On the Conditions to Extend Ricci Flow

Consider {(M, g(t)), 0 ≤ t < T < ∞} as an unnormalized Ricci flow solution: dgij dt = −2Rij for t ∈ [0, T ). Richard Hamilton shows that if the curvature operator is uniformly bounded under the flow for all t ∈ [0, T ) then the solution can be extended over T . Natasa Sesum proves that a uniform bound of Ricci tensor is enough to extend the flow. We show that if Ricci is bounded from below, then a scalar curvature integral bound is enough to extend flow, and this integral bound condition is optimal in some sense. 1 When can Ricci flow be extended? In (7), R. Hamilton introduces Ricci flow which deforms Riemannian metrics in the direction of the Ricci tensor. One hopes that the Ricci flow will deform any Riemannian metric to some canonical metrics, such as Einstein metrics. One can even understand geometric and topological structure of the underlying differential manifold by this sort of deformation. The idea is best illustrated in (7) where Hamilton proves that in any simply connected 3 manifold without boundary, any Riemannian metric with positive Ricci curvature can be deformed into a positive space form (up to scaling). Consequently, R. Hamilton proves that the underlying manifold is indeed diffeomorphic to S. This fundamental work sparks a great interest of many mathematicians in Ricci flow. In a series of work, R. Hamilton introduces an ambitious program to prove the Poincarè conjecture via Ricci flow (cf. (9) for Hamilton’s program and early references in Ricci flow.). The celebrated work of G. Perelman (14), (15) and (16) indeed proves the Poincarè conjecture which states that every simply connected 3 manifold is S. We refer the readers to (12), (13) for more information. After Perelman’s work in the Ricci flow, there is a renewed interest in Ricci flow and its application around the world. We will refer readers to the book (4) for more updated references. In this note, we want to concentrate in studying some basic issue on Ricci flow: the maximal existence time of Ricci flow and the geometric conditions that might affect the maximal existence time. One notes that Ricci flow is a weak Parabolic flow. R. Hamilton first proves that for any smooth initial data, the flow will exist for a short time in (7). In (6), Hamilton’s proof is simplified greatly by a clever choice of gauge. The next immediate question is the so called “maximal existence time” for the Ricci flow (with respect to initial metric). In (9), Hamilton proves that if T < ∞ is the maximal existence time of a closed Ricci flow solution {(M, g(t)), 0 ≤ t <