Communications in Analysis and Geometry

Volume 28 (2020)

Number 7

Stable fixed points of the Einstein flow with positive cosmological constant

Pages: 1533 – 1576



David Fajman (Faculty of Physics, University of Vienna, Austria)

Klaus Kröncke (Department of Mathematics, University of Hamburg, Germany)


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.

Received 5 June 2015

Accepted 19 March 2018

Published 7 December 2020