Ancient Ricci Flow Solutions on Bundles

Abstract

We generalize the circle bundle examples of ancient solutions of the Ricci flow discovered by Bakas, Kong, and Ni to a class of principal torus bundles over an arbitrary finite product of Fano Kähler-Einstein manifolds studied by Wang and Ziller in the context of Einstein geometry. As a result, continuous families of $\hat{I}º$-collapsed and $\hat{I}º$-noncollapsed ancient solutions of type I are obtained on circle bundles for all odd dimensions $\ge 7$. In dimension $7$ such examples moreover exist on pairs of homeomorphic but not diffeomorphic manifolds. Continuous families of $\hat{I}º$-collapsed ancient solutions of type I are also obtained on torus bundles for all dimensions $\ge 8$.

40 pages, no figure