Paper detail

Uniformization of branched surfaces and Higgs bundles

Given a compact Riemann surface $Σ$ of genus $g_Σ\, \geq\, 2$, and an effective divisor $D\, =\, \sum_i n_i x_i$ on $Σ$ with $\text{degree}(D)\, <\, 2(g_Σ-1)$, there is a unique cone metric on $Σ$ of constant negative curvature $-4$ such that the cone angle at each $x_i$ is $2πn_i$ (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 $Σ$ parametrized by a nonempty open subset of $H^0(Σ,\,K_Σ^{\otimes 2}\otimes{\mathcal O}_Σ(-2D))$ that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin&#39;s results in [Hi1], for the case $D\,=\, 0$.