Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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G. Catino - P. Mastrolia - A. Roncoroni

Two rigidity results for stable minimal hypersurfaces

created by catino on 17 Oct 2023



Inserted: 17 oct 2023
Last Updated: 17 oct 2023

Year: 2022

ArXiv: 2209.10500 PDF


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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