Uniformization of branched surfaces and Higgs bundles

Abstract

Given a compact Riemann surface $\Sigma$ of genus $g_{\Sigma }\ge 2$, 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 $D=0$.

minor changings; to appear in International Journal of Mathematics