On $\delta$-Stable Minimal Hypersurfaces in $\mathbb{R}^{n+1}$

TL;DR

This paper generalizes key regularity, compactness, and Bernstein theorems for stable minimal hypersurfaces to the broader class of $oldsymbol{ extdelta}$-stable hypersurfaces in Euclidean space, establishing optimal stability ranges.

Contribution

It extends classical results to $oldsymbol{ extdelta}$-stable hypersurfaces, including regularity, compactness, and Bernstein theorems, with optimal stability bounds.

Findings

Regularity and compactness theorems for $oldsymbol{ extdelta}$-stable hypersurfaces

Bernstein theorem for $oldsymbol{ extdelta}$-stable hypersurfaces in dimensions 3 and 4

Optimal stability range identified with the $n$-dimensional catenoid

Abstract

In this paper, we extend several results established for stable minimal hypersurfaces to $\delta$-stable minimal hypersurfaces. These include the regularity and compactness theorems for immersed $\delta$-stable minimal hypersurfaces in $\mathbb{R}^{n+1}$ when $n \geq 3$ and $\delta > \frac{n-2}{n}$, as well as the $\delta$-stable Bernstein theorem for $n=3$ and $n=4$ for properly immersion. The range of $\delta$ is optimal, as the $n$-dimensional catenoid in $\mathbb{R}^{n+1}$ is $\frac{n-2}{n}$-stable.