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Having classified all asymptotic gradient shrinking solitons in three and fewer dimensions in the previous lecture, we now use this classification, combined with extensive use of compactness and contradiction arguments, as well as the comparison geometry of complete Riemannian manifolds of non-negative curvature, to understand the structure of -solutions in these dimensions, with the aim being to state and prove precise versions of Theorem 1 and Corollary 1 from Lecture 12.
The arguments are particularly simple when the asymptotic gradient shrinking soliton is compact; in this case, the rounding theorems of Hamilton show that the -solution is a (time-shifted) round shrinking spherical space form. This already classifies -solutions completely in two dimensions; the only remaining case is the three-dimensional case when the asymptotic gradient soliton is a round shrinking cylinder (or a quotient thereof by an involution). To proceed further, one has to show that the -solution exhibits significant amounts of curvature, and in particular that one does not have bounded normalised curvature at infinity. This curvature (combined with comparison geometry tools such as the Bishop-Gromov inequality) will cause asymptotic volume collapse of the -solution at infinity. These facts lead to the fundamental Perelman compactness theorem for -solutions, which then provides enough geometric control on such solutions that one can establish the structural theorems mentioned earlier.
The treatment here is a (slightly simplified) version of the arguments in Morgan-Tian’s book, which is based in turn on Perelman’s paper and the notes of Kleiner-Lott (see also the paper of Cao-Zhu for a slightly different treatment of this theory).
In the previous lecture, we showed that every -solution generated at least one asymptotic gradient shrinking soliton . This soliton is known to have the following properties:
- It is ancient: t ranges over .
- It is a Ricci flow.
- M is complete and connected.
- The Riemann curvature is non-negative (though it could theoretically be unbounded).
- is non-negative.
- M is -noncollapsed.
- M is not flat.
- It obeys the gradient shrinking soliton equation
for some smooth f.
The main result of this lecture is to classify all such solutions in low dimension:
Theorem 1. (Classification of asymptotic gradient shrinking solitons) Let be as above, and suppose that the dimension d is at most 3. Then one of the following is true (up to isometry and rescaling):
- d=2,3 and M is a round shrinking spherical space form (i.e. a round shrinking , , , or for some finite group acting freely on ).
- d=3 and M is the round shrinking cylinder or the oriented or unoriented quotient of this cylinder by an involution.
The case d=2 of this theorem is due to Hamilton; the compact d=3 case is due to Ivey; and the full d=3 case was sketched out by Perelman. In higher dimension, partial results towards the full classification (and also relaxing many of the hypotheses 1-8) have been established by Petersen-Wylie, by Ni-Wallach, and by Naber; these papers also give alternate proofs of Perelman’s classification.
To prove this theorem, we induct on dimension. In 1 dimension, all manifolds are flat and so the claim is trivial. We will thus take d=2 or d=3, and assume that the result has already been established for dimension d-1. We will then split into several cases:
- Case 1: Ricci curvature has a zero eigenvector at some point. In this case we can use Hamilton’s splitting theorem to reduce the dimension by one, at which point we can use the induction hypothesis.
- Case 2: Manifold noncompact, and Ricci curvature is positive and unbounded. In this case we can take a further geometric limit (using some Toponogov theory on the asymptotics of rays in a positively curved manifold) which is a round cylinder (or quotient thereof), and also a gradient steady soliton. One can easily rule out such an object by studying the potential function of that soliton on a closed loop.
- Case 3: Manifold noncompact, and Ricci curvature is positive and bounded. Here we shall follow the gradient curves of f using some identities arising from the gradient shrinking soliton equation to get a contradiction.
- Case 4: Manifold compact, and curvature positive. Here we shall use Hamilton’s rounding theorem to show that one is a round shrinking sphere or spherical space form.
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.
We continue our study of -solutions. In the previous lecture we primarily exploited the non-negative curvature of such solutions; in this lecture and the next, we primarily exploit the ancient nature of these solutions, together with the finer analysis of the two scale-invariant monotone quantities we possess (Perelman entropy and Perelman reduced volume) to obtain a important scaling limit of -solutions, the asymptotic gradient shrinking soliton of such a solution.
The main idea here is to exploit what I have called the infinite convergence principle in a previous post: that every bounded monotone sequence converges. In the context of -solutions, we can apply this principle to either of our monotone quantities: the Perelman entropy
where is the backwards time variable and
or the Perelman reduced volume
where is a fixed base point. As pointed out in Lecture 11, these quantities are related, and both are non-increasing in .
The reduced volume starts off at when , and so by the infinite convergence principle it approaches some asymptotic limit as . (We will later see that this limit is strictly between 0 and .) On the other hand, the reduced volume is invariant under the scaling
in the sense that
Thus, as we send , the reduced volumes of the rescaled flows (which are also -solutions) converge pointwise to a constant .
Suppose that we could somehow “take a limit” of the flows (or perhaps a subsequence of such flows) and obtain some limiting flow . Formally, such a flow would then have a constant reduced volume of . On the other hand, the reduced volume is monotone. If we could have a criterion as to when the reduced volume became stationary, we could thus classify all possible limiting flows , and thus obtain information about the asymptotic behaviour of -solutions (at least along a subsequence of scales going to infinity).
We will carry out this program more formally in the next lecture, in which we define the concept of an asymptotic gradient-shrinking soliton of a -solution.
In this lecture, we content ourselves with a key step in this program, namely to characterise when the Perelman entropy or Perelman reduced volume becomes stationary; this requires us to revisit the theory we have built up in the last few lectures. It turns out that, roughly speaking, this only happens when the solution is a gradient shrinking soliton, thus at any given time one has an equation of the form for some and . Our computations here will be somewhat formal in nature; we will make them more rigorous in the next lecture.
In previous lectures, we have established (modulo some technical details) two significant components of the proof of the Poincaré conjecture: finite time extinction of Ricci flow with surgery (Theorem 4 of Lecture 2), and a -noncollapsing of Ricci flows with surgery (which, except for the surgery part, is Theorem 2 of Lecture 7). Now we come to the heart of the entire argument: the topological and geometric control of the high curvature regions of a Ricci flow, which is absolutely essential in order for one to define surgery on these regions in order to move the flow past singularities. This control is intimately tied to the study of a special type of Ricci flow, the -solutions to the Ricci flow equation; we will be able to use compactness arguments (as well as the -noncollapsing results already obtained) to deduce control of high curvature regions of arbitrary Ricci flows from similar control of -solutions. A secondary compactness argument lets us obtain that control of -solutions from control of an even more special type of solution, the gradient shrinking solitons that we already encountered in Lecture 8.
[Even once one has this control of high curvature regions, the proof of the Poincaré conjecture is still not finished; there is significant work required to properly define the surgery procedure, and then one has to show that the surgeries do not accumulate in time, and also do not disrupt the various monotonicity formulae that we are using to deduce finite time extinction, -noncollapsing, etc. But the control of high curvature regions is arguably the largest single task one has to establish in the entire proof.]
The next few lectures will be devoted to the analysis of -solutions, culminating in Perelman’s topological and geometric classification (or near-classification) of such solutions (which in particular leads to the canonical neighbourhood theorem for these solutions, which we will briefly discuss below). In this lecture we shall formally define the notion of a -solution, and indicate informally why control of such solutions should lead to control of high curvature regions of Ricci flows. We’ll also outline the various types of results that we will prove about -solutions.
Our treatment here is based primarily on the book of Morgan and Tian.
Having established the monotonicity of the Perelman reduced volume in the previous lecture (after first heuristically justifying this monotonicity in Lecture 9), we now show how this can be used to establish -noncollapsing of Ricci flows, thus giving a second proof of Theorem 2 from Lecture 7. Of course, we already proved (a stronger version) of this theorem already in Lecture 8, using the Perelman entropy, but this second proof is also important, because the reduced volume is a more localised quantity (due to the weight in its definition and so one can in fact establish local versions of the non-collapsing theorem which turn out to be important when we study ancient -noncollapsing solutions later in Perelman’s proof, because such solutions need not be compact and so cannot be controlled by global quantities (such as the Perelman entropy).
The route to -noncollapsing via reduced volume proceeds by the following scheme:
Non-collapsing at time t=0 (1)
Large reduced volume at time t=0 (2)
Large reduced volume at later times t (3)
Non-collapsing at later times t (4)
The implication is the monotonicity of Perelman reduced volume. In this lecture we discuss the other two implications , and ).
Our arguments here are based on Perelman’s first paper, Kleiner-Lott’s notes, and Morgan-Tian’s book, though the material in the Morgan-Tian book differs in some key respects from the other two texts. A closely related presentation of these topics also appears in the paper of Cao-Zhu.
Having completed a heuristic derivation of the monotonicity of Perelman reduced volume (Conjecture 1 from the previous lecture), we now turn to a rigorous proof. Whereas in the previous lecture we derived this monotonicity by converting a parabolic spacetime to a high-dimensional Riemannian manifold, and then formally applying tools such as the Bishop-Gromov inequality to that setting, our approach here shall take the opposite tack, finding parabolic analogues of the proof of the elliptic Bishop-Gromov inequality, in particular obtaining analogues of the classical first and second variation formulae for geodesics, in which the notion of length is replaced by the notion of -length introduced in the previous lecture.
The material here is primarily based on Perelman’s first paper and Müller’s book, but detailed treatments also appear in the paper of Ye, the notes of Kleiner-Lott, the book of Morgan-Tian, and the paper of Cao-Zhu.
We now turn to Perelman’s second scale-invariant monotone quantity for Ricci flow, now known as the Perelman reduced volume. We saw in the previous lecture that the monotonicity for Perelman entropy was ultimately derived (after some twists and turns) from the monotonicity of a potential under gradient flow. In this lecture, we will show (at a heuristic level only) how the monotonicity of Perelman’s reduced volume can also be “derived”, in a formal sense, from another source of monotonicity, namely the relative Bishop-Gromov inequality in comparison geometry (which has already been alluded to in previous lectures). Interestingly, in order to obtain this connection, one must first reinterpret parabolic flows such as Ricci flow as the limit of a certain high-dimensional Riemannian manifold as the dimension becomes infinite; this is part of a more general philosophy that parabolic theory is in some sense an infinite-dimensional limit of elliptic theory. Our treatment here is a (liberally reinterpreted) version of Section 6 of Perelman’s paper.
In the next few lectures we shall give a rigorous proof of this monotonicity, without using the infinite-dimensional limit and instead using results related to the Li-Yau-Hamilton Harnack inequality. (There are several other approaches to understanding Perelman’s reduced volume, such as Lott’s formulation based on optimal transport, but we will restrict attention in this course to the methods that are in Perelman’s original paper.)
It is well known that the heat equation
using the inner product associated to the volume measure . Indeed, if we evolve f in time at some arbitrary rate , a simple application of integration by parts (equation (29) from Lecture 1) gives
from which we see that (1) is indeed the gradient flow for (3) with respect to the inner product. In particular, if f solves the heat equation (1), we see that the Dirichlet energy is decreasing in time:
Thus we see that by representing the PDE (1) as a gradient flow, we automatically gain a controlled quantity of the evolution, namely the energy functional that is generating the gradient flow. This representation also strongly suggests (though does not quite prove) that solutions of (1) should eventually converge to stationary points of the Dirichlet energy (2), which by (3) are just the harmonic functions (i.e. the functions f with ).
As one very quick application of the gradient flow interpretation, we can assert that the only periodic (or “breather”) solutions to the heat equation (1) are the harmonic functions (which, in fact, must be constant if M is compact, thanks to the maximum principle). Indeed, if a solution f was periodic, then the monotone functional E must be constant, which by (4) implies that f is harmonic as claimed.
It would therefore be desirable to represent Ricci flow as a gradient flow also, in order to gain a new controlled quantity, and also to gain some hints as to what the asymptotic behaviour of Ricci flows should be. It turns out that one cannot quite do this directly (there is an obstruction caused by gradient steady solitons, of which we shall say more later); but Perelman nevertheless observed that one can interpret Ricci flow as gradient flow if one first quotients out the diffeomorphism invariance of the flow. In fact, there are infinitely many such gradient flow interpretations available. This fact already allows one to rule out “breather” solutions to Ricci flow, and also reveals some information about how Poincaré’s inequality deforms under this flow.
The energy functionals associated to the above interpretations are subcritical (in fact, they are much like ) but they are not coercive; Poincaré’s inequality holds both in collapsed and non-collapsed geometries, and so these functionals are not excluding the former. However, Perelman discovered a perturbation of these functionals associated to a deeper inequality, the log-Sobolev inequality (first introduced by Gross in Euclidean space). This inequality is sensitive to volume collapsing at a given scale. Furthermore, by optimising over the scale parameter, the controlled quantity (now known as the Perelman entropy) becomes scale-invariant and prevents collapsing at any scale – precisely what is needed to carry out the first phase of the strategy outlined in the previous lecture to establish global existence of Ricci flow with surgery.
We now set aside our discussion of the finite time extinction results for Ricci flow with surgery (Theorem 4 from Lecture 2), and turn instead to the main portion of Perelman’s argument, which is to establish the global existence result for Ricci flow with surgery (Theorem 2 from Lecture 2), as well as the discreteness of the surgery times (Theorem 3 from Lecture 2).
As mentioned in Lecture 1, local existence of the Ricci flow is a fairly standard application of nonlinear parabolic theory, once one uses de Turck’s trick to transform Ricci flow into an explicitly parabolic equation. The trouble is, of course, that Ricci flow can and does develop singularities (indeed, we have just spent several lectures showing that singularities must inevitably develop when certain topological hypotheses (e.g. simple connectedness) or geometric hypotheses (e.g. positive scalar curvature) occur). In principle, one can use surgery to remove the most singular parts of the manifold at every singularity time and then restart the Ricci flow, but in order to do this one needs some rather precise control on the geometry and topology of these singular regions. (In particular, there are some hypothetical bad singularity scenarios which cannot be easily removed by surgery, due to topological obstructions; a major difficulty in the Perelman program is to show that such scenarios in fact cannot occur in a Ricci flow.)
In order to analyse these singularities, Hamilton and then Perelman employed the standard nonlinear PDE technique of “blowing up” the singularity using the scaling symmetry, and then exploiting as much “compactness” as is available in order to extract an “asymptotic profile” of that singularity from a sequence of such blowups, which had better properties than the original Ricci flow. [The PDE notion of a blowing up a solution around a singularity, by the way, is vaguely analogous to the algebraic geometry notion of blowing up a variety around a singularity, though the two notions are certainly not identical.] A sufficiently good classification of all the possible asymptotic profiles will, in principle, lead to enough structural properties on general singularities to Ricci flow that one can see how to perform surgery in a manner which controls both the geometry and the topology.
However, in order to carry out this program it is necessary to obtain geometric control on the Ricci flow which does not deteriorate when one blows up the solution; in the jargon of nonlinear PDE, we need to obtain bounds on some quantity which is both coercive (it bounds the geometry) and either critical (it is essentially invariant under rescaling) or subcritical (it becomes more powerful when one blows up the solution) with respect to the scaling symmetry. The discovery of controlled quantities for Ricci flow which were simultaneously coercive and critical was Perelman’s first major breakthrough in the subject (previously known controlled quantities were either supercritical or only partially coercive); it made it possible, at least in principle, to analyse general singularities of Ricci flow and thus to begin the surgery program discussed above. (In contrast, the main reason why questions such as Navier-Stokes global regularity are so difficult is that no controlled quantity which is both coercive and critical or subcritical is known.) The mere existence of such a quantity does not by any means establish global existence of Ricci flow with surgery immediately, but it does give one a non-trivial starting point from which one can hope to make progress.