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arXiv · 2209.10500

Two rigidity results for stable minimal hypersurfaces

Giovanni Catino·Paolo Mastrolia·Alberto Roncoroni
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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.

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Giovanni Catino, Paolo Mastrolia, Alberto Roncoroni. 2022-09-21. Two rigidity results for stable minimal hypersurfaces. https://arxiv.org/abs/2209.10500

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