Conformal positive mass theorems

We show the following two extensions of the standard positive mass theorem (one for either sign): Let (N,g) and (N,g') be asymptotically flat Riemannian 3-manifolds with compact interior and finite mass, such that g and g' are C-2,C-alpha and related via the conformal rescaling g' = phi(4)g with a C-2,C-alpha -function phi > 0. Assume further that the corresponding Ricci scalars satisfy R +/- phi(4) R' greater than or equal to 0. Then the corresponding masses satisfy m +/- m' greater than or equal to 0. Moreover, in the case of the minus sign, equality holds iff g and g' are isometric, whereas equality holds for the plus sign iff both (N,g) and (N,g') are flat Euclidean spaces. While the proof of the case with the minus signs is rather obvious, the case with the plus signs requires a subtle extension of Witten's proof of the standard positive mass theorem. The idea for this extension is due to Masood-ul-Alam who, in the course of an application, proved the rigidity part m + m' = 0 of this theorem, for a special conformal factor. We observe that Masood-ul-Alam's method extends to the general situation.