CMC Graphs with Planar Boundary in $\mathbb {H}^{2}\times \mathbb {R}$

Abstract

Let $\Omega \subset \mathbb {R}^{2}$ be an unbounded convex domain. It is proven in López (J Diff Equ 171:54–62, 2001) and Ripoll (Pac J Math 198(1):175–196, 2001) that, given $H>0$, there exists a graph $G\subset \mathbb {R}^{3}$ of constant mean curvature H over $\Omega$ with $\partial G=\partial \Omega$ if and only if $\Omega$ is included in a strip of width 1/H. In this paper we obtain results in $\mathbb {H}^{2}\times \mathbb {R}$ in the same direction: given $H\in \left( 0,1/2\right)$, if $\Omega$ is included in a region of $\mathbb {H}^{2}\times \left\{ 0\right\}$ bounded by two hypercircles equidistant $\ell (H)$ to a same geodesic, we show that, if the geodesic curvature of $\partial \Omega$ is bounded from below by $-1,$ then there is an H-graph G over $\Omega$ with $\partial G=\partial \Omega$. We also present more refined existence results involving the curvature of $\partial \Omega$, which can also be less than $-1.$