Parabolic metrics with conical singularities on compact Riemann surfaces

TL;DR

This paper proves the existence and uniqueness of parabolic conical metrics with generalized angles on Riemann surfaces, providing explicit examples and extending classical formulas like Schwarz-Christoffel.

Contribution

It introduces a new existence and uniqueness theorem for parabolic conical metrics with generalized angles on Riemann surfaces, expanding previous results.

Findings

Established existence and uniqueness of parabolic conical metrics with real angles

Provided explicit examples on the two-sphere

Generalized Schwarz-Christoffel formula for these metrics

Abstract

In this paper, we prove the existence and uniqueness theorem for parabolic conical metrics on Riemann surfaces in the situation of generalized real angles, positive, zero and negative, by complex analysis, and give an example of this theorem to clarify concrete expressions of parabolic metrics on the two-sphere and generalize the well-known Schwarz-Christoffel formula.