Global Regularity for the 2+1 Dimensional Equivariant Einstein-Wave Map System

TL;DR

This paper proves global existence for the equivariant 2+1 dimensional Einstein-wave map system under certain conditions, with implications for 3+1 vacuum Einstein equations with symmetry.

Contribution

It establishes global regularity results for the Einstein-wave map system satisfying the Grillakis condition, extending understanding of spacetime evolution.

Findings

Global existence for the Einstein-wave map system under Grillakis condition

Reduction of 3+1 vacuum Einstein equations to 2+1 system with hyperbolic target

Applicability to equivariant spacetimes with spacelike translational symmetry

Abstract

In this paper we consider the equivariant 2+1 dimensional Einstein-wave map system and show that if the target satisfies the so called Grillakis condition, then global existence holds. In view of the fact that the 3+1 vacuum Einstein equations with a spacelike translational Killing field reduce to a 2+1 dimensional Einstein-wave map system with target the hyperbolic plane, which in particular satisfies the Grillakis condition, this work proves global existence for the equivariant class of such spacetimes.