Non linear stability of einsteinian spacetimes with U(1) isometry group

TL;DR

This paper proves the global stability of certain Einstein spacetimes with U(1) symmetry, showing that small, symmetric vacuum initial data lead to complete, expanding solutions without restrictive spectral assumptions.

Contribution

It establishes nonlinear stability and completeness of Einstein spacetimes with U(1) symmetry without imposing previous eigenvalue restrictions.

Findings

Global completeness in the expanding direction.

Decay of total energy over time.

Stability results hold without restrictive eigenvalue assumptions.

Abstract

We prove global completeness in the expanding direction of spacetimes satisfying the vacuum Einstein equations on a manifold of the form $\Sigma \times S^{1}\times R$ where $\Sigma $ is a compact surface of genus $G>1.$ The Cauchy data are supposed to be invariant with respect to the group $S^{1}$ and sufficiently small, but we do not impose a restrictive hypothesis made in gr-qc 0112049 on the lowest eigenvalue of a relevant Laplacian. The total energy decay still holds, but its rate depends of the asymptotic value of this eigenvalue.