CONFORMAL METRICS WITH CONSTANT CURVATURE ONE AND FINITELY MANY CONICAL SINGULARITIES ON COMPACT RIEMANN SURFACES

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\{singularities}, 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 Mobius transformation on f, the logarithmic differential d(log f) 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 Sigma 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.