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Ancient solutions arise in the study of Ricci flow singularities. Motivatedby the work of Fateev on 3-dimensional ancient solutions we construct highdimensional ancient solutions to Ricci flow on spheres and complex projectivespaces as well as the twistor spaces over a compact quaternion-Kahler manifold.Differing from Fateev's examples most of our examples are non-collapsed. Theconstruction of this paper, different from the ad hoc anastz of Fateev, issystematic, generalizing (as well as unifying) the previous constructions ofEinstein metrics by Bourguignon-Karcher, Jensen, and Ziller in the sense thatthe existence problem to a backward nonlinear parabolic PDE is reduced to thestudy of nonlinear ODE systems. The key step of solving the reduced nonlinearODE system is to find suitable monotonic and conserved quantities under Ricciflow with symmetry. Besides supplying new possible singularity models for Ricciflow on spheres and projective spaces, our solutions are counter-examples tosome folklore conjectures on ancient solutions of Ricci flow on compactmanifolds. As a by-product, we infer that some nonstandard Einstein metrics onspheres and complex projective spaces are unstable fixed points of the Ricciflow.

Ancient solutions of Ricci flow on spheres and generalized Hopf fibrations