# On metrics of constant positive curvature with four conic singularities on the sphere

## TL;DR

This paper proves that for four specified points on a sphere with given non-multiple-of-2π angles, the number of constant positive curvature metrics with conic singularities at these points is finite.

## Contribution

It establishes the finiteness of such metrics under specified conditions, advancing understanding of geometric structures with conic singularities.

## Key findings

- Number of metrics is finite for given four points and prescribed angles.
- Angles are not multiples of 2π, ensuring finiteness.
- Provides a classification framework for metrics with conic singularities.

## Abstract

We show that for given four points on the sphere and prescribed angles at these points, which are not multiples of $2\pi$, the number of metrics of curvature 1 having conic singularities with these angles at these points is finite.