Uniformization of branched surfaces and Higgs bundles

Indranil Biswas, Steven Bradlow, Sorin Dumitrescu, Sebastian Heller

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Article numberA152
JournalInternational Journal of Mathematics
Issue number13
StatePublished - Dec 1 2021


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

ASJC Scopus subject areas

  • General Mathematics


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