Curvature pinching and eigenvalue rigidity for minimal submanifolds
type: Item
instance of: scholarly article
title: Curvature pinching and eigenvalue rigidity for minimal submanifolds
zbMATH Open document ID: 0563.53046
DOI: 10.1007/BF01162343
author: Antonio Ros
author: Francisco Urbano
author: Sebastián Montiel
published in: Mathematische Zeitschrift
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173697
review text: Simon's formula is a basic and useful tool in the study of some problems of global rigidity for minimal submanifolds immersed in kind Riemannian manifolds. In this paper a modified version of this formula is obtained, that even though this essentially coincides with the classic version, it allows a more controlled treatment and from new view points. We give sharp applications in two different contexts: First we obtain a complete solution to the pinching problem for the Ricci curvature of n-dimensional compact totally real sumanifolds immersed minimally into the n- dimensional complex projective space, and we characterize in this way six nice submanifolds of this family. Second we characterize the compact symmetric spaces of rank one among all compact Riemannian manifolds which admit a minimal immersion into a sphere by the first eigenfunctions as the extreme cases in several inequalities involving only the simplest spectral invariants: The first and the second eigenvalue of the Laplacian, the volume, the integral of the scalar curvature and the multiplicity of the first eigenvalue.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 3897778
zbMATH Keywords: Simons formula
zbMATH Keywords: global rigidity
zbMATH Keywords: minimal submanifolds
zbMATH Keywords: pinching problem
zbMATH Keywords: Ricci curvature
zbMATH Keywords: totally real submanifolds
zbMATH Keywords: complex projective space
zbMATH Keywords: symmetric spaces
zbMATH Keywords: spectral invariants
MaRDI profile type: Publication
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OpenAlex ID: W2035406101
description: scientific article; zbMATH DE number 3897778