Conformal Spectrum and Harmonic maps

Details Conformal Spectrum and Harmonic maps Conformal Spectrum and Harmonic maps

2010-07-19

arxiv.org/abs/1007.3104v3

Mathematics

Differential Geometry

Scientific paper

This paper is devoted to the study of the conformal spectrum (and more precisely the first eigenvalue) of the Laplace-Beltrami operator on a smooth connected compact Riemannian surface without boundary, endowed with a conformal class. We give a constructive proof of a critical metric which is smooth except at some conical singularities and maximizes the first eigenvalue in the conformal class of the background metric. We also prove that the map associating a finite number of eigenvectors of the maximizing $\lambda_1$ into the sphere is harmonic, establishing a link between conformal spectrum and harmonic maps.

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