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# The Spectrum and Isometric Embeddings of Surfaces of Revolution

Published:2006-06-01
Printed: Jun 2006
• Martin Engman
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## 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 $(\R^3,\can)$. As a corollary we prove: If the first four distinct eigenvalues have even multiplicities then the metric cannot be equivariantly, isometrically embedded in $(\R^3,\can)$. This leads to generalizations of some classical results in the theory of surfaces.
 MSC Classifications: 58J50 - Spectral problems; spectral geometry; scattering theory [See also 35Pxx] 58J53 - Isospectrality 53C20 - Global Riemannian geometry, including pinching [See also 31C12, 58B20] 35P15 - Estimation of eigenvalues, upper and lower bounds

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