Abstract
We complement a recent work on the stability of fixed points of the CMC-Einstein-Λ flow. In particular, we modify the utilized gauge for the Einstein equations and remove a restriction on the fixed points whose stability we are able to prove by this method, and thereby generalize the stability result. In addition, we consider the notion of the reduced Hamiltonian, originally introduced by Fischer and Moncrief for the standard CMC-Einstein flow. For the analog version of the flow in the presence of a positive cosmological constant we identify the stationary points and relate them to the long-time behavior of the flow on manifolds of different Yamabe types. This entails conjectures on the asymptotic behaviour and potential attractors.
Highlights
Determining the long-time behaviour of the Einstein flow on compact manifolds without boundary is one of the central objectives of mathematical cosmology
This behaviour is independent of the topology of the spatial hypersurfaces of spacetime due to the fast expansion rate in the case Λ > 0, which causes a localization of perturbations in space
In a recent paper [FK15], the authors provide an alternative proof of this stability result using the constant-mean-curvature-spatial-harmonic gauge, originally introduced in the work of Andersson and Moncrief [?, AM11]. This approach leads to a very concise proof by a suitably arranged energy estimate. This method does not cover the full result of Ringström but only those cases where the perturbed spacetime allows for a CMC foliation with the mean curvature being a time-function
Summary
Determining the long-time behaviour of the Einstein flow on compact manifolds without boundary is one of the central objectives of mathematical cosmology. In a recent paper [FK15], the authors provide an alternative proof of this stability result using the constant-mean-curvature-spatial-harmonic gauge, originally introduced in the work of Andersson and Moncrief [?, AM11]. This approach leads to a very concise proof by a suitably arranged energy estimate. This modified spatial harmonic gauge is introduced in this paper Another question concerning the CMC-Einstein-Λ flow regards the existence of other attractors except for the spatial Einstein geometries. We draw some conclusions on the possibility for existence of data not evolving to a spatial Einstein geometry
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
