Spectral properties of reducible conical metrics

TL;DR

This paper establishes a link between the reducibility of monodromy in spherical conical metrics and real eigenfunctions of the associated Laplace operator, providing new insights into their spectral properties and classification.

Contribution

It introduces a novel connection between complex analysis and PDE methods in analyzing spherical conical metrics, including eigenvalue bounds and monodromy classification.

Findings

Monodromy reducibility corresponds to real eigenfunctions with eigenvalue 2.

Provides a lower bound for the first nonzero eigenvalue.

Classifies eigenspace dimensions based on monodromy properties.

Abstract

We show that the monodromy of a spherical conical metric is reducible if and only if it has a real-valued eigenfunction with eigenvalue 2 in the holomorphic extension of the associated Laplace--Beltrami operator. Such an eigenfunction produces a meromorphic vector field, which is then related to the developing maps of the conical metric. We also give a lower bound of the first nonzero eigenvalue, and a complete classification of the eigenspace dimension depending on the monodromy. This paper can be seen as a new connection between the complex analysis method and the PDE approach in the study of spherical conical metrics.