A gap theorem for hypersurfaces of the sphere with constant scalar curvature one
type: Item
instance of: scholarly article
title: A gap theorem for hypersurfaces of the sphere with constant scalar curvature one
zbMATH Open document ID: 1032.53045
author: Hilário Alencar
author: Walcy Santos
author: Manfredo P. do Carmo
published in: Commentarii Mathematici Helvetici
publication date: 2002-12-01T00:00:00Z
review text: Let \( x \) be an immersion of a \( n \)-dimensional (connected and oriented) manifold \( M^{n} \) into a space form of dimension \( n+1 \) and curvature \( c \), denoting by \( S_{r} \) the \( r \)-th elementary symmetric function of the principal curvatures \( k_{i} \). An identity of H. Weyl for the Euclidean surface case \( c = 0, n = 2 \) has been extended to the general situation, stating \( \triangle S_{r} = L_{r-1}(S_{1})+\Omega_{r}+\Pi_{r} \), where \( \triangle \) is the Laplace/Beltrami operator of the first fundamental form, \( L_{r-1} \) the second order differential operator of the corresponding Newtonian \( P_{r-1} \), \( \Omega_{r} \) a certain symmetric polynomial in the \( k_{i} \), and \( \Pi_{r} \) a mixed invariant of the shape operator \( A \) and its covariant derivative \( \nabla A \). The \( \Omega_{r} \) can be brought to the invariant form \( \Omega_{r} = \Omega_{r}^{(0)}+c\Omega_{r}^{(1)} \) with \( \Omega_{r}^{(0)}:= 2rS_{2}S_{r}-(r+1)S_{1}S_{r+1}-(r-1)S_{1}^{2}S_{r} \) and \( \Omega_{r}^{(1)}:=nrS_{r}-(n-r+1)S_{1}S_{r-1} \). See for this the reviewer's paper [Math. Ann. 270, 125-145 (1985; Zbl 0536.53054)] which also contains applications to compact hypersurfaces with one constant \( S_{r} \). Various other consequences have been drawn from special cases of the Weyl identities; see the references in this paper. For instance, \textit{S. Y. Cheng} and \textit{S. T. Yau} showed that a compact hypersurface in the unit sphere \( S^{n+1} \) of nonnegative sectional curvature with \( S_{2} = \text{ const.} \geq 0 \) is either a small or great subsphere or a Clifford torus [Math. Ann. 225, 195--204 (1977; Zbl 0349.53041)]. The Newtonians were considered by \textit{K. Voß} in connection with the variation of the \( S_{r} \) [Math. Ann. 131, 180--218 (1956; Zbl 0073.38401)]. Recently, related results and extensions have been established by \textit{P. Kohlmann}, electronically available under www.mathematik.uni-dortmund.de/lsvii/Preprints/. The pr…
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1837406
zbMATH Keywords: scalar curvature
zbMATH Keywords: sphere
zbMATH Keywords: Clifford torus
zbMATH Keywords: index one
zbMATH Keywords: projective space
reviewed by: Rolf Walter
MaRDI profile type: Publication
full work available at URL: s00014-002-8351-1
OpenAlex ID: W2013669607
DOI: 10.1007/S00014-002-8351-1
Recommended article: Rotational linear Weingarten hypersurfaces in the Euclidean sphere \(\mathbb S^{n+1}\)
Recommended article: The convex-concave principle and uniqueness for hypersurfaces in space forms
Recommended article: Compact graphs over a sphere of constant second order mean curvature
Recommended article: Integral formula of Minkowski type and new characterization of the Wulff shape
Recommended article: The first eigenvalue for compact hypersurfaces in space form \(S^{n+1}(c)\)
Recommended article: A note on compact Weingarten hypersurfaces embedded in \(\mathbb{R}^{n + 1}\)
Recommended article: Q4887620
Recommended article: Characterizations of Euclidean spheres
Recommended article: A function of direction in a Weyl hypersurface
Recommended article: Q5089702
description: scientific article; zbMATH DE number 1837406