2025/02/27 by Yu Feng, Xu Bin, Feng, Yu +1
Mathematics · #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2502.19789
openalex publication_date 2025/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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 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 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.