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

Abstract

It is known that for $\Omega \subset {\mathbb{R}}^2$ an unbounded convex domain and $H>0$, 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 equidistant hypercycles $\ell (H)$ apart, 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 $-1.$

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