  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.
