A conformal approach to the stability of Einstein spaces with spatial sections of negative scalar curvature

TL;DR

This paper develops a conformal method to analyze the non-linear stability of Einstein spaces with negative scalar curvature, leveraging conformal Gaussian systems and hyperbolic reductions to establish stability results.

Contribution

It introduces a conformal approach using conformal geodesics and Gaussian systems to study the stability of negatively curved Einstein spaces, extending previous methods.

Findings

Demonstrates the use of conformal Gaussian systems for hyperbolic reduction.

Establishes conditions for the global stability of certain Einstein spaces.

Reframes the stability problem as finite existence time for conformal evolution.

Abstract

In this article, it is shown how the extended conformal Einstein field equations and a gauge based on the properties of conformal geodesics can be used to analyse the non-linear stability of de Sitter-like spacetimes with spatial sections of negative scalar curvature. This class of spacetimes admits a smooth conformal extension with a space-like conformal boundary. Central to the analysis is the use of conformal Gaussian systems to obtain a hyperbolic reduction of the conformal Einstein field equations for which standard Cauchy stability results for symmetric hyperbolic systems can be employed. The use of conformal methods allows us to rephrase the question of the global existence of solutions to the Einstein field equations into considerations of finite existence time for the conformal evolution system.