Conformally maximal metrics for Laplace eigenvalues on surfaces

Understand

The paper is concerned with the maximization of Laplace eigenvalues on surfaces of given volume with a Riemannian metric in a fixed conformal class.

• A significant progress on this problem has been recently achieved by Nadirashvili-Sire and Petrides using related, though different methods.
• In particular, it was shown that for a given $k$, the maximum of the $k$-th Laplace eigenvalue in a conformal class on a surface is either attained on a metric which is smooth except possibly at a finite number of conical singularities, or it is attained in the limit while a "bubble tree" is formed on a surface.
• Geometrically, the bubble tree appearing in this setting can be viewed as a union of touching identical round spheres.