Heinz type estimates for graphs in Euclidean space

TL;DR

This paper extends Heinz's estimates for mean and scalar curvature to higher-dimensional graphs in Euclidean space, providing bounds and conditions related to Ricci curvature and second fundamental form.

Contribution

It generalizes Heinz's curvature estimates to n-dimensional graphs and explores implications for negative Ricci curvature, partially answering a longstanding question.

Findings

Upper bounds for infimum of mean and scalar curvature on graphs.

Infimum of second fundamental form length is zero for entire graphs with negative Ricci curvature.

Provides partial answer to a question by Smith-Xavier about curvature bounds.

Abstract

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].