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Creators/Authors contains: "Gu, Guangze"

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  1. ; ; ; (, International Mathematics Research Notices)
    Abstract 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. 
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