Some characterization, uniqueness and existence results for Euclidean graphs of constant mean curvature with planar boundary

We establish the existence and uniqueness of solutions to the Dirichlet pro blem for the cmc surface equation, including the minimal one, for zero boun dary data, in certain domains of the plane. We obtain results that characte rize the sphere and cmc graphs among compact embedded cmc surfaces with pla nar boundary satisfying certain geometric conditions. We also find conditio ns that imply that a compact embedded cmc surface which is a graph near the boundary is indeed a global graph.