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 Rn ! R; in particular,M is diffeomorphic to Rn.
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.