Mikhail Gromov: How Does He Do It?

MI KHAI LGROMOV: HOW DOES HEDO I T? b yJ e f fChe e g e r

From Gromov’s Abel Prize Citation: • A decisive role in the creation of modern global Riemannian geometry. • One of the founders of symplectic geometry, in particular, he created the theory of J-holomorphic curves, which led to the creation of symplectic topology and became linked to quantum field theory. • His solution of the conjecture that groups of polynomial growth are almost nilpotent introduced ideas which forever changed the way a discrete infinite group is viewed and his geometrical approach rendered combinatorial ideas much more natural and powerful.

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Further influence, broad view point. Apart from group theory, other fields such as partial differential equations have been strongly influenced by Gromov’s introduction of a geometric perspective. He has a deep and detailed understanding of many areas which are seemingly far from geometry — ask anyone who has attended a lecture with Gromov in the audience. As an example, the finitely generated group discussion includes, not only hyperbolic groups (incorporating a synthetic asymptotic generalization of negative curvature) but also, “random groups” and ideas from algorithmic complexity.

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Early influences. Gromov cites the “obviously nonsensical” work of Nash on the isometric imbedding problem and Smale on turning the 2-sphere inside out, as strong early influences. These led to his work far reaching work on the “h-principle” and “convex integration”. Another strong influence was the work of Kazhdan-Margulis which associated nontrivial nilpotent subgroups to the “thin” parts of locally homogeneous spaces. This led to his work on “almost flat manifolds” and subsequently, to many other works in geometry and discrete groups.

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The h-principle. It asserts very roughly, that for “most under determined” partial differential equations arising, the “obvious” obstructions to the existence of a solution are the only ones and the solutions are rather dense in a appropriate function spaces. Intuition derived from classical equations of mathematical physics, makes the above statement seem totally counterintuitive. Gromov invented a general tool, called “convex integration”, which can be used for verify the h-principle in many specific cases. These ideas, which were later elaborated in Gromov’s book: “Partial differential relations”, have slowly been assimilated, though the full effects are likely yet to be felt.

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A startling application. Every open manifold admits a (generally incomplete) Riemannian metric of positive curvature and also one of negative curvature. Gromov (age 26) included this striking but puzzling result in his talk at the 1970 ICM held in Nice. Prior to his arrival in Stony Brook in 1974, this was the theorem for which Gromov was primarily known in the west.

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Almost flat manifolds. If the metric, g, of a Riemannian manifold is multiplied by a constant η 2 > 0, then the curvature gets multiplied by η −2, K(M n,η 2·g) = η −2 · K(M n,g) . In almost all cases, as η → 0, diam(M n, η 2 · g) → 0 , and the curvature blows up. The only exception is the case of flat manifolds, K(M n,g) ≡ 0 . Bieberbach’s theorem states that the fundamental group of a flat manifold has a free abelian subgroup of finite index.

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By the 1960’s, members of the Russian school were aware that there exist compact smooth manifolds with the following startling properties: • They admit a sequence of Riemannian metrics, gλ, such that as λ → 0, the diameter goes to zero and the curvature goes to zero. • The fundamental groups of these manifolds are nilpotent, but have no abelian subgroup of finite index. From the second property and Bieberbach’s theorem, it follows that these manifolds admit no flat metric. Next we describe the simplest example.

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The 3-dimensional Heisenberg group. Let H denote R3 = (a, b, c) viewed as the nilpotent matrix group 



1 a c   0 1 b  0 0 1 For any fixed λ > 0, the set of all 

1 λa λ2c

 0

0

1 0



λb  1 

with a, b, c ∈ Z, is a subgroup, Γλ, which, up to isomorphism, is independent of λ. The quotients, H/Γλ are compact and mutally diffeomorphic, with nilpotent fundamental group Γλ = Γ1.

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Equip H with a Riemannian metric which is right-invariant and hence, has bounded curvature. The curvature of the induced metric on H/Γλ is bounded independent of λ. As λ → 0, diam(H/Γλ) ∼ λ → 0 , while the curvature stays bounded. After multiplying the metric on H/Γλ by a −1 2

factor, λ

, one obtains a family for which

diam(H/Γλ) ∼

1 λ2

→ 0,

|KM n | ∼ λ → 0 .

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By the early 1970’s Gromov had proved a striking converse: Theorem. (Gromov) Every manifold admitting a sequence of metrics such that the diameter and curvature go to zero is finitely covered by a nilmanifold.

The proof introduced many new ideas and techniques which were subsequently used in describing the general phenomonenon of “collapse with bounded curvature”. Even today, the proof is not easy.

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Remark: In 1982, E. Ruh, introduced analytic techniques into the discussion and made the nature of the finite covering precise (as had been suggested by Gromov).

Remark. It was in the context of almost flat manifolds, that Gromov invented the “Gromov-Hausdorff distance”. His motivation was to describe precisely, the phenomenon of higher dimensional spaces converging geometrically to lower ones.

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Arrival in Stony Brook, 1974. The experience of meeting in person, the man who was known primarily for the strange result on positive and negative curvature, remains vivd in my mind after 35 years. Initial curiosity rapidly gave way to shock. After some weeks of listening to Misha, I remarked to Dennis Sullivan: “I have the impression that more than half of what is known in Riemannian geometry is known only to Gromov.” A bit later Detlef Gromoll said to me: “Misha is one of the great minds of the century, I don’t know how he does it, he understands everything in the simplest possible way.” 13

Already visible characteristics. A strongly geometric perspective, also applied in other fields. A pronounced interest in discrete groups. Introduction of “rough” notions in geometry (Gromov-Hausdorff distance). Thinking in terms of structures. Identification inside the work of others, of the simple essential principles, with far reaching consequences. Strikingly original results.

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Pronouncements worth pondering. Next we look at some statements of Misha. They seem to reflect in part, a continuation of themes that have already been noted.

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Oral communication: • “Quite often, famous problems are famous primarily because they have remained open for a long time, Intrinsically they may not be so interesting. When they are finally solved, the really significant point is often some statement which remains buried inside the proof.”

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From “Spaces and questions”. • “A common way to generate questions (not only) in geometry is to confront properties of objects specific to different categories: what is a possible topology (e.g. homology) of a manifold with a given type of curvature? ... These seduce us by simplicity and apparent naturality, sometimes leading to new ideas and structures ... but often the mirage of naturality lures us into a featureless desert, where the solution, even if found, does not quench our thirst for structural mathematics.”

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Oral communication: “Many people don’t really think about what they do.”

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Continuation of the discussion from “Spaces and questions”. • “Another approach consists in interbreeding (rather than intersecting) categories and ideas. For example, random graphs, differential topology, p-adic analysis, . . . This has a better chance for a successful outcome, with questions following (rather than preceding) construction of new objects.”

Remark. Compare also “hyperbolic groups”, “random groups”, etc.

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From “Stability and pinching”. • “What may be new and interesting for non-experts is an exposition of the stability/pinching philosopy which lies behind the basic results and methods in the field and which is rarely (if ever) presented in print. This common and unfortunate fact of the lack of adequate presentation of basic ideas and motivations of almost any mathematical theory is probably due to the binary nature of mathematical perception. Either you have no inkling of and idea or, once you have understood it, the very idea appears so embarassingly obvious that you feel reluctant to say it aloud ...”

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Oral communication (concerning the GromovHausdorff distance): • “I knew it for a long time, but it seemed too trivial to write. Sometimes you just have to say it.”

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Structural thinking versus technique. Oral communication: • “There is a temptation for people who are extremely powerful to rely on technique rather than structural thinking, because for them, it is so much easier. At the highest level of technique, structure can emerge, as with say Jacobi and certain of his present day counterparts. Both structural thinking and technique are necessary; most people are more naturally inclined to one or the other.”

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Dennis Sullivan (oral communication): “Sometimes Gromov secretly computes. He computes by logic.”

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Soft and hard structures. The relation between “soft” and “hard” structures plays a major role in much of Gromov’s work. From his 1986 ICM talk: “Soft and hard sympectic geometry”: • “Intuitively, hard refers to a strong and rigid structure of a given object, while soft refers to some weak general property of a vast class of objects.” For further discussion, see also “Spaces and questions”.

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J-holomorphic curves. A symplectic structure on a smooth manifold, M 2n, is a closed 2-form, ω, of maximal rank i.e. the n-fold wedge product of ω with itself is nonzero. By Darboux’s theorem, in suitable local coordinates, we can always write ω = dx1 ∧ dx2 + · · · + dx2n−1 ∧ dx2n . Therefore, locally, the subject is completely soft i.e. all symplectic forms are locally equivalent.

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Gromov revolutionized symplectic geometry by introducing elliptic methods which “hardened” or “rigidified” the structure. His bold stroke was to choose a Riemannian metric g and an almost complex structure J such that one can write ω(x, y) = g(Jx, y) .

Remark. This must have been noticed before and judged not not to be helpful, since in general, J can not be chosen to be integrable.

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Let J1 denote the standard complex structure on C and let f : C → M 2n. Gromov observed that the equation, df ◦ J1 = J ◦ df , has the same linearization as that of the Cauchy-Riemann operator for maps f : C → Cn . This enabled him to associate moduli spaces of J-holomorphic curves to (M 2k , ω), whose essential properties were independent of the choice of J. Here we suppress a lot that is crucial, including the role of “positivity” of the metric in “taming” the almost complex structure.

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The “nonsqueezing” theorem. Let V1 denote the ball of radius r in Cn and let V2 denote the R-tubular neighborhood of Cn−1 ⊂ Cn. Let zj = xj + iyj and let ω denote the standard symplectic form, ω = dx1 ∧ dx2 + · · · + dx2n−1 ∧ dx2n .

Theorem. If there exists a simplectic embedding, f : V1 → V2, i.e. f ∗(ω) = ω , then r < R.

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The Gromov-Hausdorff distance. In proving the polynomial growth conjecture, Gromov employed a soft geometric tool, the Gromov-Hausdorff distance, in the solution of a discrete algebraic problem. Subsequently, the Gromov-Hausdorff distance, has been used in Riemannian geometry to the study the shapes of manifolds with Ricci curvature bounded below. In particular, it has been used to study degenerations of Einstein metrics which, by definition, are solutions of the highly nonlinear elliptic system RicM n = λ · g .

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Definition. If X, Y are compact subsets of a metric space Z, the Hausdorff distance, dH (X, Y ), is defined as the infimal ǫ, such that: X is contained in the ǫ-tubular neighborhood of Y , and Y is contained in the ǫ-tubular neighborhood of X. More generally, if X, Y are compact metric spaces, define their Gromov-Hausdorff distance, dGH (X, Y ), to be: The infimum of the collection of Hausdorff distances obtained from pairs isometric embeddings of X and Y into the same metric space Z.

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Inuition. Intuitively, dGH (X, Y ) is small if X, Y are hard to distinguish with the naked eye, although they may look entirely different under the microscope. Thus, with respect to dGH , a finite segment of a thin cylinder is close to a line segment. Remark. We should consider isometry classes, since what we defined is actually a pseudodistance.

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Gromov’s compactness theorem. Let d > 0, and N (ǫ) : (0, 1] → Z+ denote some function. Let X (d, N (ǫ)) denote the collection of isometry classes of compact metric spaces, X, with diam(X) ≤ d , and such that for all ǫ > 0, there an ǫ-dense subset with ≤ N (ǫ) members The collection, X (d, N (ǫ)), is said to be uniformly totally bounded.

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Theorem. The collection, X (d, N (ǫ)), is compact with respect to the topology induced by dGH . Proof: Use the pigeon hole principle and a diagonal argument.

This compactness theorem is an elementary result, whose proof is not difficult. It nonetheless, the theorem constitutes a powerful organizing principle.

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Ricci curvature bounded below. The Bishop-Gromov inequality in Riemannian geometry controls ratios of volumes of concentric metric balls for manifolds with a definite lower bound on Ricci curvature RicM n ≥ (n − 1)H · g .

It implies a doubling condition, for r ≤ R, Vol(B2r (x)) ≤ c(n, H, R) · Vol(Br (x)) . Here, Br (x) := {y | dist(y, x) < r} .

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Let M(d, H, n) denote the collection of isometry classes of Riemannian manifolds M n, with diam(M n) ≤ d , RicM n ≥ (n − 1)H · g .

By a well known easy consequence of the doubling condition: For all M n ∈ M(d, H, n), there is an ǫ-dense set with ≤ N (ǫ, c(n, H, d)) members. Thus, M(d, H, n) is uniformly totally bounded.

Corollary. M(d, H, n) is precompact with repect to the topology induced by dGH .

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Application to potential bad behavior. Consider the possible existence of a sequence of manifolds, Min, in M(d, H, n), exhibiting some specific sort of arbitrarily bad geometric behavior as i → ∞. After passing to a subsequence, we find a dGH n convergent subsequence, Mj −→ M∞.

By analogy with the theory of distributions or Sobolev spaces, we think of M∞ as some kind of generalized riemannian manifold with bounded diameter and Ricci tensor bounded below. Suppose next, that we actually know some properties of M∞ — the analog of a Sobolev embedding theorem.

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In favorable cases be able to conclude that the putative arbitrarily badly behaving sequence could not have existed. But a priori, we have virtually no idea at all what the limiting objects M∞ look like. Indeed, the possible existence of such potentially bizarre objects arising from Riemannin geometry was initially quite disturbing. Even worse, it seems like the only way of getting information on M∞ is to have uniform information on the sequence Mj . Thus, the program looks circular.

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However. It is true that getting some initial control over M∞ requires uniform information on Mjn. But once it has been obtained, it can be used to argue directly on M∞, to obtain more properties. This in turn, gives new information on the sequence, Mjn, etc.

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An exception that proves the rule. Over and over, Gromov has invented new techniques enabling him to deal with problems which otherwise would have been completely out of reach. But in at least one instance, things went differently. For simplicity, we state a special case of the estimate he proved.

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The Betti number estimate. Let bi(M n) denote the i-th Betti number of the manifold M n with coefficients some arbitrary fixed field F . Theorem. There is a constant, c(n), such that if M n denotes a complete Riemannian manifold with nonnegative sectional curvature, then n X

bi(M n) ≤ c(n) .

i=0

Remark. It is still conceivable that on can choose c(n) = 2n, as holds for the n-torus.

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Critical points of distance functions. In Riemannian geometry, distance functions, distp(x) := dist(x, p) , are, of course Lipschitz, but need not be smooth. K Grove and K. Shiohama observed that for these functions, there is a notion of critical point for which the Isotopy Lemma of Morse theory holds. They used it to prove a beautiful generalization of Berger’s sphere theorem. Unfortunately, in general, there is no analogue of the Morse Lemma.

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When Misha announced that he had used the Grove-Shiohama technique to obtain the Betti number estimate, I was stunned, Was there in fact, an analog of the Morse Lemma? No, it turned out that he had invented a new method of estimating Betti numbers based on the Isotopy Lemma! In the whole proof, nonnegative curvature was used only once, in a key lemma, whose rather standard proof, took only a few lines. But as far as I know, Misha’s method has had no further applications.

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This proves that no one is perfect!

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