We now begin using the theory established in the last two lectures to rigorously extract an asymptotic gradient shrinking soliton from the scaling limit of any given -solution. This will require a number of new tools, including the notion of a geometric limit of pointed Ricci flows , which can be viewed as the analogue of the Gromov-Hausdorff limit in the category of smooth Riemannian flows. A key result here is Hamilton’s compactness theorem: a sequence of complete pointed non-collapsed Ricci flows with uniform bounds on curvature will have a subsequence which converges geometrically to another Ricci flow. This result, which one can view as an analogue of the Arzelá-Ascoli theorem for Ricci flows, relies on some parabolic regularity estimates for Ricci flow due to Shi.
Next, we use the estimates on reduced length from the Harnack inequality analysis in Lecture 13 to locate some good regions of spacetime of a -solution in which to do the asymptotic analysis. Rescaling these regions and applying Hamilton’s compactness theorem (relying heavily here on the -noncollapsed nature of such solutions) we extract a limit. Formally, the reduced volume is now constant and so Lecture 14 suggests that this limit is a gradient soliton; however, some care is required to make this argument rigorous. In the next section we shall study such solitons, which will then reveal important information about the original -solution.
Our treatment here is primarily based on Morgan-Tian’s book and the notes of Ye. Other treatments can be found in Perelman’s original paper, the notes of Kleiner-Lott, and the paper of Cao-Zhu. See also the foundational papers of Shi and Hamilton, as well as the book of Chow, Lu, and Ni.
— Geometric limits —
To develop the theory of geometric limits for pointed Ricci flows , we begin by studying such limits in the simpler context of pointed Riemannian manifolds , i.e. a Riemannian manifold together with a point , which we shall call the origin or distinguished point of the manifold. To simplify the discussion, let us restrict attention to complete Riemannian manifolds (though for later analysis we will eventually have to deal with incomplete manifolds).
Definition 1. (Geometric limits) A sequence of pointed d-dimensional connected complete Riemannian manifolds is said to converge geometrically to another pointed d-dimensional connected complete Riemannian manifold if there exists a sequence of connected neighbourhoods of increasing to (i.e. ) and a sequence of smooth embeddings mapping to such that
- The closure of each is compact and contained in (note that this implies that every compact subset of will be contained in for sufficiently large n);
- The pullback metric converges in the topology to (i.e. all derivatives of the metric converge uniformly on compact sets).
Example 1. The pointed round d-sphere of radius R converges geometrically to the pointed Euclidean space as . Note how this example shows that the geometric limit of compact manifolds can be non-compact.
Example 2. If (M,g) is Hamilton’s cigar (Example 3 from Lecture 8), and is a sequence on M tending to infinity, then converges geometrically to the pointed round 2-cylinder.
Example 3. The d-torus of length 1/n does not converge to a geometric limit as , despite being flat. More generally, the sequence needs to be locally uniformly non-collapsed in order to have a geometric limit.
Exercise 1. Show that the geometric limit of a sequence , if it exists, is unique up to (pointed) isometry.
Geometric limits, as their name suggests, tend to preserve all (local) “geometric” or “intrinsic” information about the manifold, although global information of this type can be lost. Here is a typical example:
Exercise 2. Suppose that converges geometrically to . Show that for every , and that we have the Fatou-type inequality . Give an example to show that the latter inequality can be strict.
Here is the basic compactness theorem for such limits.
Theorem 1. (Compactness theorem) Let be a sequence of connected complete Riemannian d-dimensional manifolds. Assume that
- (Uniform bounds on curvature and derivatives) For all , one has the pointwise bound on the ball for all sufficiently large n and some constant .
- (Uniform non-collapsing) For every there exists such that for all and , and all sufficiently large n.
Then, after passing to a subsequence if necessary, the sequence has a geometric limit.
Proof. (Sketch) Let be an arbitrary radius. From Cheeger’s lemma (Theorem 1 from Lecture 7) and hypothesis 2, we know that the injectivity radius on is bounded from below by some small for sufficiently large n. Also, from the curvature bounds and Bishop-Gromov comparison geometry (Lemma 1 from Lecture 9) we know that the volume of is uniformly bounded from above for sufficiently large n.
Now find a maximal -net of , thus the balls are disjoint and the balls cover . Volume counting shows that k is bounded for all sufficiently large n; by passing to a subsequence we may assume that it is constant. Similarly we may assume that all the distances converge to a limit. Using the exponential map and some arbitrary identification of tangent spaces with , we can identify each ball with the standard Euclidean ball of radius . Any pair of separation less than induces a smooth transition map from the Euclidean ball of radius into some subset of , which can be shown by comparison geometry to be uniformly bounded in norms; applying the ( version of the) Arzelá-Ascoli theorem we may thus pass to a subsequence and assume that all these transition maps converge in to a limit. It is then a routine matter to glue together all the limit transition maps to fashion an incomplete manifold to which the balls converge geometrically (up to errors of at the boundary). Furthermore, as one increases , one can show (by a modification of Exercise 1) that these limits are compatible. Now letting go to infinity (and using the usual diagonalisation trick on all the subsequences obtained), and then gluing together all the incomplete limits obtained, one can create the full geometric limit.
Remark 1. One could use ultrafilters here in place of subsequences, but this does not significantly affect any of the arguments.
Now we turn to geometric limits of pointed Ricci flows (Ricci flows with a specified origin ).
Definition 2. Let be a sequence of pointed d-dimensional complete connected Ricci flows, each on its own time interval . We say that a pointed d-dimensional complete connected Ricci flow on a time interval I is a geometric limit of this sequence if
- Every compact subinterval of I is contained in for all sufficiently large n.
- There exists neighbourhoods of as in Definition 1, compact time intervals increasing to I, and smooth embeddings preserving the origin such that the pullback of the flow to converges in spacetime to .
Exercise 3. Show that if a sequence of -noncollapsed Ricci flows (with a uniform value of ) converges geometrically to another Ricci flow, then the limit flow is also -noncollapsed.
Now we present Hamilton’s compactness theorem for Ricci flows, which requires less regularity hypotheses than Theorem 1 due to the parabolic smoothing effects of Ricci flow (as captured by Shi’s estimates, see Theorem 3).
Theorem 2. (Hamilton compactness theorem) Let and be as in Definition 2, and let I be an open interval obeying hypothesis 1 of that definition. Let be a time. Suppose that
- For every compact subinterval J of I containing and every , one has the curvature bound on the cylinder for some and all sufficiently large n; and
- One has the non-collapsing bound for some and , and all sufficiently large n.
- (Uniform lower bound on curvature) For any compact J, there is a K such that for any , one has the curvature lower bound on for all sufficiently large n. (This is not quite implied by 1. because the curvature bound K there is allowed to depend on r, whereas here we require that K is independent of r.)
Then some subsequence of converges geometrically to a limit on I.
Condition 3 is technical (and was erroneously omitted in some of the literature), but it was recently observed by Topping that the claim fails without some hypothesis of this form. Fortunately, in the applications to the Poincare conjecture one always has a uniform lower bound on Ricci curvature, and so Condition 3 is not difficult to verify in practice.
Proof. By Shi’s estimates (Theorem 3) we can upgrade the bound on curvature in hypothesis 1 to bounds on derivatives of curvature. Indeed, these estimates imply that for any J, r as in that hypothesis, and any , we have for some and sufficiently large n.
Now we restrict to the time slice and apply Theorem 1. Passing to a subsequence, we can assume that converges geometrically to a limit .
For any radius r and any compact J in I containing , we can pull back the flow to a (spatially incomplete) flow on the cylinder for sufficiently large n. By construction, converges in norm to ; in particular, it is uniformly bounded in each of the seminorms of this space. Also, each is a Ricci flow with uniform bounds on any derivative of curvature for sufficiently large n.
Exercise 4. Using these facts, show that the sequence of flows is uniformly bounded in each of the seminorms of for each fixed J, r, and for n sufficiently large.
By using the Arzelá-Ascoli theorem as before, we may thus pass to a further subsequence and assume that converges in to a limiting flow . Clearly this limit is a Ricci flow. Letting and pasting together the resulting limits one obtains the desired geometric limit. (One has to verify that every geodesic in starting from can be extended to any desired length, thus establishing completeness by the Hopf-Rinow theorem, but this is easy to establish given all the uniform bounds on the metric and curvature, and their derivatives. It is here that hypothesis 3 is used to prevent the metric from shrinking too rapidly as one goes backwards in time. )
— Locating an asymptotic gradient shrinking soliton —
We now return to the study of -solutions . We pick an arbitrary point and consider the reduced length function . Recall (see equation (18) from Lecture 11) that we had
for every . (This bound was obtained from the parabolic inequality and the maximum principle.) Thus we can find a sequence with such that
Now recall that as a consequence of Hamilton’s Harnack inequality, we have the pointwise estimates
(see equations (37), (38) from Lecture 13). From these bounds and Gronwall’s inequality, one easily sees that we can extend (2) to say that
for any in the cylinder and any and . Applying (3) once more, together with the hypothesis of non-negative curvature more, we also obtain bounded normalised curvature on this cylinder:
If we thus introduce the rescaled flow by setting , , and , we see that these flows obey hypothesis 1 of Theorem 2. Also, since the original -solutions are -noncollapsed, so are their rescalings, which (in conjunction with hypothesis 1) gives us hypothesis 2. We can thus invoke Theorem 2 and assume (after passing to a subsequence) that the rescaled flows converge geometrically to an ancient Ricci flow on the time interval . From Exercise 3 we see that this limit is also -noncollapsed. Since the rescaled flows have non-negative curvature, the limit flow has non-negative curvature also. (Note however that we do not expect in general that has bounded curvature (for instance, if the original -solution was a round shrinking sphere terminating at the unit radius sphere, the limit object would be a round shrinking sphere terminating at a point). In particular we do not expect to be a -solution.)
Let be the rescaled length function, thus . From (5) we see that is uniformly bounded on compact subsets of for n sufficiently large (where we identify compact subsets of with subsets of for n large enough). By the rescaled versions of (4) and (5) we also see that is also uniformly bounded on such compact sets for sufficiently large n; thus the are uniformly Lipschitz on each compact set. Applying the Arzelá-Ascoli theorem and passing to a subsequence, we may thus assume that the converge uniformly on compact sets to some limit , which is then locally Lipschitz.
Remark 2. We do not attempt to interpret as a reduced length function arising from some point at time t=0; indeed we expect the limiting flow to develop a singularity at this time.
We know that the reduced volume is non-increasing in and ranges between 0 and , and so converges to a limit between 0 and . This limit cannot equal since this would mean that the -solution is flat (by Theorem 1 from Lecture 14), which is absurd. The limit cannot be zero either, since the bounds (5) and the non-collapsing ensure a uniform lower bound on the reduced volume. By rescaling, we conclude that
for each fixed .
Let us now argue informally, and then return to make the argument rigorous later. Formally taking limits in (7), we conclude that
On the other hand, from the proof of the monotonicity of reduced volume from Lecture 10 we have (formally, at least)
and hence by rescaling
Formally taking limits, we obtain
We can rewrite this as the assertion that is a subsolution of the backwards heat equation:
This (formally) implies that the left-hand side of (8) is non-increasing in . On the other hand, this quantity is constant in ; and so (12) must be obeyed with equality, and thus
Also, recall from Lecture 10 that
Rescaling and taking limits, we formally conclude that the same is true for ;
From (14) and (15) we obtain that the Perelman -functional
Combining this with the vanishing of (16) we thus conclude that
and thus is a gradient shrinking soliton as desired.
— Making the argument rigorous I. Spatial localisation —
Now we turn to the (surprisingly delicate) task of justifying the steps from (7) to (18).
The first task is to deduce (8) from (7). From the dominated convergence theorem it is not difficult to show that
for any fixed and r; the difficulty is to prevent the escape of mass of to spatial infinity. (Fatou’s lemma will tell us that the left-hand side of (8) is less than or equal to the right, but this is not enough for our application.)
In order to prevent such an escape, one needs a lower bound on when is large. (Note that estimates such as (3), (4) only provide upper bounds on .) The problem is equivalent to that of upper bounding in terms of . To do this we need some control on quantities related to the distance function at extremely large distances. Remarkably, such bounds are possible. We begin with a lemma of Perelman (related to an earlier argument of Hamilton).
Lemma 1. Let be a d-dimensional Riemannian manifold, let , and let . Suppose that on the balls and . Then for any minimising geodesic connecting x and y, we have , where is the velocity field.
Proof. We may assume that , since the claim is trivial otherwise. We recall the second variation formula
whenever one deforms a geodesic along a vector field Y (see equation (17) of Lecture 10). Since is minimising, the left-hand side of (20) is non-negative when Y vanishes at the endpoints of . Now let v be any unit vector at x, transported by parallel transport along . Setting Y(t) to equal tv/r when , equal to v when , and equal to (d(x,y)-t)v/r when , we conclude that
Letting v vary over an orthonormal frame and summing, we soon obtain the claim.
The above lemma, combined with the Ricci flow equation, gives an upper bound as to how rapidly the distance function can grow as one goes backwards in time.
Corollary 1. Let be a d-dimensional Ricci flow, let , let t be a time, and let . Suppose that on and on . Then (in the sense of forward difference quotients).
Using this estimate, we can now obtain a bound on distance in terms of reduced length.
Proposition 1. Let be a d-dimensional -solution, let , and . Then
Proof. We use an argument of Ye. Write A for the expression inside the on the right-hand side, and let be minimising -geodesics from to p, p’ respectively. By the fundamental theorem of calculus, we have
Using (3) and the -Gauss lemma we see that move at speed , and that all curvature tensors are in a -neighbourhood of either curve. Applying Corollary 1, the chain rule, and the Gauss lemma we conclude that
inserting this into (24) we obtain the claim.
Combining this with (5) and rescaling we see that we have a bound of the form
for all x and some ; taking limits we also obtain
On the other hand, from the Bishop-Gromov inequality we know that balls of radius r in either or have volume . These facts are enough to establish that the portion of (7) or (8) outside of the ball of radius r decays exponentially fast in r, uniformly in n, and this allows us to take limits in (19) as to deduce (8) from (7).
— Making the argument rigorous II. Parabolic inequality for —
The next major task in making the previous arguments rigorous is to justify the passage from (10) to (11). First of all, because of the -cut locus, (10) is only valid in the sense of distributions. We would like to take limits and conclude that (11) holds in the sense of distributions as well. There is no difficulty taking limits with the linear terms in (10), or in the zeroth order terms ; the only problem is in justifying the limit from to . We know that the are uniformly locally Lipschitz, and converge locally uniformly to ; but this is unfortunately not enough to ensure that converges in the sense of distributions to , due to possible high frequency oscillations in . To give a toy counterexample, the one-dimensional functions are uniformly Lipschitz and converge uniformly to zero, but converges in the distributional sense to rather than zero.
Since is bounded and converges distributionally to zero, it will be locally asymptotically orthogonal . From this and Pythagoras’ theorem we obtain
in the sense of distributions, where we pass to a subsequence in order to make the limits on both sides exist. (Note that converges locally uniformly to g and so there is no difficulty passing back and forth between those metrics.) The task is now to show that there is not enough oscillation to cause the second term on the right-hand side to be non-vanishing.
To do this, we observe that (10) provides an upper bound on ; indeed on any fixed compact set in , we have . This one-sided bound on the Laplacian is enough to rule out the oscillation problem. Indeed, as converges locally uniformly to , we see that
for any non-negative bump function and chosen so that on the support of . Integrating by parts and disposing of a lower order term, we conclude that
On the other hand, since is bounded converges weakly to zero, one has
One can easily replace and here by and . Combining (30) and (31) we conclude that the second term on the RHS of (28) is non-positive in the sense of distributions. But it is clearly also non-negative, and so it vanishes as required.
This gives (11); as a by-product of the argument we have also established the useful fact
in the sense of distributions. Combining this with the growth bounds (26), (27) on and $l_\infty$ from the previous section (which give exponential decay bounds on , and their first derivatives), it is not too difficult to then justify the remaining steps (12)-(18) of the argument rigorously; see Section 9.2 of Morgan-Tian for full details. (Note that once one reaches (13), one has a nonlinear heat equation for , and it is not difficult to use the smoothing effects of the heat kernel to then show that the locally Lipschitz function is in fact smooth.)
— The asymptotic gradient shrinking soliton is not flat —
Finally, we show that the asymptotic gradient shrinking soliton is non-trivial in the sense that its curvature is not identically zero at some time. For if the curvature did vanish everywhere at time t, then the equation (18) simplifies to . On the other hand, being flat, is the quotient of Euclidean space by some discrete subgroup. Lifting up to this space, we thus see that f is quadratic, and more precisely is equal to plus an affine-linear function. Thus f has no periodicity whatsoever and so the above-mentioned discrete subgroup is trivial. If we now apply (8) we see that . But on the other hand, as the original -solution was not flat, its reduced volume was strictly less than by Theorem 1 from Lecture 14, a contradiction. Thus the asymptotic gradient soliton is not flat.
— Appendix: Shi’s derivative estimates —
The purpose of this appendix is to prove the following estimate of Shi.
Theorem 3. Suppose that is a complete d-dimensional Ricci flow on the time interval , and that on the cylinder one has the pointwise curvature bound . Then on any slightly smaller cylinder one has the curvature bounds for any , where the implied constant depend on .
Proof. (Sketch) We induct on k. The case is trivial, so suppose that and that the claim has already been proven for all smaller values of k. We allow all implied constants in the O() notation to depend on . We refer to and as the “large cylinder” and “small cylinder” respectively.
We make some reductions. It is easy to see that we can take and T to be small.
Since on the cylinder, we see that the metric at later times of the large cylinder is comparable to the initial metric up to multiplicative constants. The curvature bound tells us that if is small, then we are inside the conjugacy radius; pulling back under the exponential map, we may thus assume that the exponential map from is injective on the large cylinder. Let be the time-varying radial coordinate; observe that the annulus will be contained between the large cylinder and small cylinder for T small enough.
Exercise 5. Show that if and T are small enough, then on the large cylinder.
Let be a smooth non-negative radial cutoff to the large cylinder that equals 1 on the small cylinder. From the above exercise, the Gauss lemma, and the chain rule, we see that .
Now we study the heat equation obeyed by the “energy densities” for various m.
Exercise 6. (Bochner-Weitzenböck type estimate) For any , show that
(Hint: start with the equation and use the product rule and the definition of curvature repeatedly.)
From this exercise and the induction hypothesis we see that
for all , with the understanding that the second term on the right-hand side is absent when m=0. Telescoping this, we can thus find an expression
for some sufficiently large positive constant , which obeys the heat equation . Also, by hypothesis we have E=O(1) at time zero. Applying the maximum principle, we obtain the claim.
Exercise 6. Suppose that in the hypotheses of Shi’s theorem that we also have for on the large cylinder at time zero. Conclude that we have on the small cylinder for all j.
Exercise 7. Let be a smooth compact manifold, and let be a bounded solution to the heat equation which obeys a pointwise bound at time zero. Establish the bounds on the spacetime and all , where the implied constant depends on (M,g), K, T, and k.
(Update, June 2: some corrections.)
(Update, Oct 18 2011: A recently discovered issue with the Hamilton compactness theorem in the case where the curvature bound is not uniform in r has been addressed.)