The spectrum and isometric embeddings of surfaces of revolution

1999/10/07 by Martin Engman, Engman, Martin
Mathematics · #58G25 #58G30 (Primary) 35P15 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:35P15 #msc:58G25 #msc:58G30

paper · pdf · doi:10.48550/arxiv.math/9910038

Latex, 11 pages New version contains a new reference and minor stylistic changes

arxiv created 1999/10/19 · arxiv updated 2009/11/30

Abstract

An upper bound on the first S¹ invariant eigenvalue of the Laplacian for invariant metrics on the 2-sphere is used to find obstructions to the existence of isometric embeddings of such metrics in (R³,can). As a corollary we prove: If the first four distinct eigenvalues have even multiplicities then the surface of revolution cannot be isometrically embedded in (R³,can). This leads to a generalization of a classical result in the theory of surfaces.

Related

• Inverse Spectral Problem for Surfaces of Revolution
• Inverse spectral theory and the Minkowski problem for the surface of revolution
• Sharp Bounds for Eigenvalues and Multiplicities on Surfaces of\n Revolution
• Certain classifications on surfaces of revolution in a semi-isotropic space
• Extremal metric for the first eigenvalue on a Klein bottle