Stable anisotropic minimal hypersurfaces in $\mathbb{R}^{5}$ and $\mathbb{R}^{6}$

TL;DR

This paper proves that under certain smoothness conditions, complete, stable anisotropic minimal hypersurfaces in five and six-dimensional Euclidean spaces must be flat, extending classical minimal surface results.

Contribution

It establishes flatness of stable anisotropic minimal hypersurfaces in higher dimensions when the anisotropic area functional is sufficiently close to the standard area functional.

Findings

Stable anisotropic minimal hypersurfaces are flat in $ extbf{R}^5$ and $ extbf{R}^6$ under smoothness conditions.

Flatness holds when the anisotropic area functional is $C^4$-close to the classical area functional.

The result generalizes classical minimal surface theory to anisotropic settings in higher dimensions.

Abstract

In this paper, we prove that a complete, two-sided, stable anisotropic minimal immersed hypersurface in $\mathbb{R}^{5}$ or $\mathbb{R}^{6}$ is flat, provided the anisotropic area functional is $C^4$-close to the area functional.