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.
The paper extends regularity, compactness, and Bernstein theory from stable minimal hypersurfaces to the weaker notion of $\delta$-stability.
Regularity and compactness
For $n\geq3$ and $\delta>(n-2)/n$, sequences with uniform area control admit varifold subsequences whose singular sets satisfy an explicit dimension bound determined by $\delta$.
Bernstein theorem
For $n=3,4$, complete two-sided $\delta$-stable immersed hypersurfaces are flat under either a stronger numerical stability range or the combination of simple connectivity, properness, and finitely many ends.
Sharp threshold
The $n$-dimensional catenoid is exactly $(n-2)/n$-stable. Simons-type cones also identify the sharp threshold for the cone-classification component of the theory.
The regularity argument combines a Caccioppoli inequality, De Giorgi iteration, an $\varepsilon$-regularity theorem for the second fundamental form, and tangent-cone analysis. A sharp classification of $\delta$-stable minimal cones supplies the dimension reduction. In low dimensions, point-picking and curvature estimates reduce the Bernstein statements to the global flatness results.
My research interests include Geometric Analysis and Partial Differential Equations.