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Conformally maximal metrics for Laplace eigenvalues on surfaces

Karpukhin, Mikhail and Nadirashvili, Nikolai and Penskoi, Alexei V. and Polterovich, Iosif (2020) Conformally maximal metrics for Laplace eigenvalues on surfaces. . (Unpublished)

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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.

Item Type:Report or Paper (Discussion Paper)
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Additional Information:Partially supported by the Simons-IUM fellowship. Partially supported by NSERC. The authors would like to thank Alexandre Girouard for valuable remarks on the earlier version of the paper. We are also thankful to Dorin Bucur and Daniel Stern for useful discussions, as well as to Leonid Polterovich for pointing out the reference [BE] and helpful comments.
Funding AgencyGrant Number
Simons FoundationUNSPECIFIED
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Classification Code:2010 Mathematics Subject Classification. 58J50, 58E11, 53C42
Record Number:CaltechAUTHORS:20201123-145639355
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:106797
Deposited By: Tony Diaz
Deposited On:23 Nov 2020 23:02
Last Modified:23 Nov 2020 23:02

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