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The Spectrum and Isometric Embeddings of Surfaces of Revolution

Published online by Cambridge University Press:  20 November 2018

Martin Engman
Martin Engman*
Affiliation:
Departamento de Ciencias y Tecnología, Universidad Metropolitana, San Juan, Puerto Rico 00928 email: um_mengman@suagm.edu
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Abstract

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A sharp upper bound on the first ${{S}^{1}}$ invariant eigenvalue of the Laplacian for ${{S}^{1}}$ invariant metrics on ${{S}^{2}}$ is used to find obstructions to the existence of ${{S}^{1}}$ equivariant isometric embeddings of such metrics in (${{\mathbb{R}}^{3}}$, can). As a corollary we prove: If the first four distinct eigenvalues have even multiplicities then the metric cannot be equivariantly, isometrically embedded in (${{\mathbb{R}}^{3}}$, can). This leads to generalizations of some classical results in the theory of surfaces.

Keywords

58J5058J5353C2035P15
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 49 , Issue 2 , 01 June 2006 , pp. 226 - 236
Copyright
Copyright © Canadian Mathematical Society 2006

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