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Peter Petersen and I have just uploaded to the arXiv our paper, “Classification of Almost Quarter-Pinched Manifolds“, submitted to Proc. Amer. Math. Soc..  This is perhaps the shortest paper (3 pages) I have ever been involved in, because we were fortunate enough that we could simply cite (as a black box) a reference for every single fact that we needed here.

The paper is related to the famous sphere theorem from Riemannian geometry.  This theorem asserts that any n-dimensional complete simply connected Riemannian manifold which was strictly quarter-pinched (i.e. the sectional curvatures all in the interval $(K/4,K]$ for some $K > 0$) must necessarily be homeomorphic to the n-sphere $S^n$.    (In dimensions 3 or less, this already follows from simple connectedness thanks to the Poincaré conjecture (and Myers theorem), so the theorem is really only interesting in higher dimensions.  One can easily drop the simple connectedness hypothesis by passing to a universal cover, but then one has to admit sphere quotients $S^n/\Gamma$ as well as spheres.)

Due to the existence of exotic spheres in higher dimensions, being homeomorphic to a sphere does not necessarily imply being diffeomorphic to a sphere.  (For instance, an example of an exotic sphere with positive sectional curvature (but not quarter-pinched) was recently constructed by Petersen and Wilhelm.)  Nevertheless, Brendle and Schoen recently proved the diffeomorphic version of the sphere theorem: every strictly quarter-pinched complete simply connected Riemannian manifold is diffeomorphic to a sphere.  The proof is based on Ricci flow, and involves three main steps:

1. A verification that if M is quarter-pinched, then the manifold $M \times {\Bbb R}^2$ has non-negative isotropic curvature.  (The same statement is true without adding the two additional flat dimensions, but these additional dimensions are very convenient for simplifying the analysis by allowing certain two-planes to wander freely in the product tangent space.)
2. A verification that the property of having non-negative isotropic curvature is preserved by Ricci flow.  (By contrast, the quarter-pinched property is not preserved by Ricci flow.)
3. The pinching theory of Böhm and Wilking, which is a refinement of the work of Hamilton (who handled the three and four-dimensional cases).

Brendle and Schoen in fact proved a slightly stronger statement in which the curvature bound K is allowed to vary with position x, but we will not discuss this strengthening here.

The quarter-pinching is sharp; the Fubini-Study metric on complex projective spaces ${\Bbb CP}^n$ is non-strictly quarter-pinched (the sectional curvatures lie in ${}[K/4,K]$ but is not homeomorphic to a sphere).  Nevertheless, by refining the above methods, an endpoint result was established by Brendle and Schoen (see also a later refinement by Seshadri): any complete simply-connected manifold which is non-strictly quarter-pinched is diffeomorphic to either a sphere or a compact rank one symmetric space (or CROSS, for short) such as complex projective space.  (In the latter case one also has some further control on the metric, which we will not detail here.)  The homeomorphic version of this statement was established earlier by Berger and by Klingenberg.

Our result pushes this further by an epsilon.  More precisely, we show for each dimension n that there exists $\varepsilon > 0$ such that any $\frac{1}{4}-\varepsilon_n$-pinched complete simply connected manifold (i.e. the curvatures lie in ${}[K (\frac{1}{4}-\varepsilon_n), K]$) is diffeomorphic to either a sphere or a CROSS.  (The homeomorphic version of this statement was established earlier in even dimensions by Berger.)  We do not know if $\varepsilon_n$ can be made independent of n.

I have just uploaded to the arXiv the second installment of my “heatwave” project, entitled “Global regularity of wave maps IV.  Absence of stationary or self-similar solutions in the energy class“.  In the first installment of this project, I was able to establish the global existence of smooth wave maps from 2+1-dimensional spacetime ${\Bbb R}^{1+2}$ to hyperbolic space ${\bf H} = {\bf H}^m$ from arbitrary smooth initial data, conditionally on five claims:

1. A construction of an energy space for maps into hyperbolic space obeying a certain set of reasonable properties, such as compatibility with symmetries, approximability by smooth maps, and existence of a well-defined stress-energy tensor.
2. A large data local well-posedness result for wave maps in the above energy space.
3. The existence of an almost periodic “minimal-energy blowup solution” to the wave maps equation in the energy class, if this equation is such that singularities can form in finite time.
4. The non-existence of any non-trivial degenerate maps into hyperbolic space in the energy class, where “degenerate” means that one of the partial derivatives of this map vanishes identically.
5. The non-existence of any travelling or self-similar solution to the wave maps equation in the energy class.

In this paper, the second of four in this series (or, as the title suggests, the fourth in a series of six papers on wave maps, the first two of which can be found here and here), I verify Claims 1, 4, and 5.  (The third paper in the series will tackle Claim 2, while the fourth paper will tackle Claim 3.)  These claims are largely “elliptic” in nature (as opposed to the “hyperbolic” Claims 2, 3), but I will establish them by a “parabolic” method, relying very heavily on the harmonic map heat flow, and on the closely associated caloric gauge introduced in an earlier paper of mine.  The results of paper can be viewed as nonlinear analogues of standard facts about the linear energy space $\dot H^1({\Bbb R}^2) \times L^2({\Bbb R}^2)$, for instance the fact that smooth compactly supported functions are dense in that space, and that this space contains no non-trivial harmonic functions, or functions which are constant in one of the two spatial directions.  The paper turned out a little longer than I had expected (77 pages) due to some surprisingly subtle technicalities, especially when excluding self-similar wave maps.  On the other hand, the heat flow and caloric gauge machinery developed here will be reused in the last two papers in this series, hopefully keeping their length to under 100 pages as well.

A key stumbling block here, related to the critical (scale-invariant) nature of the energy space (or to the failure of the endpoint Sobolev embedding $\dot H^1({\Bbb R}^2) \not \subset L^\infty({\Bbb R}^2)$) is that changing coordinates in hyperbolic space can be a non-uniformly-continuous operation in the energy space.  Thus, for the purposes of making quantitative estimates in that space, it is preferable to work as covariantly (or co-ordinate free) manner as possible, or if one is to use co-ordinates, to pick them in some canonical manner which is optimally adapted to the tasks at hand.  Ideally, one would work with directly with maps $\phi: {\Bbb R}^2 \to {\bf H}$ (as well as their velocity field $\partial_t \phi: {\Bbb R}^2 \to T{\bf H}$) without using any coordinates on ${\bf H}$, but then it becomes to perform basic analytical operations on such maps, such as taking the Fourier transform, or (even more elementarily) taking the difference of two maps in order to measure how distinct they are from each other.

In the previous lecture, we studied high curvature regions of Ricci flows $t \mapsto (M,g(t))$ on some time interval ${}[0,T)$, and concluded that (as long as a mild topological condition was obeyed) they all had canonical neighbourhoods. This is enough control to now study the limits of such flows as one approaches the singularity time T. It turns out that one can subdivide the manifold M into a continuing region C in which the geometry remains well behaved (for instance, the curvature does not blow up, and in fact converges smoothly to an (incomplete) limit), and a disappearing region D, whose topology is well controlled. (For instance, the interface $\Sigma$ between C and D will be a finite union of disjoint surfaces homeomorphic to $S^2$.) This allows one (at the topological level, at least) to perform surgery on the interface $\Sigma$, removing the disappearing region D and replacing them with a finite number of “caps” homeomorphic to the 3-ball $B^3$. The relationship between the topology of the post-surgery manifold and pre-surgery manifold is as is described way back in Lecture 2.

However, once surgery is completed, one needs to restart the Ricci flow process, at which point further singularities can occur. In order to apply surgery to these further singularities, we need to check that all the properties we have been exploiting about Ricci flows – notably the Hamilton-Ivey pinching property, the $\kappa$-noncollapsing property, and the existence of canonical neighbourhoods for every point of high curvature – persist even in the presence of a large number of surgeries (indeed, with the way the constants are structured, all quantitative bounds on a fixed time interval [0,T] have to be uniform in the number of surgery times, although we will of course need the set of such times to be discrete). To ensure that surgeries do not disrupt any of these properties, it turns out that one has to perform these surgeries deep in certain $\varepsilon$-horns of the Ricci flow at the singular time, in which the geometry is extremely close to being cylindrical (in particular, it should be a $\delta$-neck and not just a $\varepsilon$-neck, where the surgery control parameter $\delta$ is much smaller than $\varepsilon$; selection of this parameter can get a little tricky if one wants to evolve Ricci flow with surgery indefinitely, although for the purposes of the Poincaré conjecture the situation is simpler as there is a fixed upper bound on the time for which one needs to evolve the flow). Furthermore, the geometry of the manifolds one glues in to replace the disappearing regions has to be carefully chosen (in particular, it has to not disrupt the pinching condition, and the geometry of these glued in regions has to resemble a $(C,\varepsilon)$-cap for a significant amount of (rescaled) time). The construction of the “standard solution” needed to achieve all these properties is somewhat delicate, although we will not discuss this issue much here.

In this, the final lecture, we shall present these issues from a high-level perspective; due to lack of time and space we will not cover the finer details of the surgery procedure. More detailed versions of the material here can be found in Perelman’s second paper, the notes of Kleiner-Lott, the book of Morgan-Tian, and the paper of Cao-Zhu. (See also a forthcoming paper of Bessières, Besson, Boileau, Maillot, and Porti.)

Having characterised the structure of $\kappa$-solutions, we now use them to describe the structure of high curvature regions of Ricci flow, as promised back in Lecture 12, in particular controlling their geometry and topology to the extent that surgery will be applied, which we will discuss in the next (and final) lecture of this class.

The material here is drawn largely from Morgan-Tian’s book and Perelman’s first and second papers; see also Kleiner-Lott’s notes and Cao-Zhu’s paper for closely related material. Due to lack of time, some details here may be a little sketchy.

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 $\kappa$-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 $\kappa$-solution is a (time-shifted) round shrinking spherical space form. This already classifies $\kappa$-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 $\kappa$-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 $\kappa$-solution at infinity. These facts lead to the fundamental Perelman compactness theorem for $\kappa$-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 $\kappa$-solution generated at least one asymptotic gradient shrinking soliton $t \mapsto (M,g(t))$. This soliton is known to have the following properties:

1. It is ancient: t ranges over $(-\infty,0)$.
2. It is a Ricci flow.
3. M is complete and connected.
4. The Riemann curvature is non-negative (though it could theoretically be unbounded).
5. $\frac{dR}{dt}$ is non-negative.
6. M is $\kappa$-noncollapsed.
7. M is not flat.
8. It obeys the gradient shrinking soliton equation

$\hbox{Ric} + \hbox{Hess}(f) = \frac{1}{2\tau} g$ (1)

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 $t \mapsto (M,g(t))$ 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):

1. d=2,3 and M is a round shrinking spherical space form (i.e. a round shrinking $S^2$, $S^3$, $\Bbb{RP}^2$, or $S^3/\Gamma$ for some finite group $\Gamma$ acting freely on $S^3$).
2. d=3 and M is the round shrinking cylinder $S^2 \times {\Bbb R}$ 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:

1. 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.
2. 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.
3. 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.
4. 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 will follow Morgan-Tian‘s treatment of Perelman’s argument; see also the notes of Kleiner-Lott, the paper of Cao-Zhu, and the book of Chow-Lu-Ni for other treatments of this argument.

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 $\kappa$-solution. This will require a number of new tools, including the notion of a geometric limit of pointed Ricci flows $t \mapsto (M, g(t), p)$, 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 $\kappa$-solution in which to do the asymptotic analysis. Rescaling these regions and applying Hamilton’s compactness theorem (relying heavily here on the $\kappa$-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 $\kappa$-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 $\kappa$-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 $\kappa$-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 $\kappa$-solutions, we can apply this principle to either of our monotone quantities: the Perelman entropy

$\displaystyle \mu(g(t),\tau) := \inf \{ {\mathcal W}(M,g(t),f,\tau): \int_M (4\pi\tau)^{-d/2} e^{-f}\ d\mu = 1 \}$ (1)

where $\tau := -t$ is the backwards time variable and

$\displaystyle {\mathcal W}(M,g(t),f,\tau) := \int_M (\tau(|\nabla f|^2 + R) + f - d) (4\pi\tau)^{-d/2} e^{-f}\ d\mu$, (2)

or the Perelman reduced volume

$\displaystyle \tilde V_{(0,x_0)}(-\tau) := \tau^{-d/2} \int_M e^{-l_{(0,x_0)}(-\tau,x)}\ d\mu(x)$ (3)

where $x_0 \in M$ is a fixed base point. As pointed out in Lecture 11, these quantities are related, and both are non-increasing in $\tau$.

The reduced volume starts off at $(4\pi)^{d/2}$ when $\tau=0$, and so by the infinite convergence principle it approaches some asymptotic limit $0 \leq \tilde V_{(0,x_0)}(-\infty) \leq (4\pi)^{d/2}$ as $\tau \to -\infty$. (We will later see that this limit is strictly between 0 and $(4\pi)^{d/2}$.) On the other hand, the reduced volume is invariant under the scaling

$g^{(\lambda)}(t) := \frac{1}{\lambda^2} g( \lambda^2 t )$, (4)

in the sense that

$\tilde V_{(0,x_0)}^{(\lambda)}(-\tau) = \tilde V_{(0,x_0)}(-\lambda^2 \tau)$. (5)

Thus, as we send $\lambda \to \infty$, the reduced volumes of the rescaled flows $t \mapsto (M, g^{(\lambda)}(t))$ (which are also $\kappa$-solutions) converge pointwise to a constant $\tilde V_{(0,x_0)}(-\infty)$.

Suppose that we could somehow “take a limit” of the flows $t \mapsto (M, g^{(\lambda)}(t))$ (or perhaps a subsequence of such flows) and obtain some limiting flow $t \mapsto (M^{(\infty)}, g^{(\infty)}(t))$. Formally, such a flow would then have a constant reduced volume of $\tilde V_{(0,x_0)}(-\infty)$. 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 $t \mapsto (M^{(\infty)}, g^{(\infty)}(t))$, and thus obtain information about the asymptotic behaviour of $\kappa$-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 $\kappa$-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 $-\tau$ one has an equation of the form $\hbox{Ric} + \hbox{Hess}(f) = \lambda g$ for some $f: M \to {\Bbb R}$ and $\lambda > 0$. Our computations here will be somewhat formal in nature; we will make them more rigorous in the next lecture.

The material here is largely based on Morgan-Tian’s book and the first paper of Perelman. Closely related treatments also appear in the notes of Kleiner-Lott and the paper of Cao-Zhu.

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 $\kappa$-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 $\kappa$-solutions to the Ricci flow equation; we will be able to use compactness arguments (as well as the $\kappa$-noncollapsing results already obtained) to deduce control of high curvature regions of arbitrary Ricci flows from similar control of $\kappa$-solutions. A secondary compactness argument lets us obtain that control of $\kappa$-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, $\kappa$-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 $\kappa$-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 $\kappa$-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 $\kappa$-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 $\kappa$-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 $e^{-l_{(0,x_0)}}$ 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 $\kappa$-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 $\kappa$-noncollapsing via reduced volume proceeds by the following scheme:

Non-collapsing at time t=0 (1)

$\Downarrow$

Large reduced volume at time t=0 (2)

$\Downarrow$

Large reduced volume at later times t (3)

$\Downarrow$

Non-collapsing at later times t (4)

The implication $(2) \implies (3)$ is the monotonicity of Perelman reduced volume. In this lecture we discuss the other two implications $(1) \implies (2)$, and $(3) \implies (4)$).

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.