Conformal positive mass theorems

Abstract

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 twice Hoelder differentiable and related via the conformal rescaling g' = (phi^4).g, with a twice Hoelder differentiable function phi>0. Assume further that the corresponding Ricci scalars satisfy either R + (phi^4).R' >= 0 or R - (phi^4).R' >= 0. Then the corresponding masses satisfy m + m' >= 0 or m - m' >= 0, respectively. Moreover, in the case of the minus signs, equality holds iff g and g' are isometric, whereas for the plus signs equality holds 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.