Existence and non-uniqueness of cone spherical metrics with prescribed singularities on a compact Riemann surface with positive genus

TL;DR

This paper investigates the existence, non-uniqueness, and classification of cone spherical metrics with prescribed singularities on compact Riemann surfaces of positive genus, revealing multiple metrics for certain divisors and dimensions of parameter spaces.

Contribution

It establishes new results on the existence and multiplicity of cone spherical metrics with prescribed singularities using polystable bundle extensions on higher genus surfaces.

Findings

Existence of multiple irreducible cone spherical metrics for certain divisors.

Identification of parameter spaces with positive Hausdorff dimension for reducible metrics.

Construction of families of reducible metrics parametrized by a real variable.

Abstract

Cone spherical metrics, defined on compact Riemann surfaces, are conformal metrics with constant curvature one and finitely many cone singularities. Such a metric is termed \textit{reducible} if a developing map of the metric has monodromy in ${\rm U(1)}$, and \textit{irreducible} otherwise. Utilizing the polystable extensions of two line bundles on a compact Riemann surface $X$ with genus $g_X>0$, we establish the following three primary results concerning these metrics with cone angles in $2\pi{\mathbb Z}_{>1}$: \begin{itemize} \item[(1)] Given an effective divisor $D$ with an odd degree surpassing $2g_X$ on $X$, we find the existence of an effective divisor $D'$ in the complete linear system $|D|$ that can be represented by at least two distinct irreducible cone spherical metrics on $X$. \item[(2)] For a generic effective divisor $D$ with an even degree and $\deg D\geq 6g_X-2$ on $X$, we can identify an arcwise connected Borel subset in $|D|$ that demonstrates a Hausdorff dimension of no less than $\big(\deg D-4g_{X}+2\big)$. Within this subset, each divisor $D'$ can be distinctly represented by a family of reducible metrics, defined by a single real parameter. \item[(3)] For an effective divisor $D$ with $\deg D=2$ on an elliptic curve, we can identify a Borel subset in $|D|$ that is arcwise connected, showcasing a Hausdorff dimension of one. Within this subset, each divisor $D'$ can be distinctly represented by a family of reducible metrics, defined by a single real parameter.