Stable parabolic Higgs bundles of rank two and singular hyperbolic metrics

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.