A simple proof of the Uniqueness of blow-up solutions of mean field equations

TL;DR

This paper presents a simplified proof of the uniqueness of blow-up solutions for regular and singular mean field equations on compact Riemann surfaces, extending previous complex proofs and making the results more accessible.

Contribution

It offers a straightforward proof for the uniqueness of blow-up solutions and extends the results to singular equations with negative poles, complementing prior work.

Findings

Simplified proof of blow-up solution uniqueness

Extension to singular equations with negative poles

Complements previous complex proofs

Abstract

For a regular mean field equation defined on a compact Riemann surface, an important work of Bartolucci-Jevnikar-Lee-Yang \cite{bart-4} proved a uniqueness theorem for blow-up solutions under non-degeneracy assumptions. However, the proof is highly nontrivial and challenging to read. In this article, we not only provide a simple proof for the regular equation but also extend our proof to the case of singular equations with negative singular poles. Our proof supplements what is not written in a recent outstanding work by Bartolucci-Yang-Zhang \cite{byz-1}.