Gradient estimates for the Green kernel under spectral Ricci bounds, and the stable Bernstein theorem in $\mathbb{R}^4$

TL;DR

This paper introduces a new method for integral inequalities on stable minimal hypersurfaces, proving that in A4-dimensional Euclidean space, such hypersurfaces are hyperplanes, based on spectral Ricci curvature bounds.

Contribution

It provides a novel approach to prove Bernstein-type theorems using spectral Ricci bounds and gradient estimates for the Green kernel.

Findings

Stable minimal hypersurfaces in A4 are hyperplanes.

Established sharp gradient estimates for the Green kernel under spectral Ricci bounds.

Extended previous work by Colding on Green kernel estimates.

Abstract

We describe a method to prove new integral inequalities for stable minimal hypersurfaces in Euclidean space. As an application, we give a simple proof that complete, two sided, stable minimal hypersurfaces in $\mathbb{R}^4$ are hyperplanes. A core part of the argument hinges on the fact that stable minimal hypersurfaces in non-negatively curved spaces are examples of manifolds with a spectral Ricci curvature lower bound; in particular, we prove a sharp pointwise gradient estimate for the Green kernel on non-parabolic manifolds with spectral Ricci lower bounds, extending previous work by Colding.