Curvature Estimates for Graphs Over Riemannian Domains

Let M-n be a completen-dimensional Riemannian manifold and Gamma(f) the graph of a C-2-function f defined on a metric ball of M-n. In the same spirit of the estimates obtained by Heinz for the mean and Gaussian curvatures of a surface in R-3 which is a graph over an open disk in the plane, we obtain in this work upper estimates for in f vertical bar R vertical bar,in f vertical bar A vertical bar and in f vertical bar H-k vertical bar, where R, vertical bar A vertical bar and H-k are, respectively, the scalar curvature, the norm of the second fundamental form and the k-th mean curvature of Gamma(f). From our estimates we obtain several results for graphs over complete manifolds. For example, we prove that if M-n, n >= 3, is a complete noncompact Riemannian manifold with sectional curvature bounded below by a constantc, and Gamma(f) is a graph over M with Ricci curvature less thanc, then inf vertical bar A <= 3(n-2)root-c. This result generalizes and improves a theorem of Chern for entire graphs in Rn+1.