Two Rigidity Results for Stable Minimal Hypersurfaces

Abstract

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 R 4 , while they do not exist in positively curved closed Riemannian ( n +1)-manifold when n ≤5; in particular, there are no stable minimal hypersurfaces in S n +1 when n ≤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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