Fundamental tone, concentration of density, and conformal degeneration on surfaces
A Girouard - Canadian Journal of Mathematics, 2009 - cambridge.org
Canadian Journal of Mathematics, 2009•cambridge.org
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.
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.
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.
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