Heinz type estimates for graphs in Euclidean space

Abstract:

Let $M^n$ be an entire graph in the Euclidean $(n+1)$-space $\mathbb R^{n+1}$. Denote by $H$, $R$ and $|A|$, respectively, the mean curvature, the scalar curvature and the length of the second fundamental form of $M^n$. We prove that if the mean curvature $H$ of $M^n$ is bounded, then $\inf _M|R|=0$, improving results of Elbert and Hasanis-Vlachos. We also prove that if the Ricci curvature of $M^n$ is negative, then $\inf _M|A|=0$. The latter improves a result of Chern as well as gives a partial answer to a question raised by Smith-Xavier. Our technique is to estimate $\inf |H|,\;\inf |R|$ and $\inf |A|$ for graphs in $\mathbb R^{n+1}$ of $C^2$ real-valued functions defined on closed balls in $\mathbb R^n$.