Université libre de Bruxelles, 25/11/2013
“Ricci flow and sphere theorems.”
A “sphere theorem” characterizes the standard sphere as the only compact, simply connected smooth manifold admitting a Riemannian metric, which is sufficiently positively curved. We state several sphere theorems and indicate their proof using Ricci flow and Lie-theoretic methods.
“Long time behaviour of Ricci flow on open 3-manifolds.”
Let M be a compact, orientable 3-manifold with toral boundary. Thurston’s hyperbolization conjecture, proved by Perelman in the closed case, and Thurston in the non-closed case, states that if M is irreducible and atoroidal, then M is hyperbolic or Seifert-fibered. We give a unified proof of this result by showing that under mild assumptions on the initial metric, Ricci flow with surgery on the interior of M is well-defined, and converges to a hyperbolic metric when M is not Seifert. This is joint work with Laurent Bessières and Gérard Besson.
“Codazzi tensors and locally conformally flat Ricci flows.”
I will discuss the applications of the Codazzi tensors to the study of Ricci solitons or more in general of (ancient) Ricci flows, in particular, under the assumption that at every time the manifolds are locally conformally flat.