#### Vol. 273, No. 1, 2015

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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
##### 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 $2\pi {ℤ}_{>1}$ on the two-sphere.

##### Keywords
conformal metric of constant curvature one, conical singularity, developing map, character 1-form
Primary: 32Q30
Secondary: 34M35