Uniqueness and Symmetry for the Mean Field Equation on Arbitrary Flat Tori.

We study the following mean field equation on a flat torus |$T:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$|⁠ : $$\begin{equation*} \varDelta u + \rho \left(\frac{e^{u}}{\int_{T}e^u}-\frac{1}{|T|}\right)=0, \end{equation*}$$ where |$ \tau \in \mathbb{C}, \mbox{Im}\ \tau>0$|⁠ , and |$|T|$| denotes the total area of the torus. We first prove that the solutions are evenly symmetric about any critical point of |$u$| provided that |$\rho \leq 8\pi $|⁠. Based on this crucial symmetry result, we are able to establish further the uniqueness of the solution if |$\rho \leq \min{\{8\pi ,\lambda _1(T)|T|\}}$|⁠. Furthermore, we also classify all one-dimensional solutions by showing that the level sets must be closed geodesics. [ABSTRACT FROM AUTHOR]