The Einstein-Lambda flow on product manifolds

TL;DR

This paper studies the Einstein flow with a positive cosmological constant on product manifolds, showing conditions for recollapse or expansion based on curvature and topology, revealing diverse long-term behaviors in higher dimensions.

Contribution

It demonstrates the existence of recollapsing and expanding models on product manifolds with positive Einstein metrics, highlighting the role of curvature and topology in Einstein flow dynamics.

Findings

Recollapsing models exist when factors admit positive Einstein metrics.

Models with no positive curvature factors expand in one direction and collapse in another.

Positive curvature of a factor is necessary for recollapse in these models.

Abstract

We consider the vacuum Einstein flow with a positive cosmological constant on spatial manifolds of product form. In spatial dimension at least four we show the existence of continuous families of recollapsing models whenever at least one of the factors or admits a Riemannian Einstein metric with positive Einstein constant. We moreover show that these families belong to larger continuous families with models that have two complete time directions, i.e. do not recollapse. Complementarily, we show that whenever no factor has positive curvature, then any model in the product class expands in one time direction and collapses in the other. In particular, positive curvature of one factor is a necessary criterion for recollapse within this class. Finally, we relate our results to the instability of the Nariai solution in three spatial dimensions and point out why a similar construction of recollapsing models in that dimension fails. The present results imply that there exist different classes of initial data which exhibit fundamentally different types of long-time behavior under the Einstein flow whenever the spatial dimension is strictly larger than three. Moreover, this behavior is related to the spatial topology through the existence of Riemannian Einstein metrics of positive curvature.