Title: On $δ$-Stable Minimal Hypersurfaces in $\mathbb{R}^{n+1}$ Show Chinese title

Title: 关于$δ$稳定的最小超曲面在$\mathbb{R}^{n+1}$中

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. Show Chinese abstract

Abstract: 在本文中，我们将已为稳定最小超曲面建立的几个结果扩展到$\delta$-稳定最小超曲面。 这些包括当$n \geq 3$和$\delta > \frac{n-2}{n}$时，浸入的$\delta$稳定极小超曲面在$\mathbb{R}^{n+1}$中的正则性和紧性定理，以及适用于$n=3$和$n=4$的适当浸入的$\delta$稳定 Bernstein 定理。 $\delta$的范围是最佳的，因为$\mathbb{R}^{n+1}$中的$n$维悬链面是$\frac{n-2}{n}$稳定的。