Y. K. Wang

We show that a strict, nearly Kähler $6$-manifold with either second or third Betti number nonzero is linearly unstable with respect to the $\nu$-entropy of Perelman and hence is dynamically unstable for the Ricci flow.

We generalize the ancient solutions of the Ricci flow on certain principal $\mathrm{SO}(3)$ bundles over compact quaternionic Kähler manifolds constructed by Bakas, Kong, and Ni to certain $R{P}^3$ fibre bundles over a product of two compact quaternionic Kähler manifolds. The ancient solutions are of Type I, $\kappa$-noncollapsed, and have positive Ricci curvature. Analogous solutions on even-dimensional non-Kählerian bundle spaces are also constructed.

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 $\kappa$-collapsed and $\kappa$-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 $\kappa$-collapsed ancient solutions of type I are also obtained on torus bundles for all dimensions $\ge 8$.