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Spherical conical metrics and harmonic maps to spheres

Karpukhin, Mikhail and Zhu, Xuwen (2022) Spherical conical metrics and harmonic maps to spheres. Transactions of the American Mathematical Society, 375 (5). pp. 3325-3350. ISSN 0002-9947. doi:10.1090/tran/8578.

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A spherical conical metric g on a surface Σ is a metric of constant curvature 1 with finitely many isolated conical singularities. The uniformization problem for such metrics remains largely open when at least one of the cone angles exceeds 2π. The eigenfunctions of the Friedrichs Laplacian Δg with eigenvalue λ=2 play a special role in this problem, as they represent local obstructions to deformations of the metric g in the class of spherical conical metrics. In the present paper we apply the theory of multivalued harmonic maps to spheres to the question of existence of such eigenfunctions. In the first part we establish a new criterion for the existence of 2-eigenfunctions, given in terms of a certain meromorphic data on Σ. As an application we give a description of all 2-eigenfunctions for metrics on the sphere with at most three conical singularities. The second part is an algebraic construction of metrics with large number of 2-eigenfunctions via the deformation of multivalued harmonic maps. We provide new explicit examples of metrics with many 2-eigenfunctions via both approaches, and describe the general algorithm to find metrics with arbitrarily large number of 2-eigenfunctions.

Item Type:Article
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URLURL TypeDescription Paper
Karpukhin, Mikhail0000-0003-1935-731X
Zhu, Xuwen0000-0003-1614-9216
Additional Information:© 2022 American Mathematical Society. Received by editor(s): May 9, 2021. Received by editor(s) in revised form: October 5, 2021. Published electronically: February 9, 2022. Additional Notes: The first author was partially supported by NSF DMS-1363432. The second author was partially supported by NSF DMS-2041823. The second author is grateful to Rafe Mazzeo and Bin Xu for many helpful discussions. We would like to thank Dima Jakobson for suggesting the collaboration and the anonymous referee for many valuable suggestions on the presentation of the article.
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Issue or Number:5
Classification Code:2020 Mathematics Subject Classification. Primary 53C43, 53C21.
Record Number:CaltechAUTHORS:20220707-977745000
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:115418
Deposited By: George Porter
Deposited On:12 Jul 2022 15:07
Last Modified:12 Jul 2022 15:07

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