Ancient solutions of Ricci flow on spheres and generalized Hopf fibrations

Abstract

Abstract. Ancient solutions arise in the study of Ricci flow singularities. Motivated by the work of Fateev on 3-dimensional ancient solutions we construct high dimensional ancient solutions to Ricci flow on spheres and complex projective spaces as well as the twistor spaces over a compact quaternion-Kähler manifold. Di¤ering from Fateev's examples most of our examples are non-collapsed. The construction of this paper, di¤erent from the ad hoc anastz of Fateev, is systematic, generalizing (as well as unifying) the previous constructions of Einstein metrics by Bourguignon-Karcher, Jenson, and Ziller in the sense that the existence problem to a backward nonlinear parabolic PDE is reduced to the study of nonlinear ODE systems. The key step of solving the reduced nonlinear ODE system is to find suitable monotonic and conserved quantities under Ricci flow with symmetry. Besides supplying new possible singularity models for Ricci flow on spheres and projective spaces, our solutions are counter-examples to some folklore conjectures on ancient solutions of Ricci flow on compact manifolds. As a by-product, we infer that some nonstandard Einstein metrics on spheres and complex projective spaces are unstable fixed points of the Ricci flow.

Locally conformally flat ancient Ricci flows

“…In the non-rotationally symmetric case, the first construction is due to Fateev [17] in dimension three. Motivated by this construction, Bakas, Kong and Ni [4] produced high dimensional compact ancient solutions to the Ricci flow which are not rotationally symmetric. In dimension two, Daskalopoulos, Hamilton and Sesum [15] have obtained a complete classification of all compact ancient solutions to the Ricci flow.…”

On truncations of the Chalker-Coddington model

“…It is interesting to study directly this model when p 1 = p 2 , in order to recover the black-hole theory in the UV limit. We proceed as follows (closely following calculations in [49]. We adopt all notations from reference [42] where we have, after setting p 1 = p 2 ,…”

Kähler-Ricci shrinkers and ancient solutions with nonnegative orthogonal bisectional curvature

“…The examples in [1] seem to suggest that the results no longer hold if we drop the assumption of the non-collapsing. However, since the example of [1] is the Ricci flow of Hermitian metrics, it remains interesting to construct examples of the Kähler-Ricci flow.…”