Conformal metrics with constant curvature one and finite conical singularities on compact Riemann surfaces

A conformal metric $g$ with constant curvature one and finite conical singularities on a compact Riemann surface $Σ$ can be thought of as the pullback of the standard metric on the 2-sphere by a multi-valued locally univalent meromorphic function $f$ on $Σ\backslash \{{\rm singularities}\}$, called the {\it developing map} of the metric $g$. When the developing map $f$ of such a metric $g$ on the compact Riemann surface $Σ$ has reducible monodromy, we show that, up to some M{\" o}bius transformation on $f$, the logarithmic differential $d\,(\log\, f)$ of $f$ turns out to be an abelian differential of 3rd kind on $Σ$, which satisfies some properties and is called a {\it character 1-form of} $g$. Conversely, given such an abelian differential $ω$ of 3rd kind satisfying the above properties, we prove that there exists a unique conformal metric $g$ on $Σ$ with constant curvature one and conical singularities such that one of its character 1-forms coincides with $ω$. This provides new examples of conformal metrics on compact Riemann surfaces of constant curvature one and with singularities. Moreover, we prove that the developing map is a rational function for a conformal metric $g$ with constant curvature one and finite conical singularities with angles in $2π\,{\Bbb Z}_{>1}$ on the two-sphere.

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