TITLE: fundamental solution of the laplace Beltrami operator
QUESTION [8 upvotes]: It is well know that $s(z,z_0)=\frac{1}{2\pi} \ln \vert z-z_0\vert$ satisfies $\Delta_z s=\delta_{z_0}$ in dimension $2$. Does any one have a reference when we consider a general metric $g$ on an open set of $R^2$? that is to say which is the solution of $(\Delta_g)_z s=\delta_{z_0}$

REPLY [5 votes]: In general there is no explicit expression, but under some mild conditions on the metric, you know they exist, and you know their qualitative behaviour. For reference, I suggest Some nonlinear problems in Riemannian geometry by Thierry Aubin.