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

Qing Chen, Wei Wang, Yingyi Wu and Bin Xu

Vol. 273 (2015), No. 1, 75–100

DOI: 10.2140/pjm.2015.273.75

Abstract

A conformal metric $g$ with constant curvature one and finitely many 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 multivalued locally univalent meromorphic function $f$ on $\Sigma \setminus \left\{singularities\right\}$, called the 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 Möbius transformation on $f$, the logarithmic differential $d\left(logf\right)$ of $f$ turns out to be an abelian differential of the third kind on $\Sigma$, which satisfies some properties and is called a character 1-form of $g$. Conversely given such an abelian differential $\omega$ of the third kind satisfying the above properties, we prove that there exists a unique 1-parameter family of conformal metrics on $\Sigma$ such that all these metrics have constant curvature one, the same conical singularities, and have $\omega$ as one of their character 1-forms. 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 finitely many conical singularities with angles in $2\pi {\mathbb{Z}}_{>1}$ on the two-sphere.

Keywords

conformal metric of constant curvature one, conical singularity, developing map, character 1-form

Mathematical Subject Classification 2010

Primary: 32Q30

Secondary: 34M35

Milestones

Received: 20 July 2013

Revised: 18 December 2013

Accepted: 25 February 2014

Published: 6 December 2014