Parabolic bundles and spherical metrics

TL;DR

This paper leverages the Kobayashi-Hitchin correspondence for parabolic bundles to reestablish key results on the existence and uniqueness of spherical metrics with prescribed cone angles on the Riemann sphere.

Contribution

It provides a new proof of Troyanov and Luo-Tian's results using parabolic bundle theory, offering a different perspective on spherical metrics with cone singularities.

Findings

Existence of spherical metrics with prescribed cone angles in (0, 2π)

Uniqueness of such metrics for given configurations

Reproves classical results via parabolic bundle methods

Abstract

We use the Kobayashi-Hitchin correspondence for parabolic bundles to reprove the results of Troyanov and Luo-Tian regarding existence and uniqueness of conformal spherical metrics on the Riemann sphere with prescribed cone angles in the interval $(0, 2\pi)$ at a given configuration of three or more points.