Ancient Ricci Flow Solutions on Bundles

TL;DR

This paper extends known ancient Ricci flow solutions on circle bundles to a broader class of torus bundles over Fano Kähler-Einstein manifolds, revealing new examples in higher dimensions and topological variations.

Contribution

It generalizes ancient Ricci flow solutions to principal torus bundles over complex manifolds, including new examples in dimensions 7 and above with diverse topologies.

Findings

Constructs continuous families of ancient solutions on circle bundles for all odd dimensions ≥7.

Provides examples of solutions on pairs of manifolds that are homeomorphic but not diffeomorphic in dimension 7.

Extends solutions to torus bundles in all dimensions ≥8.

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 K\"ahler-Einstein manifolds studied by Wang and Ziller in the context of Einstein geometry. As a result, continuous families of $\kappa$-collapsed and $\kappa$-noncollapsed ancient solutions of type I are obtained on circle bundles for all odd dimensions $\geq 7$. In dimension $7$ such examples moreover exist on pairs of homeomorphic but not diffeomorphic manifolds. Continuous families of $\kappa$-collapsed ancient solutions of type I are also obtained on torus bundles for all dimensions $\geq 8$.