Conformal Positive Mass Theorems

Abstract

We show the following two extensions of the standard positive mass theorem (one for either sign): Let $\mathcal{N}$,g) and $\mathcal{N}$,g' be asymptotically flat Riemannian 3-manifolds with compact interior and finite mass, such that g and g' are C ^2,α and related via the conformal rescaling g' = φ ⁴ g with a C ^2,α -function φ > 0. Assume further that the corresponding Ricci scalars satisfy R ± φ ⁴ R'≥; 0. Then the corresponding masses satisfy m±m' ≥ 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 $\mathcal{N}$,g) and $\mathcal{N}$,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.