Curvature estimates for hypersurfaces of constant curvature in hyperbolic space

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 $\kappa=(\kappa_1,\ldots,\kappa_n)$ 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 $1 \leq k \leq n$.