Global existence and convergence of a flow to Kazdan-Warner equation with non-negative prescribed function
Linlin Sun and Jingyong Zhu
Contact the author: Please use for correspondence this email.
Submission date: 01. Jun. 2020
MSC-Numbers: 35B33, 58J35
Keywords and phrases: Kazdan-Warner equation, mean field type flow, global existence, global convergence
Download full preprint: PDF (221 kB)
Link to arXiv: See the arXiv entry of this preprint.
We consider an evolution problem associated to the Kazdan-Warner equation on a closed Riemann surface (Σ,g)
− Δgu = 8π
where the prescribed function h ≥ 0 and maxΣh > 0. We prove the global existence and convergence under additional assumptions such as
Δg lnh(p0) + 8π − 2K(p0) > 0
for any maximum point p0 of the sum of 2lnh and the regular part of the Green function, where K is the Gaussian curvature of Σ. In particular, this gives a new proof of the existence result by Yang and Zhu [Proc. Amer. Math. Soc. 145(2017), no. 9, 3953-3959] which generalizes existence result of Ding, Jost, Li and Wang [Asian J. Math. 1(1997), no. 2, 230-248] to the non-negative prescribed function case.