| Two mass-like quantities arising from the Green's function for the Laplacian operator on surfaces are considered, along with a higher dimensional analog.; On a surface, the Green's function for the Laplacian has a logarithmic singularity, and the Robin's mass is obtained by regularizing that singularity. Two regularizations for the Green's function are compared, and as a result, the Robin's mass is shown to be a density for a spectral invariant. Additionally, a variational formula is computed for the change of the Robin's mass corresponding to a conformal change of metric.; On spheres, 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', which is a smooth function on the sphere. 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 minimized at the standard round metric.; The results for the Robin's mass and the geometrical mass hold in even dimensions n ≥ 2. When n > 2, the Laplacian is replaced by the Paneitz operator, an nth order elliptic operator. |
