Stable minimal hypersurfaces in $\mathbb R^6$

TL;DR

This paper proves that in six-dimensional Euclidean space, any complete, two-sided, stable minimal hypersurface must be a flat hyperplane, extending the understanding of stability and flatness in higher dimensions.

Contribution

The paper establishes the flatness of stable minimal hypersurfaces in $ eal^6$, combining previous strategies with new volume estimates to resolve a specific case in geometric analysis.

Findings

Stable minimal hypersurfaces in $ eal^6$ are flat.

The proof relies on volume estimates and stability properties.

Extends known results to six dimensions.

Abstract

Following the strategy developed by Chodosh, Li, Minter and Stryker, and using the volume estimate of Antonelli and Xu, we prove that, in $\mathbb R^6$, a complete, two-sided, stable minimal hypersurfaces is flat.