Existence of prescribed mean curvature graphs in hyperbolic space

Abstract

In this paper we are concerned with questions of existence and uniqueness of graph-like prescribed mean curvature hypersurfaces in hyperbolic space ?n+1. In the half-space setting, we will study radial graphs over the totally geodesic hypersurface \(\). We prove the following existence result: Let \(\) be a bounded domain of class \(\) and let \(\). If \(\) everywhere on \(\), where \(\) denotes the hyperbolic mean curvature of the cylinder over \(\), then for every \(\) there is a unique graph over \(\) with mean curvature \(\) attaining the boundary values \(\) on \(\). Further we show the existence of smooth boundary data such that no solution exists in case of \(\) for some \(\) under the assumption that \(\) has a sign.