On the asymptotic Plateau problem for CMC hypersurfaces in hyperbolic space

Abstract

Let ${\mathbb{R}}_{+}^{n+1}$ \ be the half-space model of the hyperbolic space ${\mathbb{H}}^{n+1}.$ It is proved that if $\Gamma \subset \left\{x_{n+1}=0\right\}\subset {\partial }_{\infty }{\mathbb{H}}^{n+1}$ is a bounded $C^0$ Euclidean graph over $\left\{x_1=0,x_{n+1}=0\right\}$ then, given $\left|H\right|<1,$ there is a complete, properly embedded, CMC $H$ hypersurface $\Sigma$ of ${\mathbb{H}}^{n+1}$ such that ${\partial }_{\infty }S=\Gamma \cup \left\{x_{n+1}=+\infty \right\}.$ This result can be seen as a limit case of the existence theorem proved by B. Guan and J. Spruck in \cite{GS} on CMC $\left|H\right|<1$ radial graphs with prescribed $C^0$ asymptotic boundary data.

This is a new version of arXiv:1309.3644 ; Improvements have been made in the proof of the main theorem