Positive mass theorems for higher dimensional Lorentzian manifolds

Abstract

We extend Witten’s proof [“A new proof of the positive energy theorem,” Commun. Math. Phys. 80, 381–402 (1981)] of the positive mass theorem to a rigorous proof for Lorentzian manifolds of any dimension. This includes the original higher-dimensional positive mass theorem proved by spinors [Parker, T. and Taubes, C., “On Witten’s proof of the positive energy theorem,” Commun. Math. Phys. 84, 223–238 (1982)] for dimension 4 and [Zhang, X., “Positive mass conjecture for five-dimensional Lorentzian manifolds,” J. Math. Phys. 40, 3540–3552 (1999)] for dimension 5. Stimulated by the article of Zhang [“Positive mass theorem for hypersurface in 5-dimensional Lorentzian manifolds,” Commun. Anal. Geom. 8, 635–652 (2000)], we weaken the spin condition on the spacelike hypersurface to require only a spinc structure and give a modified positive mass theorem for Lorentzian manifolds for dimensions 4, 5, and 6.