A local curvature estimate for the Ricci flow
Abstract.
We show that the norm of the Riemann curvature tensor of any smooth solution to the Ricci flow can be explicitly estimated in terms of its initial values on a given ball, a local uniform bound on the Ricci tensor, and the elapsed time. This provides a new, direct proof of a result of Šesǔm, which asserts that the curvature of a solution on a compact manifold cannot blow up while the Ricci curvature remains bounded, and extends its conclusions to the noncompact setting. We also prove that the Ricci curvature must blow up at least linearly along a subsequence at a finite time singularity.
1. Introduction
Let be a smooth dimensional manifold and a solution to the Ricci flow
(1.1) 
on defined on a maximal interval with . When is compact, Hamilton’s longtime existence criterion [H1] asserts that either or the the maximum of the norm of the Riemann curvature tensor blows up at , that is, . In other words, provided the sectional curvatures of a solution are uniformly bounded on a finite interval , the solution may be extended to a larger interval .
The arrival of a finitetime singularity for a solution on a compact manifold is therefore characterized by the blowup of . It is of considerable interest to try to express this criterion in terms of a quantity simpler than the norm of the full curvature tensor. One of the first improvements in this direction was made by Šesǔm [Se], who proved that, in the above situation, if , then . Her result has since been generalized in a number of directions. It has been conjectured, in fact, that the scalar curvature must also blow up in this case, that is, one must actually have . In dimension three, this is a consequence of the HamiltonIvey estimate [H2, I], and it is true for all Kähler solutions by a theorem of Zhang [Z]. Recent efforts have made considerable progress toward resolving this conjecture and clarifying the relationship between scalar curvature and singularity formation. See, for example, [BZ], [CT], [EMT], [He], [Kn], [LS], [Si1, Si2], [W1, W2], and the references therein.
On noncompact , the loss of a uniform curvature bound is no longer necessarily coincident with the arrival of a singularity. There are by now many constructions of smooth, complete solutions whose curvature is unbounded on every timeslice, and Giesen and Topping [GT] and CabezasRivas and Wilking [CW] have separately constructed examples of solutions which are defined smoothly for and possess a uniform curvature bound on , but which have unbounded curvature on for some . It is possible, even, that there exist solutions for which but for all , although there are currently no complete examples known. A recent result of Topping [T] implies that they do not occur in two dimensions. The preservation of a uniform curvature bound is closely related to the uniqueness of Shi’s solutions [Sh], which, in general dimensions, is only presently known to hold within the class of complete solutions of bounded curvature ([CZ]; cf. [Ko], [T]).
Hamilton’s longtime existence criterion nevertheless has a partial analog for noncompact : it is still true that if is a complete solution on with and , then can be extended smoothly to a solution on . Here, through the fundamental estimates of Bando [B] and Shi [Sh], one can parlay the uniform curvature bound into instantaneous uniform bounds on all derivatives of the curvature. With these, one can show that converges smoothly to a complete limit metric of bounded curvature, and, from there, use the shorttime existence result of Shi [Sh] to restart the flow.
As before, it is desirable to express this criterion in terms of a simpler object than the full curvature tensor. The first question to ask is whether an analog of Šesǔm’s theorem is valid. As we have noted, we must now contend with the new possibility that the Riemann curvature tensor may become unbounded instantaneously. This is an obstacle to directly adapting the argumentbycontradiction in [Se], which relies on the extraction of a limit of a sequence of solutions rescaled by factors comparable to the spatial maximum of curvature. (See, however, [MC] for an approach along these lines.) Recently, it was proven in [Ko] that, if the curvature of a smooth complete solution is initially bounded, and if the Ricci curvature is uniformly bounded along the flow, then the Riemann curvature tensor must also remain uniformly bounded for a short time. However, this result is obtained indirectly, as a consequence of a uniqueness theorem, and the length of the interval on which the curvature of the solution is guaranteed to remain bounded is nonexplicit.
The statements of the shorttime curvature bound in [Ko] and the longtime existence criterion in [Se] raise the question of whether it is possible to simply estimate the growth of the curvature tensor locally and explicitly in terms of the Ricci curvature, and thereby simultaneously obtain effective proofs of both of these results. Such estimates have some precedent for the Ricci flow: in [MW], it is shown that, on a gradient shrinking soliton, a bound on implies a polynomial growth bound on . In the present paper, we show that the integral estimates in [MW] which are the basis of these bounds can be adapted to general smooth solutions to the Ricci flow. Our main result is the following local estimate.
Theorem 1.
Let be a smooth solution to the Ricci flow defined for . Suppose that there exist constants , and a point such that the ball is compactly contained in and
Then there are constants , , , depending only on the dimension , such that
where
This estimate gives a new, direct, proof of Šesǔm’s theorem in the compact case, and extends its conclusion to solutions on noncompact manifolds.
Corollary 1.
Suppose that is a smooth solution to the Ricci flow defined for , satisfying
Then

There is a smooth metric on , uniformly equivalent to , to which converges, locally smoothly, as .

If, in addition, is complete and
then
and extends smoothly to a complete solution on for some depending on , , , and .
Part (a) may be proven by an argument analogous to that for compact solutions, by considering the convergence on compact sets. See Section 14 of [H1] and Section 6.7 of [CK]. This demonstrates that, on an arbitrary smooth solution, the curvature tensor cannot blow up at a fixed point so long as the Ricci tensor remains bounded in a surrounding neighborhood. The uniform bound in (b) follows directly from the application of Theorem 1 to balls of fixed radius. It asserts, in particular, that the curvature tensor cannot instantaneously become unbounded in space so long as the solution continues to move with bounded speed. This bound is also Theorem 1.4 in the paper [MC]. The proof there (which is based on a blowup argument), however, makes use of an implicit assumption that the curvature tensor remains bounded for a short time [Ch].
There are a variety of other applications which one may obtain from the implication that an initial bound on and a uniform bound on guarantees a uniform bound on . For example, it follows from Theorem 18.2 in [H2], that if a complete solution has the property that as , then it continues to do so as long as remains bounded. Our estimates also immediately imply an improvement of the dependencies of the constants in Shi’s derivative estimates [Sh].
Corollary 2.
Let be a smooth solution to Ricci flow defined for . Suppose that is compactly contained in for some and and let
Then, for all ,
Our proof of Theorem 1 is based on the following integral estimate, which may itself be of some independent interest.
Proposition 1.
Let be a smooth solution to the Ricci flow defined for . Assume that there exist constants , and a point such that the ball is compactly contained in and that
Then, for any there exists so that for all
The proof of this proposition is an adaptation of the method in [MW]. In that reference, the reduction from to is made possible by the soliton identities. Here, it is made possible by the fact that the evolution of can be expressed purely in terms of the second covariant derivatives of , see Lemma 1 below. An analog of the above inequality is also valid for , see Remark 1 below. We note that Xu [X] (cf. [Ya]) has also obtained local curvature estimates for the Ricci flow by related integral methods.
We prove Proposition 1 in the following section by the means of an energy estimate. In Section 3, we combine it with a standard iteration argument to prove Theorem 1.
In Section 4, we use a variation on the above methods to investigate the possible rates of blowup of the Ricci curvature at a finitetime singularity. Note that Part (b) of Corollary 1 implies that, if a complete solution to the Ricci flow is defined on a maximal interval with and for all , then the Ricci curvature must blowup as , i.e., there must exist a sequence of times along which . The theorem does not say anything, however, about how rapidly the blowup must occur. By contrast, for the Riemann curvature tensor, it is known not only that one actually has at , but that, in fact, . This is a simple consequence of the parabolic maximum principle; see, e.g., Lemma 8.7 of [CLN]. It is natural to ask whether there is a corresponding minimal rate of blowup for the Ricci curvature. Our methods here allow us to give at least a partial answer to this question. In Theorem 2 of Section 4, using a somewhat more careful iteration argument, we prove that in the situation just described, we must at least have
along a subsequence for some depending only on the dimension.
2. The main estimate
The following differential inequality is the primary ingredient in the proof of Proposition 1. Let be a positive function on .
Proposition 2.
Let be a smooth solution to the Ricci flow on , defined for , satisfying the uniform Ricci bound
on some open , for all . Then, for any , there are constants , depending only on and , such that
(2.1) 
for any Lipschitz function with support in .
2.1. Proof of Proposition 2
The proof requires a bit of computation. We first recall the evolution equations for the curvature see, e.g, Chapter 6 of [CK].
Lemma 1.
There exists a constant such that
(2.2)  
(2.3)  
(2.4) 
on .
In the argument below, we will use to represent a positive constant depending only on and , and will simply write for the various tensor norms induced by . All integrals are taken relative to the evolving Riemannian measure , and we use the standard convention that represents some linear combination of contractions of the tensor product relative to the metric . As above, we use to denote the scalar curvature of .
To begin, we use equation (2.4) to write
Since the volume form evolves by , we then have
for some . Integrating by parts, we find that
from which we readily obtain that
(2.5) 
We may estimate the first term on the right of (2.5) by
(2.6) 
and the second term, similarly, by
(2.7) 
Using (2.6) and (2.7) in (2.5), it follows that
(2.8) 
We now set about to estimate the first two terms on the right of (2.8). Using equation (2.2) from Lemma 1, we get
Hence, integrating by parts, it follows that
(2.9) 
Since on the support of , for the first term in (2.9), we have
(2.10) 
Furthermore, as in (2.7), we have
(2.11) 
for the second term in (2.9). Hence, using (2.10) and (2.11), equation (2.9) implies
(2.12) 
Applying again equation (2.4) of Lemma 1, we may estimate the third term on the right of (2.12) by
(2.13) 
where we have used that .
Integrating by parts on the first term of the right hand side of (2.13), it follows that
Hence, using again that and also that , we can estimate the above by
Using this estimate in (2.13) it follows that
(2.14) 
Using the inequality
together with inequality (2.11), we then obtain
(2.15) 
This completes our estimate of the first term on the right of (2.8). Updated, inequality (2.8) now reads
(2.17) 
It remains only to estimate the second term on the right of (2.17). This may be done more simply. Using (2.3) of Lemma 1, we find first that
(2.18) 
Integrating by parts, and using that , we get that
and, furthermore, that
Therefore, it follows from (2.18) that
(2.19) 
Inserting (2.19) into (2.17), we conclude that, for any , there exist constants and , depending only on and , such that
Since
and
this completes the proof of Proposition 2.
2.2. Proof of Proposition 1
Proposition 1 now follows easily. Our assumptions are that is a smooth solution defined on , and that there exist constants , and there exists so that the ball is compactly contained in and we have
We wish to prove that, for any there exists so that
for all . Due to the spacetime invariance of the Ricci flow, it suffices to prove the proposition for . We will assume below, then, that
(2.20) 
It follows that
(2.21) 
for all . Consider the cutoff function
(2.22) 
which is Lipschitz with support .
Thus, from (2.20) we see that on the support of . Furthermore, from (2.21) and (2.22), we see that
(2.23) 
Hence, we may apply Proposition 2 to obtain that
(2.24) 
We now estimate the rightmost term in (2.24) using (2.23) and Young’s inequality, obtaining
(2.25) 
Consider the function
Since, by (2.21), we have that for all
(2.26) 
from (2.24) and (2.25), we see that satisfies the differential inequality
on . It follows that
(2.27) 
However, again using Young’s inequality and (2.26), we have that
Therefore, there exists a constant so that
for all . This completes the proof of Proposition 1.
Remark 1.
Our application to Theorem 1 only requires the validity of the above estimate for sufficiently large , and the proof above requires . However, an analogous estimate can be proven for by a slightly more detailed analysis of the terms in the evolution equation for . The idea is to write
where , and use the Tachibanatype identity
to estimate in place of (2.5).
3. Proof of Theorem 1
It is now straightforward to obtain a pointwise estimate from Proposition 1 using an iteration argument.
Proof of Theorem 1.
As before, we may assume . Let us fix some . Since on , we have from Proposition 1 that
(3.1) 
By the BishopGromov volume comparison theorem and (2.21) we get
Consequently, it follows from (2.26) and (3.1) that for any
(3.2) 
where
Now, by (2.3) of Lemma 1, we have
(3.3) 
We proceed to apply De GiorgiNashMoser iteration to (3.3) using (3.2) for depending only on the dimension . Note that the Ricci curvature bound and (2.21) imply a uniform bound on the Sobolev constant of on . From the argument in [L], Ch. 19, slightly modified as the metrics evolve in , see e.g. [Ye], we then obtain
for some depending only on . This proves the theorem. ∎
4. On the rate of blowup of Ricci curvature
Theorem 1 implies that if is complete, with bounded curvature, and the Ricci flow exists on but cannot be extended past time , then the Ricci curvature cannot be uniformly bounded on . The following theorem strengthens this conclusion.
Theorem 2.
Let be a smooth solution of the Ricci flow defined on , so that each is complete and has bounded curvature. Assume that the Ricci flow cannot be extended past time . Then there exists a constant , depending only on , and a sequence so that
Proof of Theorem 2.
We argue by contradiction. Fix some