Stable fixed points of the Einstein flow with positive cosmological constant

TL;DR

This paper proves the nonlinear stability of certain Einstein solutions with positive cosmological constant using the CMC Einstein flow, and explores the development of initial data and stability under perturbations.

Contribution

It establishes nonlinear stability for Einstein solutions with positive cosmological constant and arbitrary dimensions, including the development of CMC hypersurfaces and stability under perturbations.

Findings

Stability holds for a large class of solutions with positive cosmological constant.

Existence of CMC hypersurfaces in developments of non-CMC initial data.

Identification of a critical parameter leading to incomplete spacetime developments.

Abstract

We prove nonlinear stability for a large class of solutions to the Einstein equations with a positive cosmological constant and compact spatial topology in arbitrary dimensions, where the spatial metric is Einstein with either positive or negative Einstein constant. The proof uses the CMC Einstein flow and stability follows by an energy argument. We prove in addition that the development of non-CMC initial data close to the background contains a CMC hypersurface, which in turn implies that stability holds for arbitrary perturbations. Furthermore, we construct a one-parameter family of initial data such that above a critical parameter value the corresponding development is future and past incomplete.