Moscow Mathematical Journal

Volume 15, Issue 1, January–March 2015  pp. 123–140.

Conformal Spectrum and Harmonic Maps

Authors:  Nikolai Nadirashvili (1) and Yannick Sire (2)
Author institution:(1) CNRS, I2M UMR 7353, Centre de Mathématiques et Informatique, Marseille, France
(2) Université Aix-Marseille, I2M UMR 7353, Marseille, France

Summary:

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 rather constructive proof of the existence of a critical metric which is smooth outside of a finite number of conical singularities and maximizes the first eigenvalue in the conformal class of the background metric. We also prove that there exists a subspace of the eigenspace associated to the first maximized eigenvalue such that the corresponding eigenvector gives a harmonic map from the surface to a Euclidean sphere.

2010 Math. Subj. Class. 35P15.

Keywords: Eigenvalues, isoperimetric inequalities.