Limiting behavior of Ricci flows

Consider the unnormalized Ricci flow (g_ij) _t = −2R_ij for t ∈ [0, T), where T < [special characters omitted] on a closed, n-dimensional manifold M. Assume that a solution of the flow exists for all times t ∈ [0, [special characters omitted]) and that the curvatures and the diameters are uniformly bounded along the flow. We will prove that for every sequence t_i → [special characters omitted] there exists a subsequence such that g(t _i + 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.

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 t_i converging to infinity, there exists a subsequence such that (M, g(t _i + t)) → (Y, g¯(t)) 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 (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)