Interior Gradient Estimates and Existence Theorems for Constant Mean Curvature Graphs in Mn × R

In this paper we establish a priori interior gradient estimates and existence theorems for n dimensional graphs S = {(x, u(x)) : x ∈ Ω} of constant mean curvature H > 0 in an n+ 1 dimensional Riemannian manifolds of the form M × R where M is simply connected and complete and Ω is a bounded domain in M. Our aim is to illustrate the use of intrinsic methods that hold in great generality to obtain apriori estimates. In particular, we shall solve the Dirichlet problem for constant mean curvature graphs analogous to the results of Serrin [14, 15].