# Spherical metrics with conical singularities on 2-spheres

## TL;DR

This paper investigates the existence of spherical metrics with specified conical singularities on 2-spheres, establishing necessary and sufficient conditions for such metrics when the cone angles are not integer multiples of 2π.

## Contribution

It proves the necessity of a known sufficient condition for the existence of spherical metrics with conical singularities, under the assumption that the cone angles are non-integers.

## Key findings

- Established the necessity of the condition for non-integer cone angles.
- Extended previous sufficient conditions to necessary ones in specific cases.
- Contributed to the classification of spherical metrics with conical singularities.

## Abstract

Suppose that $\theta_1,\theta_2,\dots,\theta_n$ are positive numbers and $n\ge 3$. Does there exist a sphere with a spherical metric with $n$ conical singularities of angles $2\pi\theta_1,2\pi\theta_2,\dots,2\pi\theta_n$? A sufficient condition was obtained by Gabriele Mondello and Dmitri Panov (arXiv:1505.01994). We show that it is also necessary when we assume that $\theta_1,\theta_2,\dots,\theta_n \not\in \mathbb{N}$.