## Abstract

Given a compact connected Riemann surface ς of genus gς ≥ 2, and an effective divisor D = inixi on ς with degree(D) < 2(gς-1), there is a unique cone metric on ς of constant negative curvature-4 such that the cone angle at each point xi is 2I ni [R. C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988) 222-224; M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793-821]. We describe the Higgs bundle on ς corresponding to the uniformization associated to this conical metric. We also give a family of Higgs bundles on ς parametrized by a nonempty open subset of H0(ς,K ς-2-ς(-2D)) that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin's results in [N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59-126] for the case D = 0.

Original language | English (US) |
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Article number | A152 |

Journal | International Journal of Mathematics |

Volume | 32 |

Issue number | 13 |

DOIs | |

State | Published - Dec 1 2021 |

## Keywords

- Branched surface
- Higgs bundle
- Hopf differential
- harmonic map
- uniformization

## ASJC Scopus subject areas

- General Mathematics