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

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 $\mathrm{U}(1)$, 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 $D^{\prime }$ 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 $°D\ge 6g_X-2$ on $X$, we can identify an arcwise connected Borel subset in $|D|$ that demonstrates a Hausdorff dimension of no less than $(°D-4g_X+2)$. Within this subset, each divisor $D^{\prime }$ can be distinctly represented by a family of reducible metrics, defined by a single real parameter. \item[(3)] For an effective divisor $D$ with $°D=2$ 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 $D^{\prime }$ can be distinctly represented by a family of reducible metrics, defined by a single real parameter.