Surveys in Differential Geometry

Volume 24 (2019)

Conformally maximal metrics for Laplace eigenvalues on surfaces

Pages: 205 – 256

DOI: https://dx.doi.org/10.4310/SDG.2019.v24.n1.a6

Authors

Mikhail Karpukhin (Department of Mathematics, University of California, Irvine, Calif., U.S.A.; and Department of Mathematics, California Institute of Technology, Pasadena, Calif., U.S.A.)

Nikolai Nadirashvili (CNRS, I2M UMR 7353, Centre de Mathématiques et Informatique, Université d’Aix-Marseille, France)

Alexei V. Penskoi (Faculty of Mathematics and Mechanics, Moscow State University, Moscow, Russia; Faculty of Mathematics, National Research University Higher School of Economics, Moscow, Russia; and Interdisciplinary Scientific Center J.-V. Poncelet, Moscow, Russia)

Iosif Polterovich (Département de mathématiques et de statistique, Université de Montréal, Québec, Canada)

Abstract

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

Published 29 December 2021