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

Abstract: A conformal metric $g$ with constant curvature one and finite conical singularities on a compact Riemann surface $\Sigma$ 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 $\Sigma\backslash {{\rm singularities}}$, called the {\it developing map} of the metric $g$. When the developing map $f$ of such a metric $g$ on the compact Riemann surface $\Sigma$ 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 $\Sigma$, which satisfies some properties and is called a {\it character 1-form of} $g$. Conversely, given such an abelian differential $\omega$ of 3rd kind satisfying the above properties, we prove that there exists a unique conformal metric $g$ on $\Sigma$ with constant curvature one and conical singularities such that one of its character 1-forms coincides with $\omega$. 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 $2\pi\,{\Bbb Z}_{>1}$ on the two-sphere.