Total scalar curvature and \(L^2\) harmonic 1-forms on a minimal hypersurface in Euclidean space
type: Item
instance of: scholarly article
title: Total scalar curvature and \(L^2\) harmonic 1-forms on a minimal hypersurface in Euclidean space
zbMATH Open document ID: 1002.53042
DOI: 10.1023/A:1014211121535
author: Gabjin Yun
published in: Geometriae Dedicata
publication date: 2002-05-30T00:00:00Z
review text: Let \(M\), \(n=\dim M\geq 3\), be a complete oriented immersed minimal hypersurface in the Euclidean space \(\mathbb R^{n+1}\). If the total scalar curvature of \(M\) is less than \((1/C_s)^{n/2}\), where \(C_s\) is the Sobolev constant for \(M\), then there are no \(L^2\) harmonic 1-forms on \(M\). Consequently, such a hypersurface contains no nontrivial harmonic functions with finite Dirichlet integral and so it has only one end. This finally implies that \(M\) is a hyperplane.
reviewed by: Zbigniew Olszak
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 58C40
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 1747813
zbMATH Keywords: hyperplane
zbMATH Keywords: minimal hypersurface
zbMATH Keywords: total scalar curvature
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1747813