Variational aspects of Laplace eigenvalues on Riemannian surfaces

We study the existence and properties of metrics maximising the first Laplace eigenvalue among conformal metrics of unit volume on Riemannian surfaces. We describe a general approach to this problem and its higher eigenvalue versions via the direct method of calculus of variations. The principal results include the general regularity properties of ${\lambda }_k$-extremal metrics and the existence of a partially regular ${\lambda }_1$-maximiser.