Constructing constant curvature metrics on Riemann surfaces with singularities

TL;DR

This paper presents an explicit method to construct constant curvature metrics with singularities on Riemann surfaces using meromorphic 1-forms, and classifies such metrics on the sphere with two conical singularities.

Contribution

It introduces a new explicit construction technique for constant curvature metrics with singularities and provides a classification on the sphere with two conical singularities.

Findings

Explicit construction of conformal metrics with constant curvature using ODEs

Classification of such metrics on the sphere with two conical singularities

Extension of Troyanov's classification result

Abstract

By constructing an ODE through a kind of meromorphic 1-forms, we will give an explicit construction of a kind of conformal metrics of constant curvature on Riemann surfaces with singularities. As an application, we will classify constant curvature one metrics on $S^{2}$ with two conical singularities, which was first proved by Troyanov in \cite{Tr89} by using projective connection.