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

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A conformal metric g with constant curvature one and finite 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 multi-valued 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 M obius transformation on f, the logarithmic differential d(f) of f turns out to be an abelian differential of 3rd kind on Σ, which satisfies some properties and is called a character 1-form of g. Conversely, given such an abelian differential ω of 3rd kind satisfying the above properties, we prove that there exists a unique conformal metric g on Σ with constant curvature one and conical singularities such that one of its character 1-forms coincides with ω. 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 finite conical singularities with angles in 2π Z>₁ on the two-sphere.