The Long-Time Behavior of the Ricci Tensor under the Ricci Flow

We show that, given an immortal solution to the Ricci flow on a closed manifold with uniformly bounded curvature and diameter, the Ricci tensor goes to zero as . We also show that if there exists an immortal solution on a closed 3-dimensional manifold such that the product of the curvature and the square of the diameter is uniformly bounded, then this solution must be of type III. 1. Introduction The Ricci flow equation, introduced by Hamilton in [1], is the nonlinear partial differential equation where is a Riemannian metric on a fixed smooth manifold . Hamilton showed that, given any Riemannian metric on a closed manifold , there exists such that (1) has a solution defined for and which satisfies . A solution to the Ricci flow equation which is defined for all is called an immortal solution. If the solution is defined for all , it is said to be eternal. In this paper, we first consider immortal solutions to the Ricci flow which have a uniform bound on the curvature and a uniform upper bound on the diameter. We will use the following notation. Let be an -dimensional closed manifold and let be an immortal solution to the Ricci flow on . For each , we set The diameter of the induced metric structure on will be denoted by for each . The following theorem basically states that, given an immortal solution with a uniform bound on the curvature and on the diameter, the Ricci tensor must tend to zero in a uniform way as . Theorem 1. Given and numbers , , there exists a function with the following properties.(1) . (2)If is an immortal solution to the Ricci flow on a closed manifold of dimension such that and for all , then for all . Corollary 2. Suppose that is an immortal solution to the Ricci flow on a closed -dimensional manifold and suppose that, for some constants , , one has and for every . Then, as . The next theorem is about the singularity type of a certain class of immortal solutions on three-dimensional closed manifolds. Recall that an immortal solution to the Ricci flow on a closed manifold is said to have a type III singularity if . Theorem 3. Suppose that is an immortal solution to the Ricci flow on a closed 3-dimensional manifold and suppose that there exists a constant such that for all . Then, the solution is a type III solution. It is still an open question whether an immortal solution on a closed 3-dimensional manifold is necessarily of type III. 2. The proof We first prove Theorem 1 by assuming in addition a uniform lower bound on the injectivity radius. The proof of this simpler case will basically give us an outline of the proof for the