Graphs of Constant Mean Curvature in Hyperbolic Space

Abstract

We study the problem of finding constant mean curvature graphs over a domainof a totally geodesic hyperplane and an equidistant hypersurface Q of hyperbolic space. We find the existence of graphs of constant mean curvature H over mean convex domains � ⊂ Q and with boundary ∂� for −H∂� 0 is the mean curvature of the boundary ∂� . Here h is the mean curvature respectively of the geodesic hyperplane (h = 0) and of the equidistant hypersurface (0 < |h| < 1). The lower bound on H is optimal.