Global Foliations of Vacuum Spacetimes with $T^2$ Isometry

TL;DR

This paper proves a global existence theorem for vacuum Einstein solutions with a $T^2$ isometry group, showing that such spacetimes with specific symmetry properties exist globally in a well-defined geometric time.

Contribution

It establishes the first global existence result for vacuum spacetimes with $T^2$ symmetry acting on $T^3$ surfaces, advancing understanding of symmetric solutions in general relativity.

Findings

Proves global existence of vacuum solutions with $T^2$ symmetry.

Demonstrates solutions are well-behaved over a geometrically-defined time.

Extends the class of known globally hyperbolic vacuum spacetimes.

Abstract

We prove a global existence theorem (with respect to a geometrically- defined time) for globally hyperbolic solutions of the vacuum Einstein equations which admit a $T^2$ isometry group with two-dimensional spacelike orbits, acting on $T^3$ spacelike surfaces.