Conformal spectrum and harmonic maps

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.