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October, 2021 A Mean Field Type Flow on a Closed Riemannian Surface with the Action of an Isometric Group
Yamin Wang
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Taiwanese J. Math. 25(5): 1053-1072 (October, 2021). DOI: 10.11650/tjm/210106

Abstract

Let $(\Sigma,g)$ be a closed Riemannian surface, $\mathbf{G} = \{ \sigma_1,\ldots,\sigma_N \}$ be an isometric group acting on it. Denote a positive integer $\ell = \min_{x \in \Sigma} I(x)$, where $I(x)$ is the number of all distinct points of the set $\{ \sigma_1(x),\ldots,\sigma_N(x) \}$. By a method of flow due to Castéras (Pacific J. Math. 2015), we prove that the solution to the mean field equation \[ -\Delta_g u = 8\pi \ell \left( \frac{he^u}{\int_{\Sigma} he^u \, dv_g} - \frac{1}{\operatorname{Vol}_g(\Sigma)} \right) \] exists under given conditions. This gives a new proof of Yang and Zhu's result in (Internat. J. Math. 2020). The case $\ell = 1$ was studied by Li and Zhu (Calc. Var. Partial Differential Equations 2019).

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Yamin Wang. "A Mean Field Type Flow on a Closed Riemannian Surface with the Action of an Isometric Group." Taiwanese J. Math. 25 (5) 1053 - 1072, October, 2021. https://doi.org/10.11650/tjm/210106

Information

Received: 5 August 2020; Accepted: 17 January 2021; Published: October, 2021
First available in Project Euclid: 21 January 2021

Digital Object Identifier: 10.11650/tjm/210106

Subjects:
Primary: 58J05

Rights: Copyright © 2021 The Mathematical Society of the Republic of China

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Vol.25 • No. 5 • October, 2021
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