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 t goes to infinity. We also show that if there exists an immortal solution on a closed 3-dimensional manifold such that the product of the square of the diameter with the norm of the curvature tensor is uniformly bounded, then the solution must be of type III.


Introduction
The Ricci flow equation, introduced by R. Hamilton in [3], is the nonlinear partial differential equation where g(t) is a Riemannian metric on a fixed smooth manifold M . Hamilton showed that, given any Riemannian metric g 0 on a closed manifold M , there exists T > 0 such that the equation (1) has a solution g(t) defined for t ∈ [0, T ) and which satisfies g(0) = g 0 . A solution to the Ricci flow equation which is defined for all t ≥ 0 is called an immortal solution. If the solution is defined for all t ∈ R, 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 M be an n-dimensional closed manifold and let g(t) be an immortal solution to the Ricci flow on M . For each t ≥ 0, we set ||Rm|| ∞ (t) = max{|Rm|(x, t) : x ∈ M } and ||Ric|| ∞ (t) = max{|Ric|(x, t) : x ∈ M }.
The diameter of the induced metric structure on M will be denoted by diam (M ; g(t)) for each t ≥ 0. 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 t → ∞.  (2) If g(t) is an immortal solution to the Ricci flow on a closed manifold M of dimension n such that ||Rm|| ∞ (t) ≤ K and diam (M ; g(t)) ≤ D for all t ≥ 0, then ||Ric|| ∞ (t) ≤ F n,K,D (t) for all t ≥ 0.
is an immortal solution to the Ricci flow on a closed n-dimensional manifold M and suppose that, for some constants The next theorem is about the type of a certain class of immortal solutions on three-dimensional closed manifolds. Recall that an immortal solution g(t) to the Ricci flow on a closed manifold M is said to be of type III if sup M ×[0,∞) t|Rm| < ∞.
is an immortal solution to the Ricci flow on a closed 3-dimensional manifold M and suppose there exists a constant C > 0 such that for all t ≥ 0. Then, the solution g(t) is a type III solution.
It is still an open question whether an immortal solution on a closed 3-dimensional manifold is nececessarily of type III.
Acknowledgements: I would like to thank my adviser John Lott for helpful discussions and comments on earlier drafts of this paper.

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 g(t) on a closed manifold M , the injectivity radius corresponding to each metric g(t) will be denoted by inj (M ; g(t)). (2) If g(t) is an immortal solution to the Ricci flow on a closed manifold M of dimension n such that inj(M ; g(t)) ≥ ι, ||Rm|| ∞ (t) ≤ K and diam (M ; g(t)) ≤ D for all t ≥ 0, then ||Ric|| ∞ (t) ≤ F n,K,D,ι (t) for all t ≥ 0.
Proof. For every t ≥ 0, we set F n,K,D,ι (t) to be the supremum of ||Ric|| ∞ (t) over all immortal solutions (M n , g(.)) 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 lim t→∞ F n,K,D,ι (t) = 0.
Suppsose that this did not hold. Then, for some ǫ > 0, there is a se- 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 [4]. After passing to a subsequence, we can assume that the sequence (M i ,g i (t), x i ) converges to an eternal Ricci flow solutionĝ(t) on a pointed n-dimensional manifold (M ,x). The uniform bound on the diameters of (M i ,g i ) imply thatM is a closed manifold.
The eternal Ricci flow solutionĝ(t) must be Ricci flat. Indeed, based on [2, Lemma 2.18], the solution is either Ricci flat or its scalar curvature is positive. SinceM is closed, the latter cannot happen for, by [2, Corollary 2.16], there would be a finite time singularity and thus would contradict the fact thatĝ(t) is an eternal solution. Therefore, the Ricci flow solution g satisfies Ric (ĝ(t)) ≡ 0 as claimed. On the other hand, the choice of the points x i ∈ M i implies that |Ric|(x, 0) > ǫ. We reach a contradiction. The function F n,K,D,ι (t) must satisfy lim t→∞ F n,K,D,ι (t) = 0. 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, in the bounded curvature case, a more general version of Hamilton's compactness theorem which was introduced by J. Lott in [5]. 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 [5,Section 5], to an eternal Ricci flow solution on a pointedétale 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 ofétale groupoids. For basic informations about groupoids, we refer to [1,6]. For information about Riemannian groupoids and their applications in Ricci flow, we refer to [5]. We will denote anétale groupoid by a pair (G, M ) where G is the space of "arrows" and M is the space of units. The source and range maps will be denoted by s : G → M and r : G → M respectively.
Lemma 5 (Weak maximum principle for scalars on groupoids). Suppose g(t), 0 ≤ t ≤ T < ∞, is a smooth one-paramater family of metrics on a smoothétale groupoid (G, M ) with compact connected orbit space W = M/G. Let X(t) be a smooth G-invariant time-dependent vector field on M and let Proof. The proof is basically the same as the corresponding proof for closed manifolds (see [7, p. 35-36]). For ǫ > 0, we consider the ODE for a new function φ ǫ : [0.T ] → R. Using basic ODE theory, we see that the existence of φ asserted in the hypotheses and the fact that T < ∞ imply that for some ǫ 0 > 0 there exists a solution φ ǫ on [0, T ] for any 0 < ǫ ≤ ǫ 0 . Furthermore, φ ǫ → φ uniformly as ǫ → 0. Thus, it suffices to show that u(., t) < φ ǫ (t) for all t ∈ [0, T ] and arbitrary ǫ ∈ (0, ǫ 0 ).
Combining these facts with the inequality (3) for u and equation (5) for φ ǫ , we get the contradiction for t ∈ [0, T ]. Fix c ∈ R. By translating time, we see that the above inequality implies that for (x, t) ∈ M × (c, ∞). Taking the limit as c → −∞, we conclude that R(x, t) ≥ 0 for all (x, t) ∈ M × R. Now suppose that R(x 0 , t 0 ) > 0 for some (x, t 0 ) ∈ M × R. By the strong maximum principle ([2, Corollary 6.55]), this implies that R(x, t) > 0 for all t ≥ t 0 and for all x in the connected component of M which contains x 0 . In fact, this holds for all x ∈ M . Indeed, since W is connected, it follows that for any x ∈ M , one can find a sequence of points x 0 = p 0 , q 0 , p 1 , q 1 , · · · , p n , q n = x such that p i and q i are in the same connected component of M , and q i−1 and p i are in the same G-orbit for each i. Since R is G-invariant, it follows that R(p 1 , t 0 ) = R(q 0 , t 0 ) > 0. So, by the maximum principle, R(x, t) > 0 for every t ≥ t 0 and for all x in the connected component of M which contains p 1 . In particular, R(q 1 , t) > 0 for all t ≥ t 0 . After applying this argument a finite number of times, we deduce that R(x, t) > 0 for all t ≥ t 0 . This proves the claim.
The G-invariant function R induces a continuous functionR on W which satifsiesR(., t) > 0 for t ≥ t 0 . Since W is compact, there exists some α > 0 such thatR(., t 0 ) ≥ α. Hence the scalar curvature satisfies R(., t 0 ) ≥ α. Just as in the closed manifold case (see [ we deduce that Ric ≡ 0. This completes the proof. Proof. (Theorem 1).We define the function F n,K,D (t) as before: for every t ≥ 0, we set F n,K,D (t) to be the supremum of ||Ric|| ∞ (t) over all immortal solutions (M n , g(.)) satisfying the conditions of the given hypothesis.We just need to prove that lim t→∞ F n,K,D (t) = 0. This will again be proved by contradiction.
Suppose that this did not hold. Then, for some ǫ > 0, there is a sequence t i → ∞ such that F n,K,D (t i ) > ǫ. This implies that there exists a sequence (M, g i , x i ) of pointed immortal solutions to the Ricci flow such that |Ric|(x i , t i ) > ǫ. Consider the new sequence (M,g i , . After passing to a subsequence, we can assume that the sequence (M i ,g i (t), x i ) converges to an eternal Ricci flow solutionĝ(t) on a pointed n-dimensionalétale groupoid (Ĝ,M , Ox) (see [5,Theorem 5.12]). The uniform bound on the diameters of (M i ,g i ) imply that the quotient space W equipped with the metric induced by g(t) is a locally compact complete length space with finite diameter. So, by the Hopf-Rinow theorem ([1, Part I Proposition 3.7]), W is a compact connected space. We can now apply Lemma 6 to deduce that the eternal Ricci flow solution on thé etale groupoid (Ĝ,M , Ox) is Ricci flat. On the other hand, the choice of the points x i ∈ M i implies that |Ric|(., 0) > ǫ on Ox. We reach a contradiction. The function F n,K,D, (t) must satisfy lim t→∞ F n,K,D (t) = 0. This completes the proof.
The proof of Theorem 3 will also follow from Lemma 6 Proof. (Theorem 3).The proof will again be by contradiction. Suppose the solution g(t) satisfies sup M ×[0,∞) t|Rm| = ∞. As outlined in [2, 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 anétale groupoid. We first choose a sequence of times T i → ∞ and then we choose a sequence (x i , t i ) ∈ M × (0, T i ) such that For each i, we set K i = |Rm|(x i , t i ) and we define the pointed Ricci flow solution (M, g i , x i ) by g i (t) = K i g(t i + t K i ). If we set α i = −t i K i and ω i = (T i − t i )K i , then it is shown in [2, Chapter 8, Section 2] that α i → −∞ and ω i → ∞, and that the curvature Rm(g i ) of the metric g i satisfies