Stable parabolic Higgs bundles of rank two and singular hyperbolic metrics

TL;DR

This paper constructs stable rank-two parabolic Higgs bundles on compact Riemann surfaces to provide an alternative proof of the existence of singular hyperbolic metrics, extending Hitchin's work to singular cases.

Contribution

It introduces a new method linking Higgs bundles to singular hyperbolic metrics, offering an alternative proof and extending Hitchin's results to singular surfaces.

Findings

Constructed stable parabolic Higgs bundles corresponding to singular hyperbolic metrics.

Provided an alternative proof of the existence of singular hyperbolic metrics.

Extended Hitchin's work to include hyperbolic metrics with singularities.

Abstract

In this paper, we construct a stable parabolic Higgs bundle of rank two, which corresponds to the uniformization associated with a conformal hyperbolic metric on a compact Riemann surface $\overline{X}$ with prescribed singularities. This provides an alternative proof of the classical existence theorem for singular hyperbolic metrics, originally established by Heins ({\it Nagoya Math. J.} 21 (1962), 1-60). We also introduce a family of stable parabolic Higgs bundles of rank two on $\overline{X}$, parametrized by a nonempty open subset of a complex vector space. These bundles correspond to singular hyperbolic metrics with the same type of singularity as the original, but are defined on deformed Riemann surfaces of $\overline{X}$. Thus, we extend partially the final section of Hitchin's celebrated work ({\it Proc. London Math. Soc.} 55(3) (1987), 59-125) to the context of hyperbolic metrics with singularities.