Future Global in time einsteinian spacetimes with U(1) isometry group

TL;DR

This paper proves the long-term existence of certain vacuum Einstein spacetimes with U(1) symmetry, expanding infinitely in time, on manifolds with a compact surface of genus greater than one.

Contribution

It establishes the global-in-time existence of Einstein vacuum solutions with U(1) symmetry on specific manifolds, extending previous results to higher genus surfaces.

Findings

Existence of solutions for infinite proper time in the expanding direction.

Solutions are invariant under U(1) symmetry.

Applicable to manifolds with genus greater than one.

Abstract

We prove that spacetimes satisfying the vacuum Einstein equations on a manifold of the form $\Sigma \times U(1)\times R$ where $\Sigma $ is a compact surface of genus $G>1$ and where the Cauchy data is invariant with respect to U(1) and sufficiently small exist for an infinite proper time in the expanding direction.