I just met a result in Lemma5.1 of ref1 which seems interesting and counterintuitive, it said that on 2-dimension torus $\Omega$, let $G$ be the Green's function to $-\Delta$ on $\Omega$, satisfying $\int_{\Omega} G(x, y) d y=0$ for all $x$. We have $$ G(x, y)=\frac{1}{2 \pi} \ln \frac{1}{|x-y|}+\gamma(x, y), $$ where $\gamma$, the regular part of $G$, is smooth on $\Omega \times \Omega$. Here, the $| \cdot |$ is the distance in Euclidean space instead of the distance on the torus, I was astonished because this is a little bit counterintuitive, so I searched the case for the 2-sphere, in ref2, the authors computed the Green function on 2-sphere and got $$G(r)=-\frac{1}{2 \pi} \log \sin \frac{r}{2}+C,$$ here $r$ is the angle, which is the distance on the sphere, so the $2sin \frac{r}{2}$ is the distance in Euclidean space.
So I began to wonder, is it true that for all Green function to $-\Delta$ on closed surface $\Omega$, the fomula for Green function is $$ \frac{1}{2 \pi} \ln \frac{1}{|x-y|}+\gamma(x, y), $$ with the $| \cdot |$ being the distance in Euclidean space? Unfortunately, I searched for days and found nothing, I would be very glad if you know something about this!