Variational aspects of Laplace eigenvalues on Riemannian surfaces

Details Variational aspects of Laplace eigenvalues on Riemannian surfaces Variational aspects of Laplace eigenvalues on Riemannian surfaces

2011-03-12

arxiv.org/abs/1103.2448v2

Mathematics

Spectral Theory

an improved version, a number of statements have been revised and re-written, more examples and explanations given in Sect.1 a

Scientific paper

We study the existence and properties of metrics maximising the first Laplace eigenvalue among conformal metrics of unit volume on Riemannian surfaces. We describe a general approach to this problem (and its higher eigenvalue versions) via the direct method of calculus of variations. The principal results include the existence of a partially regular maximiser for the first eigenvalue and the characterisation of its complete regularity.

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