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

By Qing Chen, Wei Wang, Yingyi Wu, Bin Xu. 2013

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A conformal metric g with constant curvature one and finitely many conical singularities on a compact Riemann surface6 can be thought of as the pullback of the standard metric on the 2-sphere by a multivalued locally univalent meromorphic function f on6nfsingularitiesg, called the developing map of the metric g. When the developing map f of such a metric g on the compact Riemann surface 6 has reducible monodromy, we show that, up to some Mobius transformation on f , the logarithmic differential d.log f/ of f turns out to be an abelian differential of the third kind on 6, which satisfies some properties and is called a character 1-form of g. Conversely given such an abelian differential ! of the third kind satisfying the above properties, we prove that there exists a unique 1-parameter family of conformal metrics on6 such that all these metrics have constant curvature one, the same conical singularities, and have! 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 Z>1 on the two-sphere.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/279244ed2e11e19b18a8e58764241ed6cf4b7e52

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In Geometry and Topology

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  2. Alexandre Eremenko
  3. Andrei Gabrielov
  4. A. Varchenko
  5. E. Mukhin

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