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

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 𝑡→∞. 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 𝜕𝑔(𝑡)𝑔(𝑡),𝜕𝑡=−2Ric (1) where 𝑔(𝑡) is a Riemannian metric on a fixed smooth manifold 𝑀. Hamilton showed that, given any Riemannian metric 𝑔₀ on a closed manifold 𝑀, there exists 𝑇>0 such that (1) has a solution 𝑔(𝑡) defined for 𝑡∈[0,𝑇) and which satisfies 𝑔(0)=𝑔₀. 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∈𝑀Ric_∞(𝑡)=max{|Ric|(𝑥,𝑡)∶𝑥∈𝑀}. (2) 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→∞.

Theorem 1. Given 𝑛∈ℕ and numbers 𝐾, 𝐷>0, there exists a function 𝐹_{𝑛,𝐾,𝐷}∶[0,∞)→[0,∞) with the following properties.(1)lim_𝑡→∞𝐹_{𝑛,𝐾,𝐷}(𝑡)=0. (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.

Corollary 2. Suppose that 𝑔(𝑡) is an immortal solution to the Ricci flow on a closed 𝑛-dimensional manifold 𝑀 and suppose that, for some constants 𝐾, 𝐷>0, one has ||Rm||_∞(𝑡)≤𝐾 and diam(𝑀;𝑔(𝑡))≤𝐷 for every 𝑡≥0. Then, ||Ric||_∞(𝑡)→0 as 𝑡→∞.

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 𝑔(𝑡) to the Ricci flow on a closed manifold 𝑀 is said to have a type III singularity if sup_{𝑀×[0,∞)}𝑡|Rm|<∞.

Theorem 3. Suppose that 𝑔(𝑡) is an immortal solution to the Ricci flow on a closed 3-dimensional manifold 𝑀 and suppose that there exists a constant 𝐶>0 such that diam𝑀;𝑔(𝑡)²‖Rm‖_∞(𝑡)<𝐶 (3) 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 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 𝑔(𝑡) on a closed manifold 𝑀, the injectivity radius corresponding to each metric 𝑔(𝑡) will be denoted by inj(𝑀;𝑔(𝑡)).

Theorem 4. Given 𝑛∈ℕ and numbers 𝐾, 𝐷, 𝜄>0, there exists a function 𝐹_{𝑛,𝐾,𝐷,𝜄}∶[0,∞)→[0,∞) with the following properties.(1)Consider lim_𝑡→∞𝐹_{𝑛,𝐾,𝐷,𝜄}(𝑡)=0.(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.
Suppose that this did not hold. Then, for some 𝜖>0, there is a sequence 𝑡_𝑖→∞ such that 𝐹_{𝑛,𝐾,𝐷,𝜄}(𝑡_𝑖)>𝜖. 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)>𝜖. Consider the new sequence (𝑀,𝑔_𝑖,𝑥_𝑖) where 𝑔_𝑖(𝑡)=𝑔_𝑖(𝑡+𝑡_𝑖) for 𝑡∈[−𝑡_𝑖,∞). 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 (𝑀_𝑖,𝑔_𝑖(𝑡),𝑥_𝑖) converges to an eternal Ricci flow solution 𝑔(𝑡) on a pointed 𝑛-dimensional manifold (𝑀,𝑥). The uniform bound on the diameters of (𝑀_𝑖,𝑔_𝑖) implies that 𝑀 is a closed manifold.
The eternal Ricci flow solution 𝑔(𝑡) must be Ricci flat. Indeed, based on [3, Lemma 2.18], the solution is either Ricci flat or its scalar curvature is positive. Since 𝑀 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 𝑔(𝑡) is an eternal solution. Therefore, the Ricci flow solution 𝑔 satisfies Ric(𝑔(𝑡))≡0 as claimed. On the other hand, the choice of the points 𝑥_𝑖∈𝑀_𝑖 implies that |Ric|(𝑥,0)>𝜖. 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 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 (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 that 𝑔(𝑡), 0≤𝑡≤𝑇<∞, is a smooth one-parameter family of metrics on a smooth étale 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 𝐹∶ℝ×[0,𝑇]→ℝ be a smooth function. Suppose that 𝑢∶𝑀×[0,𝑇]→ℝ is a smooth 𝒢-invariant function which solves 𝜕𝑢𝜕𝑡≤Δ_𝑔(𝑡)𝑢+⟨𝑋(𝑡),∇𝑢⟩+𝐹(𝑢,𝑡). (4) Suppose further that 𝜙∶[0,𝑇]→ℝ solves 𝑑𝜙𝜙(𝑡),𝜙(0)𝑑𝑡=𝐹,𝑡=𝛼∈ℝ. (5)
If 𝑢(⋅,0)≤𝛼, then 𝑢(⋅,𝑡)≤𝜙(𝑡) for all 𝑡∈[0,𝑇].

Proof. The proof is basically the same as the corresponding proof for closed manifolds (see [7, page 35-36]). For 𝜖>0, we consider the ODE 𝑑𝜙_𝜖𝜙𝑑𝑡=𝐹_𝜖(𝑡)𝜙,𝑡+𝜖,_𝜖(0)=𝛼+𝜖∈ℝ, (6) for a new function 𝜙_𝜖∶[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 𝜖₀>0 there exists a solution 𝜙_𝜖 on [0,𝑇] for any 0<𝜖≤𝜖₀. Furthermore, 𝜙_𝜖→𝜙 uniformly as 𝜖→0. Thus, it suffices to show that 𝑢(⋅,𝑡)<𝜙_𝜖(𝑡) for all 𝑡∈[0,𝑇] and arbitrary 𝜖∈(0,𝜖₀).
If this were not true, then we could choose 𝜖∈(0,𝜖₀) and 𝑡₀∈(0,𝑇], where 𝑢(⋅,𝑡₀)<𝜙_𝜖(𝑡₀) fails. We may assume that 𝑡₀ is the earliest such time and pick 𝑥∈𝑀 such that 𝑢(𝑥,𝑡₀)=𝜙_𝜖(𝑡₀). Indeed, such a pair (𝑥,𝑡₀) exists because the 𝒢-invariant function 𝑢 descends to a continuous function 𝑢∶𝑊×[0,𝑇]→ℝ. Based on the hypothesis that 𝑊 is compact and connected, we can find 𝑡₀ such that it is the earliest time, where 𝑢(⋅,𝑡₀)<𝜙_𝜖(𝑡₀) fails, and pick an orbit 𝑂_𝑥 so that 𝑢(𝑂_𝑥,𝑡₀)=𝜙_𝜖(𝑡₀). This gives the required pair (𝑥,𝑡₀). Now, we know that 𝑢(𝑥,𝑠)−𝜙_𝜖(𝑠) is negative for 𝑠∈[0,𝑡₀) and zero for 𝑠=𝑡₀. Hence, we must have 𝜕𝑢𝜕𝑡𝑥,𝑡₀−𝜙^′_𝜖𝑡₀≥0. (7) Moreover, since 𝑢(⋅,𝑡₀) achieves a maximum at 𝑥, we have ∇𝑢(𝑥,𝑡₀)=0 and Δ𝑢(𝑥,𝑡₀)≤0.
Combining these facts with the inequality (4) for 𝑢 and (6) for 𝜙_𝜖, we get the contradiction 0≥𝜕𝑢𝜕𝑡−Δ_𝑔(𝑡)𝑢−⟨𝑋,∇𝑢⟩−𝐹(𝑢,⋅)𝑥,𝑡₀⟹0≥𝜙^′_𝜖𝑡₀−𝐹𝜙_𝜖𝑡₀,𝑡₀=𝜖>0. (8)
This completes the proof.

If 𝑔(𝑡) is a solution to the Ricci flow on a smooth étale groupoid (𝒢,𝑀), then, by definition, the elements of 𝒢 act by isometries for each Riemannian metric 𝑔(𝑡). More precisely, recall that, for any étale groupoid (𝒢,𝑀), there is an associated pseudogroup of local diffeomorphisms of 𝑀. The elements of this pseudogroup are the local diffeomorphisms 𝜙∶𝑈→𝑉 of the form 𝜙=𝑟∘𝛼, where 𝛼∶𝑈→𝒢 is a local section of the source map 𝑠; that is, 𝛼 satisfies 𝑠∘𝛼=𝐼𝑑. Given a Riemannian metric 𝑔 on 𝑀, the elements of 𝒢 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 𝑔(𝑡) are 𝒢-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 (𝒢,𝑀,𝑔(𝑡)) is an eternal solution to the Ricci flow on a smooth étale groupoid with compact connected orbit space 𝑊. 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 [0,𝑇] the scalar curvature satisfies 𝑛𝑅≥−2𝑡 (9) for 𝑡∈[0,𝑇]. Fix 𝑐∈ℝ. By translating time, we see that the previously mentioned inequality implies that 𝑅()𝑛𝑥,𝑡≥−2(𝑡−𝑐) (10) 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(𝑥₀,𝑡₀)>0 for some (𝑥,𝑡₀)∈𝑀×ℝ. By the strong maximum principle [3, Corollary 6.55], this implies that 𝑅(𝑥,𝑡)>0 for all 𝑡≥𝑡₀ and for all 𝑥 in the connected component of 𝑀 which contains 𝑥₀. 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 𝑥₀=𝑝₀,𝑞₀,𝑝₁,𝑞₁,…,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(𝑝₁,𝑡₀)=𝑅(𝑞₀,𝑡₀)>0. So, by the maximum principle, 𝑅(𝑥,𝑡)>0 for every 𝑡≥𝑡₀ and for all 𝑥 in the connected component of 𝑀 which contains 𝑝₁. In particular, 𝑅(𝑞₁,𝑡)>0 for all 𝑡≥𝑡₀. After applying this argument a finite number of times, we deduce that 𝑅(𝑥,𝑡)>0 for all 𝑡≥𝑡₀. This proves the claim.
The 𝒢-invariant function 𝑅 induces a continuous function 𝑅 on 𝑊 which satisfies 𝑅(⋅,𝑡)>0 for 𝑡≥𝑡₀. Since 𝑊 is compact, there exists some 𝛼>0 such that 𝑅(⋅,𝑡₀)≥𝛼. Hence, the scalar curvature satisfies 𝑅(⋅,𝑡₀)≥𝛼. 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 𝑅≡0. Based on the evolution equation of the scalar curvature 𝜕𝑅||𝜕𝑡=Δ𝑅+2Ric², (11) we deduce that Ric≡0. This completes the proof.

Proof of 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 𝜖>0, there is a sequence 𝑡_𝑖→∞ such that 𝐹_{𝑛,𝐾,𝐷}(𝑡_𝑖)>𝜖. 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)>𝜖. Consider the new sequence (𝑀,𝑔_𝑖,𝑥_𝑖), where 𝑔_𝑖(𝑡)=𝑔_𝑖(𝑡+𝑡_𝑖) for 𝑡∈[−𝑡_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 solution 𝑔(𝑡) on a pointed 𝑛-dimensional étale groupoid (𝒢,𝑀,𝑂𝑥) (see [4, Theorem 5.12]). The uniform bound on the diameters of (𝑀_𝑖,𝑔_𝑖) implies 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 ([5, 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 the étale groupoid (𝒢,𝑀,𝑂𝑥) is Ricci flat. On the other hand, the choice of the points 𝑥_𝑖∈𝑀_𝑖 implies that |Ric|>𝜖 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 of Theorem 3. The proof will again be by contradiction. Suppose that the solution 𝑔(𝑡) satisfies sup_{𝑀×[0,∞)}𝑡|Rm|=∞. 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 𝑇_𝑖→∞ and then we choose a sequence (𝑥_𝑖,𝑡_𝑖)∈𝑀×(0,𝑇_𝑖) such that 𝑡_𝑖𝑇_𝑖−𝑡_𝑖|Rm|𝑥_𝑖,𝑡_𝑖=sup_{𝑀×0,𝑇𝑖}𝑡𝑇_𝑖−𝑡|Rm|(𝑥,𝑡). (12) 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_𝑖𝑔(𝑡_𝑖+(𝑡/𝐾_𝑖)). If we set 𝛼_𝑖=−𝑡_𝑖𝐾_𝑖 and 𝜔_𝑖=(𝑇_𝑖−𝑡_𝑖)𝐾_𝑖, then it is shown in [3, Chapter 8, Section 2] that 𝛼_𝑖→−∞ and 𝜔_𝑖→∞ and that the curvature Rm(𝑔_𝑖) of the metric 𝑔_𝑖 satisfies Rm𝑔_𝑖)≤𝛼(𝑥,𝑡^𝑖𝛼_𝑖𝜔−𝑡𝜔_𝑖−^𝑖𝑡 (13) for (𝑥,𝑡)∈𝑀×[𝛼_𝑖,𝜔_𝑖]. Furthermore, condition (3) implies that diam𝑀;𝑔_𝑖(0)²=𝐾_𝑖diam𝑀;𝑔𝑡_𝑖²<𝐶. (14) Hence, after passing to a subsequence if necessary, the pointed sequence (𝑀,𝑔_𝑖,𝑥_𝑖) converges to an eternal Ricci flow solution 𝑔(𝑡) on a pointed 3-dimensional étale groupoid (𝒢,𝑀,𝑂𝑥) with compact connected orbit space. By construction, |Rm(𝑔)|(⋅,0)=1 on the orbit 𝑂𝑥. But, by Lemma 6, the solution 𝑔(𝑡) is flat. We get a contradiction. Therefore, the immortal solution 𝑔(𝑡) 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.