Uniformization of branched surfaces and Higgs bundles

TL;DR

This paper explores the uniformization of branched surfaces with cone metrics and characterizes the associated Higgs bundles, extending Hitchin's work to conical metrics on Riemann surfaces with divisors.

Contribution

It establishes a correspondence between conical constant negative curvature metrics and specific Higgs bundles, generalizing Hitchin's uniformization results to surfaces with divisors.

Findings

Unique cone metrics with prescribed cone angles are constructed.

A family of Higgs bundles parametrized by sections of a line bundle is described.

The results extend Hitchin's uniformization to conical metrics on Riemann surfaces.

Abstract

Given a compact Riemann surface $\Sigma$ of genus $g_\Sigma\, \geq\, 2$, and an effective divisor $D\, =\, \sum_i n_i x_i$ on $\Sigma$ with $\text{degree}(D)\, <\, 2(g_\Sigma -1)$, there is a unique cone metric on $\Sigma$ of constant negative curvature $-4$ such that the cone angle at each $x_i$ is $2\pi n_i$ (see McOwen and Troyanov [McO,Tr]). We describe the Higgs bundle corresponding to this uniformization associated to the above conical metric. We also give a family of Higgs bundles on $\Sigma$ parametrized by a nonempty open subset of $H^0(\Sigma,\,K_\Sigma^{\otimes 2}\otimes{\mathcal O}_\Sigma(-2D))$ that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin's results in [Hi1], for the case $D\,=\, 0$.