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Stable parabolic Higgs bundles of rank two and singular hyperbolic metrics

Yu Feng, Bin Xu

TL;DR

The work bridges parabolic Higgs bundle theory and singular hyperbolic metrics to address the problem of constructing hyperbolic metrics with prescribed cone and cusp singularities on compact Riemann surfaces. By building a stable rank-two parabolic Higgs bundle on the surface and solving Hitchin’s equations, the authors obtain a conformal hyperbolic metric representing the prescribed divisor, providing an alternative proof of Heins’ existence result. They further deform the Higgs field to generate a family of stable parabolic Higgs bundles parameterized by an open subset of $H^{0}(\overline{X},K^{2}\otimes\mathcal{O}_{\overline{X}}(D-\tilde{D}))$, yielding metrics on deformed Riemann surfaces that preserve the same singularity type; under certain angle conditions, this recovers and extends prior results (e.g., BBDH 2021) and yields an isometric classification of cone metrics with angles in $(0,\pi]$. Overall, the paper extends Hitchin’s framework to a broader class of singular hyperbolic metrics and demonstrates a robust correspondence between parabolic Higgs data and geometric structures on (possibly deformed) surfaces.

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.

Stable parabolic Higgs bundles of rank two and singular hyperbolic metrics

TL;DR

The work bridges parabolic Higgs bundle theory and singular hyperbolic metrics to address the problem of constructing hyperbolic metrics with prescribed cone and cusp singularities on compact Riemann surfaces. By building a stable rank-two parabolic Higgs bundle on the surface and solving Hitchin’s equations, the authors obtain a conformal hyperbolic metric representing the prescribed divisor, providing an alternative proof of Heins’ existence result. They further deform the Higgs field to generate a family of stable parabolic Higgs bundles parameterized by an open subset of , yielding metrics on deformed Riemann surfaces that preserve the same singularity type; under certain angle conditions, this recovers and extends prior results (e.g., BBDH 2021) and yields an isometric classification of cone metrics with angles in . Overall, the paper extends Hitchin’s framework to a broader class of singular hyperbolic metrics and demonstrates a robust correspondence between parabolic Higgs data and geometric structures on (possibly deformed) surfaces.

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 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 , 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 . 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.

Paper Structure

This paper contains 10 sections, 6 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Background of singular hyperbolic metrics
  3. Main results
  4. Outline
  5. Preliminaries
  6. Proofs of Lemma \ref{['lemma:construction of parabolic Higgs bundle']} and Theorem \ref{['thm:rat']}
  7. Constructing parabolic Higgs stable bundles
  8. Proof of Theorem \ref{['thm:rat']}
  9. Deformations of the Higgs field
  10. Acknowledgments

Key Result

Theorem 1.1

(Heins Heins:1962) There exists a hyperbolic metric representing an $\mathbb{R}$-divisor $\mathrm{\hat{D}} = \sum _{i=1}^{m} (\theta _{i} - 1) p_{i}$ with $0 \leq \theta _{i} \neq 1$ on a compact Riemann surface $\overline{X}$ if and only if $\chi (\overline{X}) + \sum _{i=1}^{m} (\theta _{i} - 1) <

Theorems & Definitions (16)

  • Theorem 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • ...and 6 more