# A singular Sphere Covering Inequality: uniqueness and symmetry of solutions to singular Liouville-type equations

## TL;DR

This paper introduces a singular Sphere Covering Inequality to analyze singular Liouville equations, leading to new uniqueness and symmetry results for solutions on spheres and bounded domains.

## Contribution

It develops a singular version of the Sphere Covering Inequality and applies it to establish novel uniqueness and symmetry results for Liouville-type equations.

## Key findings

- New uniqueness results for singular mean field equations
- Symmetry results for spherical Onsager vortex equation
- Alternative proofs for convex polytope uniqueness

## Abstract

We derive a singular version of the Sphere Covering Inequality which was recently introduced in [42], suitable for treating singular Liouville-type problems with superharmonic weights. As an application we deduce new uniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains, as well as new self-contained proofs of previously known results, such as the uniqueness of spherical convex polytopes first established in [56]. Furthermore, we derive new symmetry results for the spherical Onsager vortex equation.