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 Σ can be thought of as the pullback of the standard metric on the 2-sphere by a multivalued locally univalent meromorphic function f on Σ{singularities}, called the 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 Möbius transformation on f, the logarithmic differential d(logf) of f turns out to be an abelian differential of the third kind on Σ, which satisfies some properties and is called a character 1-form of g. Conversely given such an abelian differential ω of the third kind satisfying the above properties, we prove that there exists a unique 1-parameter family of conformal metrics on Σ such that all these metrics have constant curvature one, the same conical singularities, and have ω 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 2π>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
Authors
Qing Chen
Wu Wen-Tsun Key Laboratory of Math, USTC, Chinese Academy of Sciences
School of Mathematical Sciences
University of Science and Technology of China
Hefei, 230026
China
Wei Wang
Department of Mathematics
University of California, Irvine
Irvine, CA 92697-3875
United States
Yingyi Wu
School of Mathematical Sciences
University of Chinese Academy of Sciences
Beijing, 100049
China
Bin Xu
Wu Wen-Tsun Key Laboratory of Math, USTC, Chinese Academy of Sciences
School of Mathematical Sciences
University of Science and Technology of China
Hefei, 230026
China