A remark on the rigidity case of the positive energy theorem

TL;DR

This paper provides an alternative, more general proof of a key aspect of the positive energy theorem, applicable in various geometrical contexts without requiring a spin structure.

Contribution

It introduces a new proof method that extends to asymptotically hyperbolic cases, works in all dimensions, and avoids the need for a spin structure.

Findings

Proof applies to all dimensions n

Generalizes to asymptotically hyperbolic manifolds

Does not require a spin structure

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.