MIKHAIL GROMOV:HOW DOES HE DO IT?

by Jeff Cheeger

From Gromov’s Abel Prize Citation:

• A decisive role in the creation of modern

global Riemannian geometry.

• One of the founders of symplectic geom-

etry, in particular, he created the theory of

J-holomorphic curves, which led to the cre-

ation 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 combi-

natorial ideas much more natural and pow-

erful.

2

Further influence, broad view point.

Apart from group theory, other fields such

as partial differential equations have been

strongly influenced by Gromov’s introduc-

tion 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 gen-

eralization of negative curvature) but also,

“random groups” and ideas from algorith-

mic complexity.

3

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 nontriv-

ial nilpotent subgroups to the “thin” parts

of locally homogeneous spaces.

This led to his work on “almost flat man-

ifolds” and subsequently, to many other

works in geometry and discrete groups.

4

The h-principle.

It asserts very roughly, that for “most un-

der determined” partial differential equa-

tions arising, the “obvious” obstructions to

the existence of a solution are the only ones

and the solutions are rather dense in a ap-

propriate function spaces.

Intuition derived from classical equations

of mathematical physics, makes the above

statement seem totally counterintuitive.

Gromov invented a general tool, called “con-

vex integration”, which can be used for ver-

ify the h-principle in many specific cases.

These ideas, which were later elaborated in

Gromov’s book: “Partial differential rela-

tions”, have slowly been assimilated, though

the full effects are likely yet to be felt.

5

A startling application.

Every open manifold admits a (generally

incomplete) Riemannian metric of positive

curvature and also one of negative curva-

ture.

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.

6

Almost flat manifolds.

If the metric, g, of a Riemannian manifold

is multiplied by a constant η2 > 0, then the

curvature gets multiplied by η−2,

K(Mn,η2·g) = η−2 · K(Mn,g) .

In almost all cases, as η → 0,

diam(Mn, η2 · g) → 0 ,

and the curvature blows up.

The only exception is the case of flat man-

ifolds,

K(Mn,g) ≡ 0 .

Bieberbach’s theorem states that the fun-

damental group of a flat manifold has a

free abelian subgroup of finite index.

7

By the 1960’s, members of the Russian

school were aware that there exist com-

pact smooth manifolds with the following

startling properties:

• They admit a sequence of Riemannian

metrics, gλ, such that as λ → 0, the diame-

ter goes to zero and the curvature goes to

zero.

• The fundamental groups of these man-

ifolds are nilpotent, but have no abelian

subgroup of finite index.

From the second property and Bieberbach’s

theorem, it follows that these manifolds ad-

mit no flat metric.

Next we describe the simplest example.

8

The 3-dimensional Heisenberg group.

Let H denote R3 = (a, b, c) viewed as the

nilpotent matrix group

1 a c0 1 b0 0 1

For any fixed λ > 0, the set of all

1 λa λ2c0 1 λb0 0 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 mu-

tally diffeomorphic, with nilpotent funda-

mental group Γλ = Γ1.

9

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

factor, λ−12, one obtains a family for which

diam(H/Γλ) ∼ λ12 → 0 ,

|KMn| ∼ λ → 0 .

10

By the early 1970’s Gromov had proved a

striking converse:

Theorem. (Gromov) Every manifold ad-

mitting 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.

11

Remark: In 1982, E. Ruh, introduced an-

alytic techniques into the discussion and

made the nature of the finite covering pre-

cise (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.

12

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 pos-

sible way.”

13

Already visible characteristics.

A strongly geometric perspective, also ap-

plied in other fields.

A pronounced interest in discrete groups.

Introduction of “rough” notions in geome-

try (Gromov-Hausdorff distance).

Thinking in terms of structures.

Identification inside the work of others, of

the simple essential principles, with far reach-

ing consequences.

Strikingly original results.

14

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.

15

Oral communication:

• “Quite often, famous problems are fa-

mous primarily because they have remained

open for a long time,

Intrinsically they may not be so interesting.

When they are finally solved, the really sig-

nificant point is often some statement which

remains buried inside the proof.”

16

From “Spaces and questions”.

• “A common way to generate questions

(not only) in geometry is to confront prop-

erties of objects specific to different cat-

egories: 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 mathemat-

ics.”

17

Oral communication:

“Many people don’t really think about what

they do.”

18

Continuation of the discussion from “Spaces

and questions”.

• “Another approach consists in interbreed-

ing (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 ob-

jects.”

Remark. Compare also “hyperbolic groups”,

“random groups”, etc.

19

From “Stability and pinching”.

• “What may be new and interesting for

non-experts is an exposition of the stabil-

ity/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 mathemat-

ical 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 ...”

20

Oral communication (concerning the Gromov-

Hausdorff distance):

• “I knew it for a long time, but it seemed

too trivial to write.

Sometimes you just have to say it.”

21

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.”

22

Dennis Sullivan (oral communication):

“Sometimes Gromov secretly computes.

He computes by logic.”

23

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”.

24

J-holomorphic curves.

A symplectic structure on a smooth man-

ifold, M2n, is a closed 2-form, ω, of maxi-

mal rank i.e. the n-fold wedge product of

ω with itself is nonzero.

By Darboux’s theorem, in suitable local co-

ordinates, 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.

25

Gromov revolutionized symplectic geome-

try 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 be-

fore and judged not not to be helpful, since

in general, J can not be chosen to be inte-

grable.

26

Let J1 denote the standard complex struc-

ture on C and let f : C → M2n.

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 (M2k, ω), whose

essential properties were independent of the

choice of J.

Here we suppress a lot that is crucial, in-

cluding the role of “positivity” of the metric

in “taming” the almost complex structure.

27

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 stan-

dard symplectic form,

ω = dx1 ∧ dx2 + · · · + dx2n−1 ∧ dx2n .

Theorem. If there exists a simplectic em-

bedding, f : V1 → V2, i.e.

f∗(ω) = ω ,

then

r < R .

28

The Gromov-Hausdorff distance.

In proving the polynomial growth conjec-

ture, Gromov employed a soft geometric

tool, the Gromov-Hausdorff distance, in the

solution of a discrete algebraic problem.

Subsequently, the Gromov-Hausdorff dis-

tance, has been used in Riemannian geom-

etry to the study the shapes of manifolds

with Ricci curvature bounded below.

In particular, it has been used to study de-

generations of Einstein metrics which, by

definition, are solutions of the highly non-

linear elliptic system

RicMn = λ · g .

29

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 neighbor-

hood of Y , and Y is contained in the

ǫ-tubular neighborhood of X.

More generally, if X, Y are compact met-

ric spaces, define their Gromov-Hausdorff

distance, dGH(X, Y ), to be:

The infimum of the collection of Hausdorff

distances obtained from pairs isometric em-

beddings of X and Y into the same metric

space Z.

30

Inuition.

Intuitively, dGH(X, Y ) is small if X, Y are

hard to distinguish with the naked eye, al-

though they may look entirely different un-

der 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 pseudo-

distance.

31

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 isom-

etry 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.

32

Theorem. The collection, X (d, N(ǫ)), is

compact with respect to the topology in-

duced by dGH.

Proof: Use the pigeon hole principle and a

diagonal argument.

This compactness theorem is an elemen-

tary result, whose proof is not difficult.

It nonetheless, the theorem constitutes a

powerful organizing principle.

33

Ricci curvature bounded below.

The Bishop-Gromov inequality in Rieman-

nian geometry controls ratios of volumes of

concentric metric balls for manifolds with a

definite lower bound on Ricci curvature

RicMn ≥ (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} .

34

Let M(d, H, n) denote the collection of isom-

etry classes of Riemannian manifolds Mn,

with

diam(Mn) ≤ d ,

RicMn ≥ (n − 1)H · g .

By a well known easy consequence of the

doubling condition:

For all Mn ∈ 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.

35

Application to potential bad behavior.

Consider the possible existence of a sequence

of manifolds, Mni , in M(d, H, n), exhibiting

some specific sort of arbitrarily bad geo-

metric behavior as i → ∞.

After passing to a subsequence, we find a

convergent subsequence, Mnj

dGH−→ 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.

36

In favorable cases be able to conclude that

the putative arbitrarily badly behaving se-

quence 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 po-

tentially bizarre objects arising from Rie-

mannin geometry was initially quite disturb-

ing.

Even worse, it seems like the only way of

getting information on M∞ is to have uni-

form information on the sequence Mj.

Thus, the program looks circular.

37

However.

It is true that getting some initial control

over M∞ requires uniform information on

Mnj .

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, Mnj , etc.

38

An exception that proves the rule.

Over and over, Gromov has invented new

techniques enabling him to deal with prob-

lems which otherwise would have been com-

pletely out of reach.

But in at least one instance, things went

differently.

For simplicity, we state a special case of

the estimate he proved.

39

The Betti number estimate.

Let bi(Mn) denote the i-th Betti number

of the manifold Mn with coefficients some

arbitrary fixed field F .

Theorem. There is a constant, c(n), such

that if Mn denotes a complete Riemannian

manifold with nonnegative sectional curva-

ture, then

n∑

i=0

bi(Mn) ≤ c(n) .

Remark. It is still conceivable that on can

choose c(n) = 2n, as holds for the n-torus.

40

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 criti-

cal point for which the Isotopy Lemma of

Morse theory holds.

They used it to prove a beautiful general-

ization of Berger’s sphere theorem.

Unfortunately, in general, there is no ana-

logue of the Morse Lemma.

41

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.

42

This proves that no one is perfect!

43