Two rigidity results for stable minimal hypersurfaces

TL;DR

This paper proves two rigidity theorems for stable minimal hypersurfaces, showing they are hyperplanes in Euclidean space and do not exist in certain positively curved manifolds for dimensions up to five.

Contribution

It establishes new rigidity results for stable minimal hypersurfaces using conformal methods, extending previous work and covering cases in dimensions up to five.

Findings

Stable minimal hypersurfaces in ^4 are hyperplanes.

No stable minimal hypersurfaces exist in ^{n+1} for n  in positively curved closed manifolds.

The results rely on a conformal method inspired by Fischer-Colbrie.

Abstract

The aim of this paper is to prove two results concerning the rigidity of complete, immersed, orientable, stable minimal hypersurfaces: we show that they are hyperplane in $\mathbb{R}^4$, while they do not exist in positively curved closed Riemannian $(n+1)$-manifold when $n\leq 5$; in particular, there are no stable minimal hypersurfaces in $\mathbb{S}^{n+1}$ when $n\leq 5$. The first result was recently proved also by Chodosh and Li, and the second is a consequence of a more general result concerning minimal surfaces with finite index. Both theorems rely on a conformal method, inspired by a classical work of Fischer-Colbrie.