AbstractWe consider an evolution problem associated to the Kazdan–Warner equation on a closed Riemann surface $$(\Sigma ,g)$$
(
Σ
,
g
)
$$\begin{aligned} -\Delta _{g}u=8\pi \left( \dfrac{he^{u}}{\int _{\Sigma }he^{u}\mathop {}\mathrm {d}\mu _{g}}-\dfrac{1}{\int _{\Sigma }\mathop {}\mathrm {d}\mu _{g}}\right) \end{aligned}$$
-
Δ
g
u
=
8
π
h
e
u
∫
Σ
h
e
u
d
μ
g
-
1
∫
Σ
d
μ
g
where the prescribed function $$h\ge 0$$
h
≥
0
and $$\max _{\Sigma }h>0$$
max
Σ
h
>
0
. We prove the global existence and convergence under additional assumptions such as $$\begin{aligned} \Delta _{g}\ln h(p_0)+8\pi -2K(p_0)>0 \end{aligned}$$
Δ
g
ln
h
(
p
0
)
+
8
π
-
2
K
(
p
0
)
>
0
for any maximum point $$p_0$$
p
0
of the sum of $$2\ln h$$
2
ln
h
and the regular part of the Green function, where K is the Gaussian curvature of $$\Sigma $$
Σ
. In particular, this gives a new proof of the existence result by Yang and Zhu (Pro Am Math Soc 145:3953–3959, 2017) which generalizes existence result of Ding et al. (Asian J Math 1:230–248, 1997) to the non-negative prescribed function case.
Behaviour of Green Lines at Royden’s Boundary of Riemann Surfaces