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 $\mathbb R^{n+1}$.