On δ-Stable Minimal Hypersurfaces in Rⁿ⁺¹

In this paper, we extend several results established for stable minimal hypersurfaces to δ-stable minimal hypersurfaces. These include the regularity and compactness theorems for immersed δ-stable minimal hypersurfaces in Rⁿ⁺¹ when n ≥ 3 and δ > n-2/n, as well as the δ-stable Bernstein theorem for $n=3$ and $n=4$ for properly immersion. The range of δ is optimal, as the n-dimensional catenoid in Rⁿ⁺¹ is n-2/n-stable.