Surveys in Differential Geometry
Volume 24 (2019)
Conformally maximal metrics for Laplace eigenvalues on surfaces
Pages: 205 – 256
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. We present another proof of this statement, developing the approach proposed by the second author and Y. Sire. As a side result, we provide explicit upper bounds on the topological spectrum of surfaces.
2010 Mathematics Subject Classification
53C42, 58E11, 58J50
A. V. Penskoi was partially supported by the Simons-IUM fellowship.
I. Polterovich was partially supported by NSERC.
Published 29 December 2021