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Uniqueness of solutions for a mean field equation on torus. (English) Zbl 1105.58005
Summary: We consider on a two-dimensional flat torus \(T\) the following equation
\[ \Delta u+\rho \biggl( \frac{e^u}{\int_G e^u}- \frac{1}{|T|}\biggr)= 0. \]
When the fundamental domain of the torus is \((0,a)\times (0,b)\) \((a\geq b)\), we establish that the constants are the unique solutions whenever
\[ \rho\leq\begin{cases} 8\pi&\text{if }\frac ab\geq \frac \pi4,\\ 32\frac ba &\text{if }\frac ba\leq \frac\pi4, \end{cases} \]
and this result is sharp if \(\frac ab\geq \frac \pi4\). A similar conclusion is obtained for general two-dimensional torus by considering the length of the shortest closed geodesic. These results are derived by comparing the isoperimetric profile of the torus \(T\) with the one of the two-dimensional canonical sphere which has same volume as \(T\).

58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
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