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

Abstract

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

37 pages. Comments are welcome