Two Rigidity Results for Stable Minimal Hypersurfaces

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 stab...

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Published inGeometric and functional analysis Vol. 34; no. 1; pp. 1 - 18
Main Authors Catino, Giovanni, Mastrolia, Paolo, Roncoroni, Alberto
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.02.2024
Springer Nature B.V
Subjects
Analysis
Hyperplanes
Hyperspaces
Mathematics
Mathematics and Statistics
Minimal surfaces
Rigidity
53C42
Rigidity
Stable Bernstein problem
Stable minimal hypersurface
53C21
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Summary: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.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
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ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-024-00662-1