Curvature Estimates for Graphs Over Riemannian Domains

Abstract

Let $$M^n$$ be a complete n-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 $${\mathbb {R}}^3$$ which is a graph over an open disk in the plane, we obtain in this work upper estimates for $$\inf |R|$$ , $$\inf |A|$$ and $$\inf |H_k|$$ , where R, |A| 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 $$M^n,\;n\ge 3,$$ is a complete noncompact Riemannian manifold with sectional curvature bounded below by a constant c, and $$\Gamma _f$$ is a graph over M with Ricci curvature less than c, then $$\inf |A|\le 3(n-2)\sqrt{-c}$$ . This result generalizes and improves a theorem of Chern for entire graphs in $$\mathbb R^{n+1}$$ .