Compact minimal hypersurfaces with index one in the real projective space
type: Item
instance of: scholarly article
title: Compact minimal hypersurfaces with index one in the real projective space
zbMATH Open document ID: 0977.53059
DOI: 10.1007/PL00000373
author: Manuel Ritoré
author: Antonio Ros
author: Manfredo P. do Carmo
published in: Commentarii Mathematici Helvetici
publication date: 2002-01-07T00:00:00Z
review text: Noting that any compact closed minimal hypersurface in a real projective space (of general dimension) cannot be stable, i.e., cannot have index zero, the authors consider the next natural question about hypersurfaces of low index: Which hypersurfaces have index one? They briefly give a bit of history about the subject that demonstrates how natural the question is. For example, analogous questions have been answered in Euclidean three-space and in spherical space and in the three-dimensional projective space and in flat three tori, with some of the answers coming from the authors themselves. The index is defined as the number of negative eigenvalues of the Jacobi operator, and for this operator to be defined on functions over the hypersurface, the authors require that the hypersurface be two-sided (this means that the hypersurface has a globally defined unit normal vector field). The main result is that the only compact two-sided minimal hypersurfaces with index one in the real projective space are the totally geodesic spheres and the minimal Clifford hypersurfaces. Clifford hypersurfaces are products of two spheres whose dimensions add up to one less than the dimension of the ambient projective space, and they are briefly described in this paper. The proof is based on a clever choice of test functions which reduce the problem to a result in a paper of \textit{S. S. Chern, M. do Carmo}, and \textit{S. Kobayashi} [Funct. Anal. Relat. Fields, Conf. Chicago 1968, 59-75 (1970; Zbl 0216.44001)]. The authors then note that totally geodesic spheres and the minimal Clifford hypersurfaces are the only surfaces in real projective space that can be volume preserving stable, which means that the Jacobi operator has no negative eigenvalues in the space of functions on the hypersurfaces which have mean zero (i.e., whose integral over the hypersurface is zero).
reviewed by: Wayne Rossman
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 1523504
zbMATH Keywords: two-sided minimal hypersurface
zbMATH Keywords: real projective space
zbMATH Keywords: index
zbMATH Keywords: stable
zbMATH Keywords: volume preserving stable
zbMATH Keywords: Clifford hypersurface
MaRDI profile type: Publication
full work available at URL: pl00000373
OpenAlex ID: W2090944342
description: scientific article; zbMATH DE number 1523504