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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.
Van Vu and I have just uploaded to the arXiv our paper “Random matrices: A general approach for the least singular value problem“, submitted to Israel J. Math.. This paper continues a recent series of papers by ourselves and also by Rudelson and by Rudelson–Vershynin on understanding the least singular value of a large random random complex matrix A. There are many random matrix models that one can consider, but here we consider models of the form , where is a deterministic matrix depending on n, and is a random matrix whose entries are iid with some complex distribution x of mean zero and unit variance. (In particular, this model is useful for studying the normalised resolvents of random iid matrices , which are of importance in the spectral theory of these matrices; understanding the least singular value of random perturbations of deterministic matrices is also important in numerical analysis, and particularly in smoothed analysis of algorithms such as the simplex method.)
In the model mean zero case , the normalised singular values of are known to be asymptotically distributed according to the Marchenko-Pastur distribution , which in particular implies that most of the singular values are continuously distributed (via a semicircular distribution) in the interval . (Assuming only second moment hypotheses on the underlying distribution x, this result is due to Yin; there are many earlier results assuming stronger hypotheses on x.) This strongly suggests, but does not formally prove, that the least singular value should be of size on the average. (To get such a sharp bound on the least singular value via the Marchenko-Pastur law would require an incredibly strong bound on the convergence rate to this law, which seems out of reach at present, especially when one does not assume strong moment conditions on x; current results such as those of Götze-Tikhomirov or Chatterjee-Bose give some upper bound on which improves upon the trivial bound of by a polynomial factor assuming certain moment conditions on x, but as far as I am aware these bounds do not get close to the optimal value of , except perhaps in the special case when x is Gaussian.) The statement that with high probability has been conjectured (in various forms) in a number of places, for instance by von Neumann, by Smale, and by Spielman-Teng.
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.
Given that this blog is currently being devoted to a rather intensive study of flows on manifolds, I thought that it might be apropos to highlight an amazing 22-minute video from 1994 on this general topic by the (unfortunately now closed) Geometry Center, entitled “Outside In“, which depicts Smale’s paradox (which asserts that an 2-sphere in three-dimensional space can be smoothly inverted without ever ceasing to be an immersion), following a construction of Thurston (who was credited with the concept for the video). I first saw this video at the 1998 International Congress of Mathematicians in Berlin, where it won the first prize at the VideoMath Festival held there. It did a remarkably effective job of explaining the paradox, its resolution in three dimensions, and the lack of a similar paradox in two dimensions, all in a clear and non-technical manner.
A (rather low resolution) copy of the first half of the video can be found here, and the second half can be found here. Some higher resolution short movies of just the inversion process can be found at this Geometry Center page. Finally, the video (and an accompanying booklet with more details and background) can still be obtained today from A K Peters, although I believe the video is only available in the increasingly archaic VHS format.
There are a few other similar such high-quality expository videos of advanced mathematics floating around the internet, but I do not know of any page devoted to collecting such videos. If any readers have their own favourites, you are welcome to post some links or pointers to them here.
We now turn to the theory of parabolic Harnack inequalities, which control the variation over space and time of solutions to the scalar heat equation
which are bounded and non-negative, and (more pertinently to our applications) of the curvature of Ricci flows
whose Riemann curvature or Ricci curvature is bounded and non-negative. For instance, the classical parabolic Harnack inequality of Moser asserts, among other things, that one has a bound of the form
whenever is a bounded non-negative solution to (1) on a complete static Riemannian manifold M of bounded curvature, are spacetime points with , and is a constant which is uniformly bounded for fixed when range over a compact set. (The even more classical elliptic Harnack inequality gives (1) in the steady state case, i.e. for bounded non-negative harmonic functions.) In terms of heat kernels, one can view (1) as an assertion that the heat kernel associated to dominates (up to multiplicative constants) the heat kernel at .
The classical proofs of the parabolic Harnack inequality do not give particularly sharp bounds on the constant . Such sharp bounds were obtained by Li and Yau, especially in the case of the scalar heat equation (1) in the case of static manifolds of non-negative Ricci curvature, using Bochner-type identities and the scalar maximum principle. In fact, a stronger differential version of (3) was obtained which implied (3) by an integration along spacetime curves (closely analogous to the -geodesics considered in earlier lectures). These bounds were particularly strong in the case of ancient solutions (in which one can send ). Subsequently, Hamilton applied his tensor-valued maximum principle together with some remarkably delicate tensor algebra manipulations to obtain Harnack inequalities of Li-Yau type for solutions to the Ricci flow (2) with bounded non-negative Riemannian curvature. In particular, this inequality applies to the -solutions introduced in the previous lecture.
In this current lecture, we shall discuss all of these inequalities (although we will not give the full details for the proof of Hamilton’s Harnack inequality, as the computations are quite involved), and derive several important consequences of that inequality for -solutions. The material here is based on several sources, including Evans’ PDE book, Müller’s book, Morgan-Tian’s book, the paper of Cao-Zhu, and of course the primary source papers mentioned in this article.
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.
I’ve just uploaded to the arXiv my paper “A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential“, submitted to Dynamics of PDE. This paper continues some earlier work of myself in an attempt to understand the soliton resolution conjecture for various nonlinear dispersive equations, and in particular, nonlinear Schrödinger equations (NLS). This conjecture (which I also discussed in my third Simons lecture) asserts, roughly speaking, that any reasonable (e.g. bounded energy) solution to such equations eventually resolves into a superposition of a radiation component (which behaves like a solution to the linear Schrödinger equation) plus a finite number of “nonlinear bound states” or “solitons”. This conjecture is known in many perturbative cases (when the solution is close to a special solution, such as the vacuum state or a ground state) as well as in defocusing cases (in which no non-trivial bound states or solitons exist), but is still almost completely open in non-perturbative situations (in which the solution is large and not close to a special solution) which contain at least one bound state. In my earlier papers, I was able to show that for certain NLS models in sufficiently high dimension, one could at least say that such solutions resolved into a radiation term plus a finite number of “weakly bound” states whose evolution was essentially almost periodic (or almost periodic modulo translation symmetries). These bound states also enjoyed various additional decay and regularity properties. As a consequence of this, in five and higher dimensions (and for reasonable nonlinearities), and assuming spherical symmetry, I showed that there was a (local) compact attractor for the flow: any solution with energy bounded by some given level E would eventually decouple into a radiation term, plus a state which converged to this compact attractor . In that result, I did not rule out the possibility that this attractor depended on the energy E. Indeed, it is conceivable for many models that there exist nonlinear bound states of arbitrarily high energy, which would mean that must increase in size as E increases to accommodate these states. (I discuss these results in a recent talk of mine.)
In my new paper, following a suggestion of Michael Weinstein, I consider the NLS equation
where is the solution, and is a smooth compactly supported real potential. We make the standard assumption (which is asserting that the nonlinearity is mass-supercritical and energy-subcritical). In the absence of this potential (i.e. when V=0), this is the defocusing nonlinear Schrödinger equation, which is known to have no bound states, and in fact it is known in this case that all finite energy solutions eventually scatter into a radiation state (which asymptotically resembles a solution to the linear Schrödinger equation). However, once one adds a potential (particularly one which is large and negative), both linear bound states (solutions to the linear eigenstate equation ) and nonlinear bound states (solutions to the nonlinear eigenstate equation ) can appear. Thus in this case the soliton resolution conjecture predicts that solutions should resolve into a scattering state (that behaves as if the potential was not present), plus a finite number of (nonlinear) bound states. There is a fair amount of work towards this conjecture for this model in perturbative cases (when the energy is small), but the case of large energy solutions is still open.
In my new paper, I consider the large energy case, assuming spherical symmetry. For technical reasons, I also need to assume very high dimension . The main result is the existence of a global compact attractor K: every finite energy solution, no matter how large, eventually resolves into a scattering state and a state which converges to K. In particular, since K is bounded, all but a bounded amount of energy will be radiated off to infinity. Another corollary of this result is that the space of all nonlinear bound states for this model is compact. Intuitively, the point is that when the solution gets very large, the defocusing nonlinearity dominates any attractive aspects of the potential V, and so the solution will disperse in this case; thus one expects the only bound states to be bounded. The spherical symmetry assumption also restricts the bound states to lie near the origin, thus yielding the compactness. (It is also conceivable that the localised nature of V also restricts bound states to lie near the origin, even without the help of spherical symmetry, but I was not able to establish this rigorously.)