Green's function for the Laplacian on surfaces is considered, and
a mass-like quantity is derived from a regularization of Green's
function. A heuristic argument, inspired by the role of the
positive mass theorem in the solution to the Yamabe problem, gives
rise to a geometrical mass that is a smooth function on a
compact surface without boundary. The geometrical mass is shown
to be independent of the point on the sphere, and it is also a
spectral invariant. Moreover, a connection to a sharp
Sobolev-type inequality reveals that it is actually minimized at
the standard round metric. The behavior of the geometrical mass
on the sphere is markedly different from that on other surfaces.