Two rigidity results for stable minimal hypersurfaces

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 $n\le 5$; in particular, there are no stable minimal hypersurfaces in ${\mathbb{S}}^{n+1}$ when $n\le 5$. 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.

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