Consider the Poincare disk $\mathbb{H}^2$, that is the unit disk endowed with the following metric $$ g_{ij}=\frac{2}{1-|x|^2}\delta_{ij}. $$ I believe there exists a global Green's function on $\mathbb{H}^2$ (is that true?), i.e., a function $G$ satisfying $$ -\Delta_{\mathbb{H}^2}G(x,y)=\delta_x(y). $$ My question is, what's the exact expression of such $G$?
If we take $x=0$. In Euclidean case, we know $G(y)=-\frac{1}{2\pi}\log |x|$. So I guess in the Hyperbolic case, $G$ should be $G=-\frac{1}{a}\log d(0,x)$, where $d(0,x)=\log \frac{1+|x|}{1-|x|}$ is the geodesic distance and $a$ is the measure of unit geodesic ball in $\mathbb{H}^2$.
But I didn't find any useful reference.
Am I right, or is there any reference paper that is related to this topic?
Any help will be appreciated a lot.