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

TL;DR

This paper characterizes conformal metrics with constant curvature one and finite conical singularities on compact Riemann surfaces, linking their developing maps to abelian differentials of third kind and providing new explicit examples.

Contribution

It establishes a correspondence between such metrics with reducible monodromy and abelian differentials of third kind, and shows the developing map is rational in special cases.

Findings

Characterization of metrics via character 1-forms of third kind.

Existence and uniqueness of metrics given abelian differentials.

Developing map is rational for certain metrics on the sphere.

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.