Abstract
Let (M, g) be a closed Riemann surface. Castéras (Pac J Math 276:321–345, 2015) introduced a mean field type flow and obtained its global existence. For \(\rho \ne 8N\pi \), assuming in addition that \(J_\rho (v(t))\ge -C\) for \(\rho >8\pi \), they showed the convergence of the flow. In this paper, we will prove the convergence of the flow for the critical case \(\rho =8\pi \) with some appropriate initial data \(v_0\) and some geometric hypothesis. In particular, this gives a new proof of Ding–Jost–Li–Wang’s result in (Asian J Math 1:230–248, 1997).
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Acknowledgements
The authors would like to thank Professor Yunyan Yang for his helpful suggestions. They also thank Professor Hao Yin for his good suggestions in our seminar, which inspire us to derive Corollary 2.2.
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Communicated by J. Jost.
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The research was supported by NSFC Nos. 11526212, 11131007 and 11721101.
