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

Haizhong Li,
Han Hong,
Gaoming Wang

July, 2024

### 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.

Publication

Submitted. 2407.03222, 2024

###### PostDoc

My research interests include Geometric Analysis and Partial Differential Equations.