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The Spectrum and Isometric Embeddings of Surfaces of Revolution
Published online by Cambridge University Press: 20 November 2018
Abstract
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.
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- Copyright © Canadian Mathematical Society 2006