Rigidity of minimal surfaces in S 3
type: Item
instance of: scholarly article
title: Rigidity of minimal surfaces in S 3
zbMATH Open document ID: 0643.53042
DOI: 10.1007/BF01258661
author: Jayakumar Ramanathan
published in: Manuscripta Mathematica
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155290
review text: Let M 2 be a two-dimensional Riemannian manifold and x: M \(2\to S\) 3 be an isometric immersion of M 2 into the unit sphere S \(3=\{x\in R\) 4; \(| x| =1\}\). The following theorem is proved: Assume that M 2 is compact and let x: M \(2\to S\) 3 be an isometric minimal immersion. Then there exist at most finitely many noncongruent minimal immersions with the same induced metric as x. The proof uses the family of minimal immersions associated to x, the fact that if the Gaussian curvature K of x satisfies \(K\not\equiv 1\), then the normal map of x is again minimal, and the fact that the coordinate functions of x are eigenfunctions of the Laplacian on M. An example is given to show that the above does not hold for minimal immersions of a torus T 2 into S 7.
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 4048578
zbMATH Keywords: rigidity
zbMATH Keywords: isometric minimal immersion
reviewed by: Manfredo P. do Carmo
MaRDI profile type: Publication
cites work: Minimal Surfaces of Constant Curvature in S n
cites work: Minimal submanifolds of low cohomogeneity
cites work: Complete minimal surfaces in \(S^ 3\)
description: scientific article; zbMATH DE number 4048578