Queen Mary’s University, London, 13th December, 2019
Natasa Sesum. Classification of non-collapsed ancient solutions to the three-dimensional Ricci flow
Perelman conjectured that the only kappa-noncollapsed three dimensional ancient solutions to the Ricci flow are either the round cylinder, or the Bryant soliton, or the round sphere or the Perelman’s solution. The last solution is not a soliton. Brendle solved the conjecture in the complete, noncompact case. In joint work with Angenent, Brendle and Daskalopoulos we show that all closed noncollapsed ancient solutions have the unique precise asymptotics. Using that result, in a joint work with Brendle and Daskalopoulos we show Perelman’s conjecture is true in the compact case as well.
Miles Simon. On the regularity of Ricci flows coming out of metric spaces
We consider solutions to Ricci flow defined on manifolds M for a time interval (0,T) whose Ricci curvature is bounded uniformly in time from below, and for which the norm of the full curvature tensor at time t is bounded by c/t for some fixed constant c >1 for all t in (0,T). From previous works, it is known that if the solution is complete for all times t>0, then there is a limit metric space (M,d_0), as time t approaches zero. We show : if there is a open region V on which (V,d_0) is *smooth*, then the solution can be extended smoothly to time zero on V.
Burkhard Wilking. A fresh look at the Ricci flow ODE
For a Ricci flow solution of the Ricci flow, the curvature operator satisfies an reaction diffusion equation R’= \Delta R+ Q(R). The maximum principle then allows to deduce certain dynmalical properties of the PDE from the ODE R’=Q(R). By reinterpreting the curvature operator not just as on endomorphism of \Lamba^2 T_pM but also as an endomorphsim of the symmtric tensorproduct S^2T_pM one can rewrite the ODE R’=Q(R) in a simpler way without using the Lie algebra structure of the former vector space. We give various applications.