Spherical conical metrics and harmonic maps to spheres

TL;DR

This paper investigates spherical conical metrics on surfaces, focusing on eigenfunctions of the Laplacian with eigenvalue 2, and introduces new criteria and explicit examples for metrics with many such eigenfunctions.

Contribution

It develops a new meromorphic criterion for the existence of 2-eigenfunctions and constructs metrics with arbitrarily many 2-eigenfunctions using harmonic map deformations.

Findings

Established a criterion for 2-eigenfunction existence based on meromorphic data.

Provided explicit descriptions of 2-eigenfunctions for metrics with up to three conical singularities.

Developed an algorithm to construct metrics with arbitrarily many 2-eigenfunctions.

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.