On minimal hypersurfaces with finite harmonic indices
type: Item
instance of: scholarly article
title: On minimal hypersurfaces with finite harmonic indices
zbMATH Open document ID: 1023.53046
DOI: 10.1215/S0012-7094-01-11021-1
author: Jiaqiang Mei
author: Senlin Xu
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: In this interesting paper, the authors introduce the concept of harmonic index and harmonic stability for complete, minimal hypersurfaces \(M\) in \(\mathbb{R}^{n+1}\), \(n\geq 3\). It is shown that a stable complete, minimal hypersurface in \(\mathbb{R}^{n+1}\) is also harmonically stable. Various results are proved. For instance, if \(e(M)\) denotes the number of ends of \(M\) and \(h(M)\) is the harmonic index of \(M\), then \(e(M)=h(M)+1\) and \[ h(M)\leq c(n)\int_M|A|^n dM, \] where \(c(n)\) is a constant dependent only on \(n\) and \(|A|\) is the norm of the second fundamental form of \(M\). If the integral of the second member of the above inequality is finite, \(M\) is said to have finite total curvature. It is proved that the only orientable complete harmonic stable hypersurfaces with finite total curvature are hyperplanes. To each end \(E_i\) in \(M\), the authors associate a nonnegative harmonic function \(u_i\). These functions span an \(e(M)\)-dimensional vector space and form a partition of unity of \(M\). Furthermore, if \(V\) is the space of bounded harmonic functions on a complete minimal hypersurface \(M\) with finite total curvature, then \(\dim V=e(M)\) and \(V=\text{span}\{u_1,u_2,\dots, u_{e(M)}\}\).
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 1820853
zbMATH Keywords: harmonic index
zbMATH Keywords: harmonic stability
zbMATH Keywords: minimal hypersurfaces
zbMATH Keywords: number of ends
zbMATH Keywords: finite total curvature
reviewed by: Manfredo P. do Carmo
MaRDI profile type: Publication
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full work available at URL: s0012-7094-01-11021-1
OpenAlex ID: W2066244362
Wikidata QID: Q125938986
description: scientific article; zbMATH DE number 1820853