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

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 $n\ge 3$ 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.