A remark on the rigidity case of the positive energy theorem

Abstract

In their proof of the positive energy theorem, Schoen and Yau showed that every asymptotically flat spacelike hypersurface M of a Lorentzian manifold which is flat along M can be isometrically imbedded with its given second fundamental form into Minkowski spacetime as the graph of a function from R^n to R; in particular, M is diffeomorphic to R^n. In this short note, we give an alternative proof of this fact. The argument generalises to the asymptotically hyperbolic case, works in every dimension n, and does not need a spin structure.