Prescribed mean curvature hypersurfaces in Hn+1(-1) with convex planar boundary, I

We study immersed prescribed mean curvature compact hypersurfaces with boundary in Hn+1 (-l). When the boundary is a convex planar smooth manifold with all principal curvatures greater than 1, we solve a nonparametric Dirichlet problem and use this, together with a general flux formula, to prove a parametric uniqueness result, in the class of all immersed compact hypersurfaces with the same boundary. We specialize this result to a constant mean curvature, obtaining a characterization of totally umbilic hypersurface caps.