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.
"A geometrical mass and its extremal properties for metrics on ." Duke Math. J. 129 (1) 63 - 86, 15 July 2005. https://doi.org/10.1215/S0012-7094-04-12913-6