Curvature estimates for hypersurfaces of constant curvature in hyperbolic space

Abstract

In this note, we prove that for every $0<\sigma <1$, there exists a smooth complete hypersurface $\Sigma$ in ${\mathbb{H}}^{n+1}$ with prescribed asymptotic boundary $\partial \Sigma =\Gamma$ at infinity, whose principal curvatures $\kappa =({\kappa }_1,\dots ,{\kappa }_n)$ lie in a general cone $K$ and satisfy $f(\kappa )=\sigma$ at each point of $\Sigma$. Previously, the problem has been studied by Guan-Spruck in [J. Eur. Math. Soc. (JEMS) 12 (2010), no. 3, 797-817], and they proved the existence result for $\sigma \in ({\sigma }_0,1)$, where ${\sigma }_0>0$. A major ingredient of our proof is a refined curvature estimate of theirs that is applicable when the curvature function $f(\kappa )$ has controllable partial derivatives, but it is adequate for our purpose; specifically, we solve the problem for $f=H_k/H_{k-1}$ in the $k$-th Garding cone where $H_k$ is the normalized $k$-th elementary symmetric polynomial and $1\le k\le n$.

arXiv comment10 pages; final version; to appear in Mathematical Research Letters