A geometrical mass and its extremal properties for metrics on S 2

Abstract

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.