Mathematics – Differential Geometry
Scientific paper
2009-06-02
Mathematics
Differential Geometry
Final version, to appear in J. Reine Angew. Math
Scientific paper
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-Kahler manifold. Differing from Fateev's examples most of our examples are non-collapsed. The construction of this paper, different from the ad hoc anastz of Fateev, is systematic, generalizing (as well as unifying) the previous constructions of Einstein metrics by Bourguignon-Karcher, Jensen, 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.
Bakas Ioannis
Kong Shengli
Ni Lei
No associations
LandOfFree
Ancient solutions of Ricci flow on spheres and generalized Hopf fibrations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Ancient solutions of Ricci flow on spheres and generalized Hopf fibrations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Ancient solutions of Ricci flow on spheres and generalized Hopf fibrations will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-193256