 -estimates, let us introduce the following two We have Lemma 1. For τ and η the following equations hold Proof. These formulae have been derived in [5] by using another model of R n+1 (−1). In fact, we can show them directly. Since (10) is an immediate consequence of (9), (11) and the Codazzi equation (5), it suffices to verify (9) and (11). Let c(ρ) = cosh(ρ) and s(ρ) = sinh(ρ). Then Noting that ∇ q η = −sz q , we have from (3) and (4) that Let us now verify (11) for any fixedū ∈ S n . Noting that the both sides of (11) are tensorial, we may assume that the local coordinates are chosen such that Thus the corresponding Christoffel symbols of S n are given by This, together with (2), gives Noting that ∇ ij z = z ij atū, we thus have Therefore But from (2) and (4) we can see that the right hand side of the above equation is exactly τ b ij + ηg ij .

C 2 -estimates
Now we are in a position to derive the C 2 estimates for any smooth positive k-admissible solutions of (1) in R n+1 (−1). Let us set We will achieve our aim by choosing suitable test function and making full use of the terms involving F ij,kl . In particular, we need the following Lemma 2. ( [1]) For any symmetric matrix (η ij ) there holds where the second term on the right-hand side must be interpreted as a limit whenever This result was, to our knowledge, first stated in [1]; for proofs one may consult [11,2].

Proof.
We will estimate the maximal principal curvature of M. Since z is kadmissible, this estimate, together with the C 0 and C 1 bounds of z and the equation (4), implies an estimate for z C 2 (S n ) . Consider the function where u ∈ S n , ξ is a unit tangent vector of M at (u, z(u)), τ and η are defined as in (8), and the function Φ and the constant β > 0 will be determined later. Suppose the maximum of W is attained at some pointū ∈ S n in the unit tangential directionξ of M at (ū, z(ū)). We may choose the local coordinates θ 1 , · · · , θ n aroundū such that g ij = δ ij and ∂g ij ∂θ k = 0 atū.
Moreover, sinceξ is the maximal principal direction of M at (ū, z(ū)), such coordinates can be chosen so that {b ij } is diagonal atū and b 11 (ū) = B(ξ,ξ). Consider the local function Z = b 11 /g 11 . By direct calculation we have atū that

It is clear that the function
has a local maximum atū. Thus atū and the matrix Since {b ij } is diagonal atū, {F ij } is also diagonal there and F ii = f i . For simplicity, we let λ i = b ii (ū) and assume λ 1 ≥ λ 2 ≥ · · · ≥ λ n , moreover we may assume λ 1 ≥ 1.
Then, see lemma 2 in [10] or lemma A.2 in [13], we have f 1 ≤ f 2 ≤ · · · ≤ f n . It follows from (15) that Now we take the covariant differentiation on (1) to get ¿From (5), (6) and (7) it follows that This shows that where T := i f i . Since the degree one homogeneity of f implies i f i λ i = ψ, the above equation together with (17) gives Plugging this into (16), noting that ψ ≥ c 0 > 0 and λ 1 ≥ 1, we therefore obtain
Combining (18), (19), (20) and (21), we thus obtain Now we will use Lemma 2, similar to the way used in [18]. Case 1. λ n < −θλ 1 for some positive constant θ (to be chosen later). In this case, using the concavity of F we may discard the last term on the right hand side of (22) since it is nonnegative. Also from (14) we have for any ε > 0 1 Therefore, from (22) it yields Using (9) we have In order to choose Φ, let a > 0 be a positive number such that τ ≥ 2a which is guaranteed by our assumption. Then we define It is easy to check that Φ ′′ − (1 + ε)(Φ ′ ) 2 < 0. Moreover, for ε = a 2 2c 1 we have Therefore we get from (24) that Since λ n ≤ −θλ 1 and f n ≥ 1 . This clearly implies λ 1 is bounded from above.