# Symmetry and uniqueness of solutions to some Liouville-type equations and systems

## TL;DR

This paper establishes symmetry and uniqueness of solutions for specific Liouville-type equations and systems in geometry and physics, utilizing the Sphere Covering Inequality to handle various nonlinearities and systems.

## Contribution

It introduces a novel application of the Sphere Covering Inequality to prove symmetry and uniqueness for multiple Liouville-type problems involving different exponential nonlinearities.

## Key findings

- Proves symmetry of solutions for the asymmetric Sinh-Gordon equation
- Establishes uniqueness of solutions for the cosmic string equation
- Demonstrates the effectiveness of the Sphere Covering Inequality in analyzing systems

## Abstract

We prove symmetry and uniqueness results for three classes of Liouville-type problems arising in geometry and mathematical physics: asymmetric Sinh-Gordon equation, cosmic string equation and Toda system, under certain assumptions on the mass associated to these problems. The argument is in the spirit of the Sphere Covering Inequality which for the first time is used in treating different exponential nonlinearities and systems.