## Spherical conical metrics and harmonic maps to spheres

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- by Mikhail Karpukhin and Xuwen Zhu PDF
- Trans. Amer. Math. Soc.
**375**(2022), 3325-3350 Request permission

## Abstract:

A spherical conical metric $g$ on a surface $\Sigma$ 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 $2\pi$. The eigenfunctions of the Friedrichs Laplacian $\Delta _g$ with eigenvalue $\lambda =2$ 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 $\Sigma$. 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.## References

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## Additional Information

**Mikhail Karpukhin**- Affiliation: Department of Mathematics, Caltech, Pasadena, California 91125
- MR Author ID: 1055445
- Email: mikhailk@caltech.edu
**Xuwen Zhu**- Affiliation: Department of Mathematics, Northeastern University, Boston, Massachussetts 02115
- MR Author ID: 1220465
- Email: x.zhu@northeastern.edu
- 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. - © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 3325-3350 - MSC (2020): Primary 53C43, 53C21
- DOI: https://doi.org/10.1090/tran/8578
- MathSciNet review: 4402663