Some estimates on stable minimal hypersurfaces in Euclidean space

TL;DR

This paper provides new estimates for stable minimal hypersurfaces in Euclidean space, highlighting limitations of existing proof methods for dimensions six and above, and suggesting potential applications of the derivation techniques.

Contribution

It introduces novel estimates for stable minimal hypersurfaces and discusses their implications for Bernstein theorems, especially in higher dimensions.

Findings

Estimates suggest existing proof methods may not extend to dimension six.

The derivation method could be applicable to other geometric problems.

Highlights limitations of current techniques for stable minimal hypersurfaces in high dimensions.

Abstract

We derive some estimates for stable minimal hypersurfaces in $R^{n+1}$. The estimates are related to recent proofs of Bernstein theorems for complete stable minimal hypersurfaces in $R^{n+1}$ for $3\le n\le 5$ by Chodosh-Li, Chodosh-Li-Minter-Stryker and Mazet. In particular, the estimates indicate that the methods in their proofs may not work for $n=6$, which is observed also by Antonelli-Xu and Mazet. The method of derivation in this work might also be applied to other problems.