Constant mean curvature graphs with prescribed asymptotic values in $\mathbb{E}(-1,\ddot{I}„)$

Abstract

In the homogeneous manifold $\mathbb{E}(-1,\ddot{I}„),$ for $-\frac{1}{2}<H<\frac{1}{2},$ {we define a new product compactification in which the slices ${\left\{t=c\right\}}_{c\in \mathbb{R}}$ are rotational $H$-surfaces. This product compatification is the natural setting where it makes sense to study the asymptotic Dirichlet Problem for the constant mean curvature equation. Indeed, for every rectifiable curve $\hat{I}“$ projecting bijectively onto $\partial\H^2$ we prove the existence of a unique entire $H$-graph that is asymptotic to $\hat{I}“$.} We also find necessary and sufficient conditions for the existence of $H$-graphs over unbounded domains having prescribed, possibly infinite boundary data.

23 pages, 9 figures