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## Variational aspects of Laplace eigenvalues on Riemannian surfaces

### Abstract

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 general regularity properties of $\lambda_k$-extremal metrics and the existence of a partially regular $\lambda_1$-maximiser.Comment: revised version, 38 pages; re-written introduction, changes taking into account referee comments made, misprints corrected, new references added, to appear in Advances in Mathematic

Topics: Mathematics - Spectral Theory, Mathematics - Differential Geometry
Year: 2014
OAI identifier: oai:arXiv.org:1103.2448