On the CMC-Einstein-Lambda flow

TL;DR

This paper extends the stability analysis of the CMC-Einstein-$ Lambda$ flow by modifying gauges and removing restrictions, and explores the flow's long-term behavior and stationary points in the presence of a positive cosmological constant.

Contribution

It generalizes the stability results of the CMC-Einstein-$ Lambda$ flow by altering gauges and analyzing stationary points related to the flow's asymptotics.

Findings

Identified stationary points of the flow with positive cosmological constant.

Related stationary points to the long-term behavior on different Yamabe types.

Formulated conjectures on asymptotic behavior and attractors.

Abstract

We complement a recent work on the stability of fixed points of the CMC-Einstein-$\Lambda$ 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.