On the conditions to extend Ricci flow(III)
1 Introduction
In this paper, we develop some estimates for the Ricci flow. Some of them are the improved versions of the estimates in [17], some of them
are purely new. These estimates are useful in the study of Ricci flow with bounded scalar curvature.
In particular, they have applications in the extension problem of Ricci flow and the convergence of the Kähler Ricci flow.
Suppose is a Ricci flow solution on a complete manifold . Following the notations of [17], we define
If the flow is obvious in the content, we will drop the subindex .
We first state the improvement of the estimates in [17].
Theorem 1 (Riemannian curvature ratio estimate).
There exists a constant with the following properties.
Suppose , is a Ricci flow solution on a complete manifold , , and for every . Suppose is the first time such that , then there exists a point and a nonzero vector such that
(1) 
In particular, we have
(2) 
Consequently, we have
(3) 
Theorem 2 (Ricci curvature estimate).
Suppose is a Ricci flow solution satisfying the following properties.

is a complete manifold of dimension .

whenever .
Then there exists a large constant such that
(4) 
The constants 1 and Theorem 2 are chosen for the simplicity of applications.
They can be replaced by other constants . Then of course , .
After we obtain these two universal dimensional constants and , we can apply them
to improve the estimates of the constants in other theorems in [17]. For simplicity, we will not list these improvements term by term.
Here we give some new applications.
in Theorem
Inspired by the proof of Theorem 1 and Theorem 2, we also study the volume ratio for Ricci flow with bounded scalar curvature and equivalent metrics (c.f. Proposition 3.1 and Proposition 2.1). It turns out that the volume ratio upper bound can be obtained even if the metric equivalence is absent. Of course, some other conditions are required as price.
Theorem 3 (volume ratio estimate).
Suppose is a Ricci flow solution on a closed manifold with the following properties.

for every , .

on .
Then we have the volume ratio estimate
(8) 
for every , . Here
Remark 1.
The organization of the paper is as follows. We prove the curvature estimates (Theorem 1 and Theorem 2) and their applications in section 2, the volume estimate (Theorem 3) and its applications in section 3.
Acknowledgment The second author would like to thank Jeffrey Streets for helpful discussions during the preparation of this paper.
2 Curvature estimates
Theorem 1 is indicated by the following Proposition, which is more general.
Proposition 2.1.
For every , there exists a constant with the following properties.
Suppose that is a Ricci flow solution on a complete manifold such that and for every . If is another Ricci flow solution on such that for every and , then we have
Proof.
If this proposition were wrong, there should exist a constant and constants with corresponding Ricci flows and violating the statements.

for every , .

for every , .

.
From the first and the second property and the definition that , we can choose points such that
(9) 
By the uniform curvature bound, the conjugate radii of under the metrics are uniformly bounded from below by . Let be the standard ball in with radius , be the exponential map from to . Clearly, is a local diffeomorphism from to . We can use to pull back the metrics:
(10) 
By Shi’s local estimate along the Ricci flow, we have
It follows that
(11) 
Note that . Combining this with the fact that implies that . In light of the metric equivalence, we know that . Therefore, the estimate of CheegerGromovTaylor (c.f. [5]) applies and we have is uniformly bounded from below for every is uniformly bounded from below for every
Now we collect the information we have
(12) 
Therefore we can take smooth convergence in the CheegerGromov sense:
(13)  
(14) 
Define to be the identity map:
In view of the third property, i.e., , we see that
(15) 
Therefore, we see that converges to an isometry map :
Since both and are smooth geodesic balls, a theorem of CalabiHartman [4] says that such an isometry must be smooth. Clearly, we have
(16) 
However, by the smooth convergence equations (13) and (14), and the last two inequalities of condition (12), we obtain
In particular, we obtain , which contradicts to equation (16). This contradiction establish the proof of Proposition 2.1. ∎
Proof of Theorem 1.
After we obtain Proposition 2.1, (1) and (2) follows from Proposition 2.1 trivially. The proof of Theorem 1 follows from the same argument as Theorem 3.1 of [17]. We shall be sketchy here. Let as in Proposition 2.1, be the first time such that . By the Gap property in Proposition 2.1, we obtain that
Choose such that . It follows that
∎
In order to prove Theorem 2, we need to prove the following lemma first.
Lemma 2.1.
There is a constant such that the following properties hold.
Suppose is a Ricci flow solution on the complete manifold , whenever , . Then
(17) 
Proof.
Choose small enough. It will be determined later that how small is.
Identify with . Let be the exponential map from to under the metric . Since , the conjugate radius of at is far greater than . Therefore, is a local diffeomorphism from , the geodesic ball of radius 100 on , to . Define for every . Clearly, . Moreover, by the estimates of Jacobi fields’ lengths, we obtain that
where is the volume of the standard unit ball in .
It is not hard to see that is a Ricci flow solution. It satisfies the following properties if we choose very small.

whenever , .

has a uniform Sobolev constant for every .

uniformly for every .
Therefore, the argument in Theorem 3.2 of [17] applies. We have
Note that is the metric lifted from . Therefore, whenever is less than the conjugate radius of under the metric , which is far greater than . It follows that
So we finish the proof of Lemma 2.1. ∎
Proof of Theorem 2.
By rescaling, we have the following equivalent property of Lemma 2.1.
Suppose is a Ricci flow solution on the complete manifold , whenever , . Then
(18) 
For every , , we have
Let , we obtain
∎
Before we prove Corollary 1, let’s first see the following gap inequality for on any ancient solution:
(19) 
Actually, by maximum principle, satisfies the equation . Choose such that . Then for every , ODE comparison implies
Then (19) follows trivially from the above inequality.
If we regard as a singular time. Then (19) suggests that this singular time behaves similar to a finite time singularity when is concerned. This observation inspires us to consider the gap of associated with the singular time . Generally, such a gap may not exist. A trivial example is the Ricciflat Ricci flow solution on a K3 surface, where . However, if we assume is uniformly bounded, then such a gap does exist.
Proof of Corollary 1.
The proof is similar to the case of finite time singularity, c.f. Theorem 1 of [17].
If this theorem failed, we could find an ancient solution such that
for some positive number . Without loss of generality, we can assume
By a standard time selecting process, for every large positive integer number , we can find a time such that
(20) 
Applying a rescaling if necessary, Theorem 1 and equation (20) imply
Therefore, we have
It follows that
since and . Contradiction! ∎
3 Volume Estimates
The following proposition is the motivation of Theorem 3.
Proposition 3.1.
Suppose is a Ricci flow solution on a closed manifold , . Suppose also that

on .

for every .
Then we have
(21) 
for every . The constant in equation (21) can be chosen as
Proof.
Fix , let be a shortest geodesic connecting under the metric with unit speed. Let be the length of . Define a curve in spacetime:
Following the notation of [13], we can bound the reduced distance.
Let . Clearly, . By the monotonicity of reduced volume, we have in
which implies
(22) 
Let . Note that . Proposition 3.1 follows from inequality (22). ∎
The idea of the previous proposition is quite simple: try to use reduced volume to estimate volume. If we assume scalar curvature is uniformly bounded, it is not important to differentiate the volume at different time slices. Therefore, the proof of Proposition 3.1 indicates the following observation. It is sufficient to prove the volume upper bound by finding a nonnegative function with the following properties.

.

on the geodesic ball .
In Proposition 3.1, , which is the Jacobi of reduced volume, plays the role of (without considering the difference of volume elements at different time slices). Generally, if such exists, we have
(23) 
Clearly, approaches the fundamental solution at , for both the operator and its conjugate operator . Since , the extra term is not important when we study the limit behavior. Therefore, there are two candidates for : the fundamental solution of and the fundamental solution of . Inspired by the work of [3], we found that the fundamental solution of works. The function can be chosen as the fundamental solution of based at the point .
Proof of Theorem 3.
From the previous discussion, it suffices to prove the following properties.

(24) 
In the ball , we have
(25)
Here is the fundamental solution of satisfying .
The first property follows from direct computation:
The second inequality is more complicated. We first need to show that is comparable to , then we apply a gradient estimate to show that is comparable to in the geodesic ball . Most of these estimates are available in the paper [3]. We only need some modifications to match our condition. For the completeness, we give a sketchy proof here.
Claim 1.
For every , we have
(26) 
In particular, at , we have
(27) 
Let . Clearly . Direct computation shows
(28) 
Let be the normalization of : where . Then equation (28) reads