Non-existence of solutions for a mean field equation on flat tori at critical parameter $16\pi$

TL;DR

This paper proves the conjecture that the mean field equation has no solutions on rectangular flat tori at the critical parameter $16 ext{pi}$ for the case when $n=2$, extending previous results for $n=1$.

Contribution

It establishes the non-existence of solutions for the mean field equation at the critical parameter on rectangular tori for the case $n=2$, a challenging open problem.

Findings

No solutions exist for $n=2$ at the critical parameter $16 ext{pi}$ on rectangular tori.

Extends non-existence results from $n=1$ to $n=2$ for the mean field equation.

Supports the conjecture of non-existence at the critical parameter for rectangular tori.

Abstract

It is known from \cite{LW} that the solvability of the mean field equation $\Delta u+e^{u}=8n\pi \delta_{0}$ with $n\in\mathbb{N}_{\geq 1}$ on a flat torus $E_{\tau}$ essentially depends on the geometry of $E_{\tau}$. A conjecture is the non-existence of solutions for this equation if $E_{\tau}$ is a rectangular torus, which was proved for $n=1$ in \cite{LW}. For any $n\in \mathbb{N}_{\geq2}$, this conjecture seems challenging from the viewpoint of PDE theory. In this paper, we prove this conjecture for $n=2$ (i.e. at critical parameter $16\pi$).