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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.

Let X be a real-valued random variable, and let $X_1, X_2, X_3, ...$ be an infinite sequence of independent and identically distributed copies of X. Let $\overline{X}_n := \frac{1}{n}(X_1 + \ldots + X_n)$ be the empirical averages of this sequence. A fundamental theorem in probability theory is the law of large numbers, which comes in both a weak and a strong form:

Weak law of large numbers. Suppose that the first moment ${\Bbb E} |X|$ of X is finite. Then $\overline{X}_n$ converges in probability to ${\Bbb E} X$, thus $\lim_{n \to \infty} {\Bbb P}( |\overline{X}_n - {\Bbb E} X| \geq \varepsilon ) = 0$ for every $\varepsilon > 0$.

Strong law of large numbers. Suppose that the first moment ${\Bbb E} |X|$ of X is finite. Then $\overline{X}_n$ converges almost surely to ${\Bbb E} X$, thus ${\Bbb P}( \lim_{n \to \infty} \overline{X}_n = {\Bbb E} X ) = 1$.

[The concepts of convergence in probability and almost sure convergence in probability theory are specialisations of the concepts of convergence in measure and pointwise convergence almost everywhere in measure theory.]

(If one strengthens the first moment assumption to that of finiteness of the second moment ${\Bbb E}|X|^2$, then we of course have a more precise statement than the (weak) law of large numbers, namely the central limit theorem, but I will not discuss that theorem here.  With even more hypotheses on X, one similarly has more precise versions of the strong law of large numbers, such as the Chernoff inequality, which I will again not discuss here.)

The weak law is easy to prove, but the strong law (which of course implies the weak law, by Egoroff’s theorem) is more subtle, and in fact the proof of this law (assuming just finiteness of the first moment) usually only appears in advanced graduate texts. So I thought I would present a proof here of both laws, which proceeds by the standard techniques of the moment method and truncation. The emphasis in this exposition will be on motivation and methods rather than brevity and strength of results; there do exist proofs of the strong law in the literature that have been compressed down to the size of one page or less, but this is not my goal here.

I’ve uploaded a new paper to the arXiv entitled “The sum-product phenomenon in arbitrary rings“, and submitted to Contributions to Discrete Mathematics. The sum-product phenomenon asserts, very roughly speaking, that given a finite non-empty set A in a ring R, then either the sum set $A+A := \{ a+b: a, b \in A \}$ or the product set $A \cdot A := \{ ab: a, b \in A \}$ will be significantly larger than A, unless A is somehow very close to being a subring of R, or if A is highly degenerate (for instance, containing a lot of zero divisors). For instance, in the case of the integers $R = {\Bbb Z}$, which has no non-trivial finite subrings, a long-standing conjecture of Erdös and Szemerédi asserts that $|A+A| + |A \cdot A| \gg_\varepsilon |A|^{2-\varepsilon}$ for every finite non-empty $A \subset {\Bbb Z}$ and every $\varepsilon > 0$. (The current best result on this problem is a very recent result of Solymosi, who shows that the conjecture holds for any $\varepsilon$ greater than 2/3.) In recent years, many other special rings have been studied intensively, most notably finite fields and cyclic groups, but also the complex numbers, quaternions, and other division algebras, and continuous counterparts in which A is now (for instance) a collection of intervals on the real line. I will not try to summarise all the work on sum-product estimates and their applications (which range from number theory to graph theory to ergodic theory to computer science) here, but I discuss this in the introduction to my paper, which has over 50 references to this literature (and I am probably still missing out on a few).

I was recently asked the question as to what could be said about the sum-product phenomenon in an arbitrary ring R, which need not be commutative or contain a multiplicative identity. Once one makes some assumptions to avoid the degenerate case when A (or related sets, such as A-A) are full of zero-divisors, it turns out that there is in fact quite a bit one can say, using only elementary methods from additive combinatorics (in particular, the Plünnecke-Ruzsa sum set theory). Roughly speaking, the main results of the paper assert that in an arbitrary ring, a set A which is non-degenerate and has small sum set and product set must be mostly contained inside a subring of R of comparable size to A, or a dilate of such a subring, though in the absence of invertible elements one sometimes has to enlarge the ambient ring R slightly before one can find the subring. At the end of the paper I specialise these results to specific rings, such as division algebras or products of division algebras, cyclic groups, or finite-dimensional algebras over fields. Generally speaking, the methods here give very good results when the set of zero divisors is sparse and easily describable, but poor results otherwise. (In particular, the sum-product theory in cyclic groups, as worked out by Bourgain and coauthors, is only recovered for groups which are the product of a bounded number of primes; the case of cyclic groups whose order has many factors seems to require a more multi-scale analysis which I did not attempt to perform in this paper.)
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This week I was in Columbus, Ohio, attending a conference on equidistribution on manifolds. I talked about my recent paper with Ben Green on the quantitative behaviour of polynomial sequences in nilmanifolds, which I have blogged about previously. During my talk (and inspired by the immediately preceding talk of Vitaly Bergelson), I stated explicitly for the first time a generalisation of the van der Corput trick which morally underlies our paper, though it is somewhat buried there as we specialised it to our application at hand (and also had to deal with various quantitative issues that made the presentation more complicated). After the talk, several people asked me for a more precise statement of this trick, so I am presenting it here, and as an application reproving an old theorem of Leon Green that gives a necessary and sufficient condition as to whether a linear sequence $(g^n x)_{n=1}^\infty$ on a nilmanifold $G/\Gamma$ is equidistributed, which generalises the famous theorem of Weyl on equidistribution of polynomials.

UPDATE, Feb 2013: It has been pointed out to me by Pavel Zorin that this argument does not fully recover the theorem of Leon Green; to cover all cases, one needs the more complicated van der Corput argument in our paper.

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).

I’ve just uploaded to the arXiv a new paper, “Global regularity of wave maps III. Large energy from ${\Bbb R}^{1+2}$ to hyperbolic spaces“, to be submitted when three other companion papers (“Global regularity of wave maps” IV, V, and VI) are finished. This project (which I had called “Heatwave”, due to the use of a heat flow to renormalise a wave equation) has a somewhat lengthy history to it, which I will now attempt to explain.

For the last nine years or so, I have been working on and off on the global regularity problem for wave maps $\phi: {\Bbb R}^{1+d} \to M$. The wave map equation $(\phi^* \nabla)^\alpha \partial_\alpha \phi=0$ is a nonlinear generalisation of the wave equation $\partial^\alpha \partial_\alpha \phi = 0$ in which the unknown field $\phi$ takes values in a Riemannian manifold $M = (M,h)$ rather than in a vector space (much as the concept of a harmonic map is a nonlinear generalisation of a harmonic function). This equation (also known as the nonlinear $\sigma$ model) is one of the simplest examples of a geometric nonlinear wave equation, and is also arises as a simplified model of the Einstein equations (after making a U(1) symmetry assumption). The global regularity problem seeks to determine when smooth initial data for a wave map (i.e. an initial position $\phi_0: {\Bbb R}^d \to M$ and an initial velocity $\phi_1: {\Bbb R}^d \to TM$ tangent to the position) necessarily leads to a smooth global solution.

The problem is particularly interesting in the energy-critical dimension d=2, in which the conserved energy $E(\phi) := \int_{{\Bbb R}^d} \frac{1}{2} |\partial_t \phi|_{h(\phi)}^2 + \frac{1}{2} |\nabla_x \phi|_{h(\phi)}^2\ dx$ becomes invariant under the scaling symmetry $\phi(t,x) \mapsto \phi(t/\lambda,x/\lambda)$. (In the subcritical dimension d=1, global regularity is fairly easy to establish, and was first done by Gu and by Ladyzhenskaya-Shubov; in supercritical dimensions $d \geq 3$, examples of singularity formation are known, starting with the self-similar examples of Shatah.)

It is generally believed that in two dimensions, singularities can form when M is positively curved but that global regularity should persist when M is negatively curved, in analogy with known results (in particular, the landmark paper of Eells and Sampson) for the harmonic map heat flow (a parabolic cousin of the wave map equation). In particular, one should always have global regularity when the target is a hyperbolic space. There has been a large number of results supporting this conjecture; for instance, when the target is the sphere, examples of singularity formation have recently been constructed by Rodnianski-Sterbenz and by Krieger-Schlag-Tataru, while for suitably negatively curved manifolds such as hyperbolic space, global regularity was established assuming equivariant symmetry by Shatah and Tahvildar-Zadeh, and assuming spherical symmetry by Christodoulou and Tahvildar-Zadeh. I will not attempt to mention all the other results on this problem here, but see for instance one of these survey articles or books for further discussion.