Conformal Spectrum and Harmonic maps

TL;DR

This paper investigates the conformal spectrum of Laplace-Beltrami operators on Riemannian surfaces, constructing a critical metric with conical singularities that maximizes the first eigenvalue and linking eigenvectors to harmonic maps.

Contribution

It provides a constructive proof of a maximizer metric with conical singularities and establishes a connection between the conformal spectrum and harmonic maps.

Findings

Existence of a smooth metric with conical singularities maximizing the first eigenvalue.

The eigenvector map into the sphere is harmonic.

Link between conformal spectrum and harmonic maps.

Abstract

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.