The Einstein−Λ flow on product manifolds

We consider the vacuum Einstein flow with a positive cosmological constant $\Lambda$ on spatial manifolds of product form $M=M_1\times M_2$. In dimensions $n=\dim M⩾4$ we show the existence of continuous families of recollapsing models whenever at least one of the factors M ₁ or M ₂ 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–$\Lambda$ 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. (paper)