A singular Kazdan-Warner problem on a compact Riemann surface

TL;DR

This paper studies a singular mean field equation on compact Riemann surfaces, proving existence results at critical parameters and analyzing blowup behavior, especially at points with minimal conical angle.

Contribution

It introduces new existence results for the singular mean field equation at critical parameters and characterizes blowup points in relation to conical angles and positivity of h.

Findings

Existence of solutions at critical parameter $ ho$ under certain conditions.

Blowup points occur where the conical angle is smallest and h is positive.

Blowup analysis reveals the geometric nature of singularities.

Abstract

Let $(M,g)$ be a compact Riemann surface with unit area, $h\in C^{\infty}(M)$ a function which is positive somewhere, $\rho>0$, $p_i\in M$ and $\alpha_i\in(-1,+\infty)$ for $i=1,\cdots,\ell$, we consider the mean field equation \begin{align*} \Delta v + 4\pi\sum_{i=1}^{\ell}\alpha_i\left(1-\delta_{p_i}\right) = \rho\left(1-\frac{he^v}{\int_Mhe^vd\mu}\right), \end{align*} on $M$, where $\Delta$ and $d\mu$ are the Laplace-Beltrami operator and the area element of $(M,g)$ respectively. Using variational method and blowup analysis, we prove some existence results in the critical case $\rho=8\pi(1+\min\{0,\min_{1\leq i\leq\ell}\alpha_i\})$. These results can be seen as partial generalizations of works of Chen-Li (J. Geom. Anal. 1: 359--372, 1991), Ding-Jost-Li-Wang (Asian J. Math. 1: 230--248, 1997), Mancini (J. Geom. Anal. 26: 1202--1230, 2016), Yang-Zhu (Proc. Amer. Math. Soc. 145: 3953--3959, 2017), Sun-Zhu (arXiv:2012.12840) and Zhu (arXiv:2212.09943). Among other things, we prove that the blowup (if happens) must be at the point where the conical angle is the smallest one and $h$ is positive, this is the most important contribution of our paper.