Global existence and convergence of a flow to Kazdan-Warner equation with non-negative prescribed function

Abstract

We consider an evolution problem associated to the Kazdan-Warner equation on a closed Riemann surface $(\hat{I}\pounds ,g)$ \begin{align*} -Δ_{g}u=8π\left(\frac{he^{u}}{\int_Σhe^{u}{\rm d}μ_{g}}-\frac{1}{\int_Σ{\rm d}μ_{g}}\right) \end{align*} where the prescribed function $h\ge 0$ and ${\max }_{\hat{I}}\pounds h>0$. We prove the global existence and convergence under additional assumptions such as \begin{align*} Δ_{g}\ln h(p_0)+8π-2K(p_0)>0 \end{align*} for any maximum point $p_0$ of the sum of $2\ln h$ and the regular part of the Green function, where $K$ is the Gaussian curvature of $\hat{I}\pounds$. 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.

23 pages