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

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 reducible if a developing map of the metric has monodromy in U(1), and irreducible otherwise. Utilizing the polystable extensions of two line bundles on a compact Riemann surface X with genus gX>0, we establish the following three primary results concerning these metrics with cone angles in 2π Z>1: {itemize} [(1)] Given an effective divisor D with an odd degree surpassing 2gX 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. [(2)] For a generic effective divisor D with an even degree and D≥ 6gX-2 on X, we can identify an arcwise connected Borel subset in $|D|$ that demonstrates a Hausdorff dimension of no less than ( D-4gX+2). Within this subset, each divisor $D'$ can be distinctly represented by a family of reducible metrics, defined by a single real parameter. [(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 $D'$ can be distinctly represented by a family of reducible metrics, defined by a single real parameter.