Karpukhin, Mikhail; Nadirashvili, Nikolai; Penskoi, Alexei V; Polterovich, Iosif; (2019) Conformally maximal metrics for Laplace eigenvalues on surfaces. Surveys in Differential Geometry , 24 (1) pp. 205-256. 10.4310/sdg.2019.v24.n1.a6.

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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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