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arXiv · 2003.02871

Conformally maximal metrics for Laplace eigenvalues on surfaces

Mikhail Karpukhin·Nikolai Nadirashvili·Alexei V. Penskoi·Iosif Polterovich
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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.

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Mikhail Karpukhin, Nikolai Nadirashvili, Alexei V. Penskoi, Iosif Polterovich. 2020-05-15. Conformally maximal metrics for Laplace eigenvalues on surfaces. https://arxiv.org/abs/2003.02871

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