Limiting behavior of Ricci flows

Abstract

Consider the unnormalized Ricci flow ...Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times ... then the solution can be extended beyond T. In the thesis we prove that if the Ricci curvature is uniformly bounded under the flow for all times ... then the curvature tensor has to be uniformly bounded as well. In particular, this means that if the Ricci tensor stays uniformly bounded up to a finite time T, a Ricci flow can not develop a singularity at T. We will give two different proofs of that result. One of them relies on Hamilton's estimates on distance changes along the flow and the other one relies on the identities for reduced distances and the monotonicity formula for reduced volumes that has been introduced and proved by Perelman in [29]. Consider the Ricci flow ... on a closed, n-dimensional manifold M. Assume that a solution of the flow exists for all times ... and that the curvatures and the diameters are uniformly bounded along the flow. We will prove that for every sequence ... there exists a subsequence such that g(ti + t) converges to a metric h(t) and h(t) is a Ricci soliton. We will also prove that if one of the limit solitons is integrable, then a soliton that we get in the limit is unique up to diffeomorphisms and the convergence toward it is exponential.

(cont.) We will also prove that in an arbitrary dimension, for a given Kähler-Ricci flow with uniformly bounded Ricci curvatures, for every sequence of times ti converging to infinity, there exists a subsequence such that ... and the convergence is smooth outside a singular set (which is a set of codimension at least 4). Moreover, g(t) is a solution of the flow off the singular set. In the case of a complex dimension 2, for any sequence of times converging to infinity we can find a subsequence of times such that we have a convergence toward a Kähler-Ricci soliton, away from finitely many isolated singularities.