Mikhail Gromov is renowned for his groundbreaking contributions across many areas of mathematics through his unique geometric perspective and ability to identify simple yet powerful underlying principles. Some of his most influential works include introducing symplectic geometry through J-holomorphic curves, proving the h-principle and developing convex integration, establishing the nonsqueezing theorem in symplectic geometry, and introducing the Gromov-Hausdorff distance. Gromov often sees the essence in problems that others miss and formulates strikingly original insights that have profound and wide-ranging impacts.
2. 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.
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3. 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.
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4. 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.
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5. 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.
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6. 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.
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7. 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 man-
ifolds,
K(M n,g) ≡ 0 .
Bieberbach’s theorem states that the fun-
damental group of a flat manifold has a
free abelian subgroup of finite index.
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8. 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.
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9. 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 1 λb
0 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.
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10. 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
factor, λ2 , one obtains a family for which
1
diam(H/Γλ) ∼ λ 2 → 0,
|KM n | ∼ λ → 0 .
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11. 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.
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12. 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.
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13. 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.”
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14. 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.
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15. 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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16. 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.”
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17. 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.”
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19. 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.
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20. 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 ...”
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21. 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.”
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22. 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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23. Dennis Sullivan (oral communication):
“Sometimes Gromov secretly computes.
He computes by logic.”
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24. 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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25. J-holomorphic curves.
A symplectic structure on a smooth man-
ifold, M 2n, 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.
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26. 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.
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27. Let J1 denote the standard complex struc-
ture 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, in-
cluding the role of “positivity” of the metric
in “taming” the almost complex structure.
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28. 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.
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29. 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
RicM n = λ · g .
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30. 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.
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31. 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.
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32. 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.
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33. 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.
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34. 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
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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35. Let M(d, H, n) denote the collection of isom-
etry 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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36. 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 geo-
metric behavior as i → ∞.
After passing to a subsequence, we find a
n dGH M .
convergent subsequence, Mj −→ ∞
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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37. 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.
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38. However.
It is true that getting some initial control
over M∞ requires uniform information on
n
Mj .
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
n
sequence, Mj , etc.
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39. 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.
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40. 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 curva-
ture, then
n
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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41. 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.
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42. 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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