The Weyl Problem With Nonnegative Gauss Curvature In Hyperbolic Space

Abstract

Abstract. In this paper, we discuss the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature. We prove a priori bounds for the trace of the second fundamental form H and extend the result to n-dimensions. We also obtain an estimate for the gradient of the smaller principal curvature in 2 dimensions.

Hyperbolization of cusps with convex boundary

“…See [CX15] for more precise results if S is homeomoprhic to the sphere, in particular if the curvature is ≥ −1. Theorem 1.10 and Theorem 1.6 immediately give the following.…”

On Isometric Embeddings into Anti-de Sitter Spacetimes

“…Note that for convex surfaces in hyperbolic space with sectional curvature −1, Chang and Xiao [4] proved an a priori bound on the mean curvature with an argument based on Pogorelov's estimate using the additional assumption that the set {K = −1} consists of only finitely many points. Our new argument does not require this finiteness condition.…”

“…HenceK ≤ (max K + κ)(max f ) 4 . We apply [10, Lemma 9.1.1] to conclude that there exists a positive constant R depending only on 1 maxK and the diameter ofΣ such that there exists a ball of radius R insideΣ.…”

Curvature estimates for immersed hypersurfaces in Riemannian manifolds

“…The Weyl isometric embedding problem in hyperbolic space was considered by Pogorelov [30], we also refer [31] and references therein for discussions of isometric embeddings of (S 2 , g) to general 3-dimensional Riemannian manifolds. In hyperbolic case, an explicit mean curvature bound was recently proved by Chang and Xiao [5] for K g ≥ −1 under the condition that the set {K g (X) = −1} is finite, and sequentially by Lin and Wang [24] for general isometric embedded surfaces in…”