The Jang equation reduction of the spacetime positive energy theorem in dimensions less than eight

TL;DR

This paper extends the Jang equation proof of the positive energy theorem to dimensions less than eight, addressing new technical challenges and weakening initial data assumptions.

Contribution

It generalizes the Jang equation approach to higher dimensions and introduces a new density argument for the positive mass theorem.

Findings

Successfully extended proof to 3 ≤ n < 8 dimensions

Weaker regularity and decay assumptions used

Provided an alternative derivation of the positive mass theorem

Abstract

We extend the Jang equation proof of the positive energy theorem due to R. Schoen and S.-T. Yau from dimension $n=3$ to dimensions $3 \leq n <8$. This requires us to address several technical difficulties that are not present when $n=3$. The regularity and decay assumptions for the initial data sets to which our argument applies are weaker than those of R. Schoen and S.-T. Yau. In recent joint work with L.-H. Huang, D. Lee, and R. Schoen we have given a different proof of the full positive mass theorem in dimensions $3 \leq n < 8$. We pointed out that this theorem can alternatively be derived from our density argument and the positive energy theorem of the present paper.