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Geometry
Volume 2013 (2013), Article ID 235436, 4 pages
http://dx.doi.org/10.1155/2013/235436
Research Article

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

Department of Mathematics, University of California-Berkeley, Berkeley, CA 94720-3840, USA

Received 17 June 2013; Accepted 19 August 2013

Academic Editor: Giovanni Calvaruso

Copyright © 2013 Christian Hilaire. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 general case. Given a Ricci flow solution on a closed manifold , the injectivity radius corresponding to each metric will be denoted by .

Theorem 4. Given and numbers , , , there exists a function with the following properties.(1)Consider .(2)If is an immortal solution to the Ricci flow on a closed manifold of dimension such that , , and for all , then for all .

Proof. For every , we set to be the supremum of over all immortal solutions satisfying the conditions of the given hypothesis. This function is well defined since the bound on the curvature gives us a bound on the Ricci curvature. We just need to prove that .
Suppose that this did not hold. Then, for some , there is a sequence such that . This implies that there exists a sequence of pointed immortal solutions to the Ricci flow such that . Consider the new sequence where for . Since we have a uniform bound on the curvature and a uniform lower bound on the injectivity radius, we can apply the Hamilton-Cheeger-Gromov compactness theorem [2]. After passing to a subsequence, we can assume that the sequence converges to an eternal Ricci flow solution on a pointed -dimensional manifold . The uniform bound on the diameters of implies that is a closed manifold.
The eternal Ricci flow solution must be Ricci flat. Indeed, based on [3, Lemma 2.18], the solution is either Ricci flat or its scalar curvature is positive. Since is closed, the latter cannot happen for, by [3, Corollary 2.16], there would be a finite-time singularity and this would contradict the fact that is an eternal solution. Therefore, the Ricci flow solution satisfies as claimed. On the other hand, the choice of the points implies that . We reach a contradiction. The function must satisfy . This completes the proof.

The proof of the previous theorem is a simple application of the following results: (1)Hamilton’s compactness theorem for solutions to the Ricci flow,(2)an eternal solution to the Ricci flow on a closed manifold is Ricci flat. This result is a simple application of the weak and strong maximum principles.

In the proof of Theorem 1, we will use a more general version of Hamilton’s compactness theorem which was introduced by Lott in [4] in the bounded curvature case. We will again construct a pointed sequence of Ricci flow solutions. But in this case, after passing to a subsequence if necessary, the pointed sequence will converge, in the sense of [4, Section 5], to an eternal Ricci flow solution on a pointed étale groupoid with compact connected orbit space. The result will then follow by a simple application of the weak and strong maximum principles in the case of étale groupoids. For basic information about groupoids, we refer to [5, 6]. For information about Riemannian groupoids and their applications in Ricci flow, we refer to [4]. We will denote an étale groupoid by a pair where is the space of “arrows” and is the space of units. The source and range maps will be denoted by and , respectively.

Lemma 5 (weak maximum principle for scalars on groupoids). Suppose that , , is a smooth one-parameter family of metrics on a smooth étale groupoid with compact connected orbit space . Let be a smooth -invariant time-dependent vector field on and let be a smooth function. Suppose that is a smooth -invariant function which solves Suppose further that solves
If , then for all .

Proof. The proof is basically the same as the corresponding proof for closed manifolds (see [7, page 35-36]). For , we consider the ODE for a new function . Using basic ODE theory, we see that the existence of asserted in the hypotheses and the fact that imply that for some there exists a solution on for any . Furthermore, uniformly as . Thus, it suffices to show that for all and arbitrary .
If this were not true, then we could choose and , where fails. We may assume that is the earliest such time and pick such that . Indeed, such a pair exists because the -invariant function descends to a continuous function . Based on the hypothesis that is compact and connected, we can find such that it is the earliest time, where fails, and pick an orbit so that . This gives the required pair . Now, we know that is negative for and zero for . Hence, we must have Moreover, since achieves a maximum at , we have and .
Combining these facts with the inequality (4) for and (6) for , we get the contradiction
This completes the proof.

If is a solution to the Ricci flow on a smooth étale groupoid , then, by definition, the elements of act by isometries for each Riemannian metric . More precisely, recall that, for any étale groupoid , there is an associated pseudogroup of local diffeomorphisms of . The elements of this pseudogroup are the local diffeomorphisms of the form , where is a local section of the source map ; that is, satisfies . Given a Riemannian metric on , the elements of are said to act by isometries if the elements of the associated pseudogroup are local isometries. Therefore, given a solution to the Ricci flow on a smooth étale groupoid, the geometric quantities derived from the metrics are -invariant; that is, they are invariant under the action of the associated pseudogroup. It follows that the basic evolution equations for the Ricci flow which are derived by applying the weak maximum principle on a closed manifold are still valid in the case of the Ricci flow on a smooth étale groupoid with compact connected orbit space.

Lemma 6. Every eternal solution to the Ricci flow on a smooth étale groupoid with compact connected orbit space is Ricci flat.

Proof. Suppose that is an eternal solution to the Ricci flow on a smooth étale groupoid with compact connected orbit space . Just as in the closed manifold case (see [7, Corollary 3.2.5]), one can apply Lemma 5 to show that for any solution to the Ricci flow defined on an interval the scalar curvature satisfies for . Fix . By translating time, we see that the previously mentioned inequality implies that for . Taking the limit as , we conclude that for all .
Now suppose that for some . By the strong maximum principle [3, Corollary 6.55], this implies that for all and for all in the connected component of which contains . In fact, this holds for all . Indeed, since is connected, it follows that, for any , one can find a sequence of points such that and are in the same connected component of and and are in the same -orbit for each . Since is -invariant, it follows that . So, by the maximum principle, for every and for all in the connected component of which contains . In particular, for all . After applying this argument a finite number of times, we deduce that for all . This proves the claim.
The -invariant function induces a continuous function on which satisfies for . Since is compact, there exists some such that . Hence, the scalar curvature satisfies . Just as in the closed manifold case (see [7, Corollary 3.2.3]), we can apply Lemma 5 to show that we have a finite-time singularity. This contradicts the hypothesis that the Ricci flow solution is eternal. Therefore, we must have . Based on the evolution equation of the scalar curvature we deduce that . This completes the proof.

Proof of Theorem 1. We define the function as before: for every , we set to be the supremum of over all immortal solutions satisfying the conditions of the given hypothesis. We just need to prove that . This will again be proved by contradiction.
Suppose that this did not hold. Then, for some , there is a sequence such that . This implies that there exists a sequence of pointed immortal solutions to the Ricci flow such that . Consider the new sequence , where for . After passing to a subsequence, we can assume that the sequence converges to an eternal Ricci flow solution on a pointed -dimensional étale groupoid (see [4, Theorem 5.12]). The uniform bound on the diameters of implies that the quotient space equipped with the metric induced by is a locally compact complete length space with finite diameter. So, by the Hopf-Rinow theorem ([5, Part I, Proposition 3.7]), is a compact connected space. We can now apply Lemma 6 to deduce that the eternal Ricci flow solution on the étale groupoid is Ricci flat. On the other hand, the choice of the points implies that on . We reach a contradiction. The function must satisfy . This completes the proof.

The proof of Theorem 3 will also follow from Lemma 6.

Proof of Theorem 3. The proof will again be by contradiction. Suppose that the solution satisfies . As outlined in [3, Chapter 8, Section 2], we can construct a sequence of pointed solutions to the Ricci flow which converges to an eternal solution to the Ricci flow on an étale groupoid. We first choose a sequence of times and then we choose a sequence such that For each , we set and we define the pointed Ricci flow solution by . If we set and , then it is shown in [3, Chapter 8, Section 2] that and and that the curvature of the metric satisfies for . Furthermore, condition (3) implies that Hence, after passing to a subsequence if necessary, the pointed sequence converges to an eternal Ricci flow solution on a pointed 3-dimensional étale groupoid with compact connected orbit space. By construction, on the orbit . But, by Lemma 6, the solution is flat. We get a contradiction. Therefore, the immortal solution must be of type III.

Acknowledgment

The author would like to thank his adviser John Lott for the helpful discussions and comments on earlier drafts of this paper.

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