The Positive Energy Theorem for Asymptotically Hyperboloidal Initial Data Sets With Toroidal Infinity and Related Rigidity Results

TL;DR

This paper proves the positive energy theorem and related rigidity results for asymptotically hyperboloidal initial data with toroidal infinity, using spacetime harmonic functions, extending previous rigidity theorems with weaker assumptions.

Contribution

It establishes the positive energy theorem and rigidity results for a new class of initial data sets with toroidal infinity, and provides a novel proof of recent rigidity theorems with relaxed conditions.

Findings

Positive energy theorem for hyperboloidal data with toroidal infinity

Rigidity results showing zero energy implies isometry to Kottler spacetime

New proof of existing rigidity theorems with weaker hypotheses

Abstract

We establish the positive energy theorem and a Penrose-type inequality for 3-dimensional asymptotically hyperboloidal initial data sets with toroidal infinity, weakly trapped boundary, and satisfying the dominant energy condition. In the umbilic case, a rigidity statement is proven showing that the total energy vanishes precisely when the initial data manifold is isometric to a portion of the canonical slice of the associated Kottler spacetime. Furthermore, we provide a new proof of the recent rigidity theorems of Eichmair-Galloway-Mendes [10] in dimension 3, with weakened hypotheses in certain cases. These results are obtained through an analysis of the level sets of spacetime harmonic functions.