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Fundamental Tone, Concentration of Density, and Conformal Degeneration on Surfaces

Published online by Cambridge University Press:  20 November 2018

Alexandre Girouard
Alexandre Girouard*
Affiliation:
Département deMathématiques et Statistique, Université de Montréal, Montréal, Canada H3C 3J7, girouard@dms.umontreal.ca
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Abstract

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We study the effect of two types of degeneration of a Riemannian metric on the first eigenvalue of the Laplace operator on surfaces. In both cases we prove that the first eigenvalue of the round sphere is an optimal asymptotic upper bound. The first type of degeneration is concentration of the density to a point within a conformal class. The second is degeneration of the conformal class to the boundary of the moduli space on the torus and on the Klein bottle. In the latter, we follow the outline proposed by N. Nadirashvili in 1996.

Keywords

35P58J
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 3 , 01 June 2009 , pp. 548 - 565
Copyright
Copyright © Canadian Mathematical Society 2009

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