Calculus of Variations and Geometric Measure Theory

G. Catino - P. Mastrolia - A. Roncoroni

Two rigidity results for stable minimal hypersurfaces

created by catino on 21 Sep 2022
modified on 27 Jul 2023


Accepted Paper

Inserted: 21 sep 2022
Last Updated: 27 jul 2023

Journal: Geom. Funct. Anal.
Year: 2023


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.