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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.
In this lecture we discuss Perelman’s original approach to finite time extinction of the third homotopy group (Theorem 1 from the previous lecture), which, as previously discussed, can be combined with the finite time extinction of the second homotopy group to imply finite time extinction of the entire Ricci flow with surgery for any compact simply connected Riemannian 3-manifold, i.e. Theorem 4 from Lecture 2.
In the previous lecture, we saw that Ricci flow with surgery ensures that the second homotopy group became extinct in finite time (assuming, as stated in the above erratum, that there is no embedded with trivial normal bundle). It turns out that the same assertion is true for the third homotopy group, at least in the simply connected case:
Theorem 1. (Finite time extinction of ) Let be a Ricci flow with surgery on compact 3-manifolds with , with M(0) simply connected. Then for all sufficiently large t, is trivial (or more precisely, every connected component of M(t) has trivial ).
[Aside: it seems to me that this theorem should also be true if one merely assumes that M(0) contains no embedded copy of with trivial bundle, as opposed to M(0) being simply connected, but I will be conservative and only state Theorem 1 with this stronger hypothesis, as this is all that is necessary for proving the Poincaré conjecture.]
Suppose we apply Ricci flow with surgery to a compact simply connected Riemannian 3-manifold (M,g) (which, by Lemma 1 from Lecture 2, has no embedded with trivial normal bundle). From the above theorem, as well as Theorem 1 from the previous lecture, we know that all components of M(t) eventually have trivial and for all sufficiently large t. Also, since M is initially simply connected, we see from Exercise 2 of Lecture 2, as well as Theorem 2.1 of Lecture 2, that all components of M(t) also have trivial . The finite time extinction result (Theorem 4 from Lecture 2) then follows immediately from Theorem 1 and the following topological result, combined with the following topological observation:
Lemma 1. Let M be a compact non-empty connected 3-manifold. Then it is not possible for , , and to simultaneously be trivial.
This lemma follows immediately from the Hurewicz theorem, but for sake of self-containedness we give a proof of it here.
There are two known approaches to establishing Theorem 1; one due to Colding and Minicozzi, and one due to Perelman. The former is conceptually simpler, but requires a certain technical concentration-compactness type property for a min-max functional which has only been established recently. This approach will be the focus of this lecture, while the latter approach of Perelman, which has also been rigorously shown to imply finite time extinction, will be the focus of the next lecture.
Returning (perhaps anticlimactically) to the subject of the Poincaré conjecture, recall from Lecture 2 that one of the key pillars of the proof of that conjecture is the finite time extinction result (see Theorem 4 from that lecture), which asserted that if a compact Riemannian 3-manifold (M,g) was initially simply connected, then after a finite amount of time evolving via Ricci flow with surgery, the manifold will be empty.
In this lecture and the next few, we will describe some of the key ideas used to prove this theorem. We will not be able to completely establish this theorem at present, because we do not have a full definition of “surgery”, but we will be able to establish some partial results, and indicate (in informal terms) how to cope with the additional technicalities caused by the surgery procedure. Hopefully, if time permits later in the class, once we have studied the surgery process, I will be able to revisit this material and flesh out these technicalities a bit more.
The proof of finite time extinction proceeds in several stages. The first stage, which was already accomplished in the previous lecture (in the absence of surgery, at least), is to establish lower bounds on the least scalar curvature . The next stage, which we discuss in this lecture, is to show that the second homotopy group of the manifold must become extinct in finite time, thus all immersed copies of the 2-sphere in M(t) for sufficiently large t must be contractible to a point. The third stage is to show that the third homotopy group also becomes extinct so that all immersed copies of the 3-sphere in M are similarly contractible. The final stage, which uses homology theory, is to show that a non-empty 3-manifold cannot have simultaneously trivial, thus yielding the desired claim (note that a simply connected manifold has trivial by definition; also, from Exercise 2 of Lecture 2 we see that all components of M remain simply connected even after surgery).
More precisely, in this lecture we will discuss (most of) the proof of
Theorem 1. (Finite time extinction of ) Let be a Ricci flow with surgery on compact 3-manifolds with , with M(0) containing no embedded copy of with trivial normal bundle. Then for all sufficiently large t, is trivial (or more precisely, every connected component of M(t) has trivial ).
The technical assumption about having no copy of with trivial normal bundle is needed solely in order to apply the known existence theory for Ricci flow with surgery (see Theorem 2 from Lecture 2).
The intuition for this result is as follows. From the Gauss-Bonnet theorem (and the fact that the Euler characteristic of the sphere is positive), we know that 2-spheres tend to have positive (Gaussian) curvature on the average, which should make them shrink under Ricci flow. (Here I am conflating Gaussian curvature with Ricci curvature; however, by restricting to a special class of 2-spheres, namely minimal surfaces, one can connect the two notions of curvature to each other (and to scalar curvature) quite nicely.) On the other hand, the presence of negative scalar curvature can counteract this by expanding these spheres. But the lower bounds on scalar curvature tell us that the negativity of scalar curvature becomes weakened over time, and it turns out that the shrinkage caused by the Gauss-Bonnet theorem eventually dominates and sends the area of all minimal immersed 2-spheres into zero, at which point one can conclude the triviality of by the Sacks-Uhlenbeck theory of minimal 2-spheres.
The arguments here are drawn from the book of Morgan-Tian and from the paper of Colding-Minicozzi. The idea of using minimal surfaces to force disappearance of various topological structures under Ricci flow originates with Hamilton (who used 2-torii instead of 2-spheres, but the idea is broadly the same).
We now begin the study of (smooth) solutions to the Ricci flow equation
particularly for compact manifolds in three dimensions. Our first basic tool will be the maximum principle for parabolic equations, which we will use to bound (sub-)solutions to nonlinear parabolic PDE by (super-)solutions, and vice versa. Because the various curvatures , , R of a manifold undergoing Ricci flow do indeed obey nonlinear parabolic PDE (see equations (31) from Lecture 1), we will be able to obtain some important lower bounds on curvature, and in particular establishes that the curvature is either bounded, or else that the positive components of the curvature dominate the negative components. This latter phenomenon, known as the Hamilton-Ivey pinching phenomenon, is particularly important when studying singularities of Ricci flow, as it means that the geometry of such singularities is almost completely dominated by regions of non-negative (and often quite high) curvature.
In order to motivate the lengthy and detailed analysis of Ricci flow that will occupy the rest of this course, I will spend this lecture giving a high-level overview of Perelman’s Ricci flow-based proof of the Poincaré conjecture, and in particular how that conjecture is reduced to verifying a number of (highly non-trivial) facts about Ricci flow.
At the risk of belaboring the obvious, here is the statement of that conjecture:
[Unless otherwise stated, all manifolds are assumed to be without boundary.]
I will take it for granted that this result is of interest, but you can read the Notices article of Milnor, the Bulletin article of Morgan, or the Clay Mathematical Institute description of the problem (also by Milnor) for background and motivation for this conjecture. Perelman’s methods also extend to establish further generalisations of the Poincaré conjecture, most notably Thurston’s geometrisation conjecture, but I will focus this course just on the Poincaré conjecture. (On the other hand, the geometrisation conjecture will be rather visibly lurking beneath the surface in the discussion of this lecture.)
In the first lecture, we introduce flows on Riemannian manifolds , which are recipes for describing smooth deformations of such manifolds over time, and derive the basic first variation formulae for how various structures on such manifolds (e.g. curvature, length, volume) change by such flows. (One can view these formulae as describing the relationship between two “infinitesimally close” Riemannian manifolds.) We then specialise to the case of Ricci flow (together with some close relatives of this flow, such as renormalised Ricci flow, or Ricci flow composed with a diffeomorphism flow). We also discuss the “de Turck trick” that modifies the Ricci flow into a nonlinear parabolic equation, for the purposes of establishing local existence and uniqueness of that flow.
Next week (starting on Wednesday, to be more precise), I will begin my class on Perelman’s proof of the Poincaré conjecture. As I only have ten weeks in which to give this proof, I will have to move rapidly through some of the more basic aspects of Riemannian geometry which will be needed throughout the course. In particular, in this preliminary lecture, I will quickly review the basic notions of infinitesimal (or microlocal) Riemannian geometry, and in particular defining the Riemann, Ricci, and scalar curvatures of a Riemannian manifold. (The more “global” aspects of Riemannian geometry, for instance concerning the relationship between distance, curvature, injectivity radius, and volume, will be discussed later in this course.) This is a review only, in particular omitting any leisurely discussion of examples or motivation for Riemannian geometry; it is impossible to compress this subject into a single lecture, and I will have to refer you to a textbook on the subject for a more complete treatment (I myself am using the text “Riemannian geometry” by my colleague here at UCLA, Peter Petersen).