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- W3103610913 abstract "A conformal metric $g$ with constant curvature one and finite 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 multi-valued locally univalent meromorphic function $f$ on $Sigmabackslash {{rm singularities}}$, called the {it 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 M{ o}bius transformation on $f$, the logarithmic differential $d,(log, f)$ of $f$ turns out to be an abelian differential of 3rd kind on $Sigma$, which satisfies some properties and is called a {it character 1-form of} $g$. Conversely, given such an abelian differential $omega$ of 3rd kind satisfying the above properties, we prove that there exists a unique conformal metric $g$ on $Sigma$ with constant curvature one and conical singularities such that one of its character 1-forms coincides with $omega$. 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 $2pi,{Bbb Z}_{>1}$ on the two-sphere." @default.
- W3103610913 created "2020-11-23" @default.
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- W3103610913 date "2015-01-01" @default.
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- W3103610913 title "Conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces" @default.
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- W3103610913 doi "https://doi.org/10.2140/pjm.2015.273.75" @default.
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