Heinz type estimates for graphs in Euclidean space

Let $M^n\subset\mathbb R^{n+1}$ be the graph of a $C^2$-real valued function defined in a closed ball of $\mathbb R^n$. In this work, we obtain upper bounds for $\inf_M|H|$ and $\inf_M|R|$, where $H$ and $R$ are, respectively, the mean curvature and the scalar curvature of $M^n$, generalizing estimates given by Heinz in the case $n=2$ [Math. Annalen 129, 451-454, 1955]. Under the assumption that $M^n$ has negative Ricci curvature, we also obtain an upper bound for $\inf_M|A|$, where $|A|$ is the length of the second fundamental form. As a consequence of this latter estimate one obtains that $\inf |A|=0$ for all entire graphs with negative Ricci curvature in Euclidean space. This gives a partial answer to a question raised by Smith-Xavier [Invent. Math. 90, 443-450, 1987].