Mikhail Gromov

How to use this course

Mikhail Gromov has reshaped several parts of geometry, often by asking what survives when a space is viewed from very far away or measured very roughly. From his thesis on the h-principle to his work on hyperbolic groups and pseudoholomorphic curves, his ideas founded fields that now have their own conferences and textbooks: geometric group theory, metric geometry in the style of Gromov–Hausdorff limits, and modern symplectic topology. The Abel Prize committee in 2009 cited his revolutionary contributions to geometry.

Each technical lesson opens with a paragraph in italics for the general reader, followed by precise statements, proof sketches and exercises with folded solutions. Gromov rarely writes complete proofs in the classical style; his papers are dense with ideas and leave details to readers. This course gives standard modern formulations, which sometimes differ from his original wording.

Timeline

Boksitogorsk and Leningrad, 1943–1965

Mikhael Leonidovich Gromov was born on 23 December 1943 in Boksitogorsk, a town about 200 km east of Leningrad, during the Second World War. His parents, Leonid Gromov and Lea Rabinovitz, were both pathologists. He studied mathematics at Leningrad State University and completed the master’s degree in 1965.

Leningrad mathematics in the 1960s had a strong geometric school. Alexander Alexandrov had built a theory of metric spaces with curvature bounded in a synthetic sense, without derivatives, and his students carried it on. Gromov’s later work shares this outlook: it treats rough metric spaces with the freedom that classical geometers gave smooth surfaces, and applies it to groups and to whole families of manifolds.

Rokhlin’s student, 1965–1973

Gromov’s doctoral advisor was Vladimir Rokhlin, a topologist and ergodic theorist whose career had been interrupted by the war and years in Soviet camps. Rokhlin’s seminar in Leningrad produced a remarkable group of topologists. Gromov defended his doctorate in 1969 and taught at the university as an assistant professor from 1967 to 1974.

His early work, partly with Yakov Eliashberg, developed a single principle that unified many results on immersions and submersions: for a large class of geometric problems, the only obstruction to a solution is topological. Stephen Smale and Morris Hirsch had proved such a statement for immersions, and Anthony Phillips for submersions. Gromov found the general mechanism and showed, among other things, that every open manifold carries metrics of positive sectional curvature and metrics of negative sectional curvature. Lesson 04 presents this h-principle.

He was invited to speak at the International Congress of Mathematicians in Nice in 1970 but was not allowed to leave the country. The text of his lecture was published in the proceedings in 1971. That year the Moscow Mathematical Society awarded him its prize. In 1973 he presented a second, higher dissertation at Leningrad.

Leaving the Soviet Union, 1974

By the early 1970s Gromov wanted to leave the Soviet Union. He applied to emigrate to Israel, the usual route for Soviet emigrants at the time, and stopped publishing while the application was pending. In 1974 permission was granted, and he went to the United States, where he became a professor at the State University of New York at Stony Brook.

The move opened the world to him. He spoke at the ICM in Helsinki in 1978. In 1977 Dennis Sullivan invited him to visit the IHÉS near Paris. In 1979 he gave a course at Paris VII, and the notes, written up by Jacques Lafontaine and Pierre Pansu, appeared in 1981 as Structures métriques pour les variétés riemanniennes. The little green book introduced the Gromov–Hausdorff distance to a wide audience and became one of the most influential texts in geometry. Its expanded English edition, Metric Structures for Riemannian and Non-Riemannian Spaces, followed in 1999.

The h-principle

For the general reader. Many geometric problems ask for a map or a shape satisfying some condition on its derivatives. Often there is a cheaper version of the problem, where the derivatives are replaced by arbitrary data that need not fit together. Gromov showed that for a large class of problems, if the cheap version can be solved, the real one can too. Geometry then reduces to topology.

Formal and genuine solutions

Let \(V\) and \(W\) be manifolds and \(J^r(V,W)\) the space of \(r\)-jets of maps \(V\to W\): at each point, a value together with candidate derivatives up to order \(r\). A differential relation of order \(r\) is a subset \(\mathcal R\subset J^r(V,W)\). A formal solution is a section of \(J^r(V,W)\to V\) with values in \(\mathcal R\); a genuine solution is a map \(f\) whose actual \(r\)-jet \(j^rf\) lies in \(\mathcal R\). The relation satisfies the h-principle if every formal solution is homotopic, through formal solutions, to a genuine one.

For immersions, \(\mathcal R\) consists of 1-jets with injective derivative, and a formal solution is a map \(f\) with a fibrewise injective bundle map \(TV\to f^*TW\).

Idea. An open manifold retracts by isotopy onto a lower-dimensional polyhedron \(K\), its core. Near \(K\) one can solve the relation by working one cell at a time, using the fact that over a small cube a formal solution can be integrated with a small, controlled loss. Diffeomorphism invariance then lets one compress \(V\) into this neighbourhood of \(K\) and pull the solution back.

Convex integration

John Nash proved in 1954, with Nicolaas Kuiper’s refinement in 1955, that any short embedding of a Riemannian manifold \(V^n\) into \(\mathbb R^q\) with \(q\ge n+1\) can be approximated in \(C^0\) by \(C^1\) isometric embeddings. A round sphere can be crumpled, with \(C^1\) regularity, into an arbitrarily small ball without changing any lengths. Gromov extracted the mechanism, adding fast oscillations whose average lies in the convex hull of the relation, and named it convex integration (1973). His 1986 book Partial Differential Relations developed it systematically; much later, Camillo De Lellis and László Székelyhidi used it to build wild weak solutions of the Euler equations.

Curvature and Gromov–Hausdorff limits

For the general reader. How far apart are two shapes? Gromov gave a distance between any two compact metric spaces, then showed that manifolds with curvature bounded below form a family that is compact for this distance. Sequences of manifolds can now be studied through their limits, which may be singular spaces.

The distance

For compact metric spaces \(X,Y\), the Gromov–Hausdorff distance is

the infimum over all metric spaces \(Z\) and isometric embeddings \(\iota\colon X\to Z\), \(\jmath\colon Y\to Z\) of the Hausdorff distance between the images. It is a metric on isometry classes of compact metric spaces. A practical formula uses correspondences \(\mathfrak R\subset X\times Y\) (relations projecting onto both factors): \(d_{GH}(X,Y)=\tfrac12\inf_{\mathfrak R}\operatorname{dis}\mathfrak R\), where \(\operatorname{dis}\mathfrak R=\sup\{|d_X(x,x')-d_Y(y,y')| : (x,y),(x',y')\in\mathfrak R\}\).

Proof sketch. By the Bishop–Gromov volume comparison, \(r\mapsto\operatorname{vol}B(x,r)/V_\kappa(r)\) is non-increasing, where \(V_\kappa(r)\) is the volume of an \(r\)-ball in the model space of constant curvature \(\kappa\). Hence a maximal \(\varepsilon\)-separated set has at most \(N(\varepsilon)=V_\kappa(D)/V_\kappa(\varepsilon/2)\) points, a bound independent of the manifold. A class of compact spaces that is uniformly totally bounded in this sense is precompact, by a diagonal argument.

Limits of such sequences, the Ricci limit spaces, were analysed by Jeff Cheeger and Tobias Colding from the 1990s on. Alexandrov spaces, limits under a lower sectional curvature bound, were studied by Yuri Burago, Gromov and Grigori Perelman (1992).

Groups of polynomial growth

For the general reader. Count the elements of a group that can be written as words of length at most \(R\) in a fixed set of generators. For some groups the count grows like a polynomial in \(R\), for others exponentially. Gromov proved in 1981 that polynomial growth forces the group to be almost nilpotent, close to commutative. His proof looked at the group from farther and farther away until it became a smooth geometric object.

Growth

Let \(G\) be generated by a finite symmetric set \(S\). The word length \(|g|_S\) is the least number of generators needed to write \(g\), and \(\beta_S(R)=\#\{g : |g|_S\le R\}\). The group has polynomial growth if \(\beta_S(R)\le CR^d\) for some \(C,d\). Changing \(S\) changes word lengths by bounded factors, so polynomial growth is a property of \(G\).

The direction “virtually nilpotent implies polynomial growth” is due to Joseph Wolf (1968); John Milnor had shown that solvable groups have either polynomial or exponential growth and asked about the general case. Hyman Bass and Yves Guivarc’h computed the exact degree: if \(G\) is nilpotent with lower central series \(G=G_1\supset G_2\supset\cdots\), then

Strategy of proof

Rescale the word metric by \(1/R\) and let \(R\to\infty\). Polynomial growth gives a uniform bound on the number of \(\varepsilon\)-balls needed to cover a ball, so by the compactness argument of Lesson 05 a subsequence of \((G,\tfrac1R d_S)\) converges in the pointed Gromov–Hausdorff sense to a locally compact limit \(Y\), what is now called an asymptotic cone. The limit is homogeneous, finite-dimensional and locally connected; by the solution of Hilbert’s fifth problem (Andrew Gleason, Deane Montgomery, Leo Zippin), its isometry group is a Lie group. The action of \(G\) gives a homomorphism to this Lie group with infinite image, and an induction on the growth degree, using results of Jacques Tits on linear groups and of Milnor and Wolf on solvable groups, produces the nilpotent subgroup of finite index. Bruce Kleiner gave a new proof in 2010 using harmonic functions, avoiding the Montgomery–Zippin theorem.

Hyperbolic groups

For the general reader. In the hyperbolic plane triangles are thin: each side stays close to the other two. Gromov turned this single property into a definition that applies to any metric space and to any finitely generated group. The resulting hyperbolic groups are, in a precise sense, the typical groups, and their study launched geometric group theory.

Definitions

A geodesic metric space \(X\) is \(\delta\)-hyperbolic if every geodesic triangle is \(\delta\)-slim: each side lies in the \(\delta\)-neighbourhood of the union of the other two. (Gromov’s original definition used the Gromov product \((y\cdot z)_x=\tfrac12\big(d(x,y)+d(x,z)-d(y,z)\big)\); the definitions agree up to changing \(\delta\). Eliyahu Rips had used thin triangles for groups earlier.) A finitely generated group is hyperbolic if its Cayley graph, with the word metric, is \(\delta\)-hyperbolic for some \(\delta\).

A map \(f\colon X\to Y\) is a quasi-isometry if there are constants \(\lambda\ge1,C\ge0\) with \(\lambda^{-1}d(x,x')-C\le d(fx,fx')\le\lambda d(x,x')+C\) and every point of \(Y\) is within \(C\) of the image.

The key to (a) is the Morse lemma: in a hyperbolic space, every quasi-geodesic stays within bounded distance of a true geodesic. Each hyperbolic group has a Gromov boundary \(\partial G\), the set of equivalence classes of geodesic rays, on which it acts. For the fundamental group of a closed hyperbolic \(n\)-manifold the boundary is \(S^{n-1}\); for a free group it is a Cantor set.

The 1987 essay Hyperbolic groups is a long collection of results and programmes. Its later companion, Asymptotic invariants of infinite groups (1993), proposed studying all finitely generated groups up to quasi-isometry. With Ilya Piatetski-Shapiro, Gromov also constructed non-arithmetic lattices in \(\mathrm{SO}(n,1)\) for every \(n\), and with Richard Schoen he developed harmonic maps into singular spaces to prove arithmeticity for lattices in quaternionic hyperbolic groups (1992).

Pseudoholomorphic curves and non-squeezing

For the general reader. Hamilton’s equations preserve volume, and for a long time it was unclear whether they preserve anything more. Gromov proved in 1985 that a ball cannot be squeezed through a narrow hole by a Hamiltonian flow, even though volume would allow it. The tool he invented, holomorphic curves in a symplectic manifold, underlies modern symplectic topology and the mirror-symmetry predictions of string theory.

Set-up

A symplectic manifold \((M,\omega)\) carries a closed non-degenerate 2-form. An almost complex structure \(J\) (a bundle map with \(J^2=-1\)) is compatible with \(\omega\) if \(g_J(u,v)=\omega(u,Jv)\) is a Riemannian metric; such \(J\) exist and form a contractible space. A \(J\)-holomorphic curve is a map \(u\colon(\Sigma,j)\to(M,J)\) from a Riemann surface with \(du\circ j=J\circ du\). Its energy equals its symplectic area:

which depends only on the homology class of \(u\) when \(\Sigma\) is closed.

Proof sketch. Suppose \(\varphi\) embeds \(B(r)\) into \(Z(R)\); compactify the image to sit in \(S^2(A)\times T^{2n-2}\) with \(A\) slightly larger than \(\pi R^2\). For the standard split structure, every class \([S^2\times pt]\) is represented by a unique holomorphic sphere through each point. Gromov compactness and a cobordism argument show that for any compatible \(J\), in particular one equal to \(\varphi_*J_0\) on the image of the ball, there is still a \(J\)-holomorphic sphere \(u\) in this class through \(\varphi(0)\). Its area is \(A\). Pulled back to the ball, \(u\) gives a properly embedded holomorphic curve through the centre of \(B(r)\), and the monotonicity formula for minimal surfaces gives it area at least \(\pi r^2\). So \(\pi r^2\le A\), and letting \(A\downarrow\pi R^2\) yields \(r\le R\).

Non-squeezing led to symplectic capacities, invariants measuring the size of symplectic sets. Gromov–Witten invariants, which count holomorphic curves, and Floer homology, which counts holomorphic strips, both grew from the 1985 paper. With Yakov Eliashberg he later developed symplectic and contact convexity (1991).

Systoles and filling

For the general reader. On a surface that cannot be shrunk to a point, some loop cannot be shrunk either. How short must the shortest such loop be, compared with the total area? Charles Loewner and Pao Ming Pu answered this for the torus and the projective plane. Gromov proved in 1983 that a universal bound holds in every dimension for a large class of manifolds.

Definitions

The systole \(\operatorname{sys}(M,g)\) of a compact Riemannian manifold is the least length of a non-contractible closed curve. A closed \(n\)-manifold \(M\) is essential if a classifying map \(M\to K(\pi_1M,1)\) cannot be deformed into the \((n-1)\)-skeleton. Tori, real projective spaces and all aspherical manifolds are essential.

Strategy. Embed \(M\) isometrically into the Banach space \(L^\infty(M)\) by the Kuratowski map \(x\mapsto d(x,\cdot)\). The filling radius is the least \(\rho\) such that \(M\) bounds in its \(\rho\)-neighbourhood there. Gromov proves two inequalities. First, for essential \(M\), \(\operatorname{sys}(M)\le6\operatorname{FillRad}(M)\): a filling at small radius would let one build a map of \(M\) into its own \((n-1)\)-skeleton, contradicting essentialness. Second, \(\operatorname{FillRad}(M)\le c_n\operatorname{vol}(M)^{1/n}\), from an isoperimetric inequality for cycles in Banach spaces proved by an induction on dimension with cone constructions. Combining them gives the theorem with an explicit, though large, constant.

In the companion paper Volume and bounded cohomology (1982) Gromov introduced the simplicial volume \(\|M\|\) and proved \(\operatorname{vol}(M)\ge c_n\|M\|\) for metrics with \(\operatorname{Ric}\ge-(n-1)\), one of several topological invariants (with minimal volume and K-area) that he attached to manifolds by minimizing geometric quantities. His student Mikhail Katz developed systolic geometry into a field.

Isoperimetry, concentration and scalar curvature

For the general reader. On a high-dimensional sphere, almost all the area lies near any equator. Gromov turned this observation into a principle: functions on large, positively curved spaces are nearly constant. He also showed, with Blaine Lawson, that the topology of a manifold can forbid it from carrying positive scalar curvature, the weakest form of positive curvature.

Paul Lévy had stated the case of the sphere itself. Gromov’s proof uses geometric measure theory: a minimizer of area for the given volume exists, is smooth away from a small singular set, and has constant mean curvature; the Heintze–Karcher comparison then bounds the volume of its tubular neighbourhoods by those in the sphere.

Concentration

A consequence is that on \(S^n\) with normalized measure \(\sigma\), for any set \(A\) with \(\sigma(A)\ge\tfrac12\), the \(\varepsilon\)-neighbourhood satisfies \(\sigma(A_\varepsilon)\ge1-\sqrt{\pi/8}\,e^{-(n-1)\varepsilon^2/2}\). Equivalently, every 1-Lipschitz function \(f\) on \(S^n\) is within \(\varepsilon\) of its median except on a set of measure exponentially small in \(n\varepsilon^2\). Gromov and Vitali Milman (1983) introduced Lévy families, sequences of spaces with this property, and used them to prove fixed-point theorems for actions of infinite-dimensional groups. Concentration of measure has since become a basic tool in probability, high-dimensional geometry and theoretical computer science.

Positive scalar curvature

Scalar curvature is the trace of the Ricci tensor, the weakest of the curvature conditions. A closed manifold \(M\) is enlargeable if for every \(\varepsilon>0\) it has a spin covering that admits an \(\varepsilon\)-contracting map of non-zero degree onto the unit sphere, constant near infinity.

The proof uses the Dirac operator twisted by a bundle pulled back from the sphere. The Lichnerowicz formula \(D^2=\nabla^*\nabla+\tfrac{s}{4}+\mathcal R^E\) forces the kernel to vanish when \(s>0\) and the twisting curvature \(\mathcal R^E\) is small, while the Atiyah–Singer index theorem shows the index is non-zero. Richard Schoen and Shing-Tung Yau proved the torus case independently by minimal hypersurfaces. Gromov and Lawson also proved in 1980 that positive scalar curvature is preserved under surgeries in codimension at least 3, which made the existence question for simply connected manifolds largely a question in bordism theory.

Paris, the IHÉS, New York

After Stony Brook, Gromov moved to the Université Paris VI in 1981 and in 1982 became a permanent professor at the Institut des Hautes Études Scientifiques at Bures-sur-Yvette, a post without teaching duties. He became a French citizen in 1992. From 1991 to 1996 he was also professor at the University of Maryland, College Park, and from 1996 he has held the Jay Gould Professorship at the Courant Institute of New York University, dividing his time between France and New York.

His doctoral students include Pierre Pansu, who went on to study the large-scale geometry of nilpotent groups, Christophe Bavard, Mikhail Katz, François Labourie, Abdelghani Zeghib, Denis Auroux and Yann Ollivier. Many more mathematicians count themselves his students through his papers and lectures. He gave invited talks at the International Congresses in Helsinki (1978) and Warsaw (1982 Congress, held in 1983), and a plenary address at Berkeley in 1986, Soft and hard symplectic geometry, whose title named the divide between the flexible h-principle and the rigid world of holomorphic curves.

His writing style is distinctive: long essays that move rapidly through ideas, with proofs sketched and conjectures scattered freely. Papers such as Hyperbolic groups (1987), Asymptotic invariants of infinite groups (1993) and Metric Structures (1999) have kept specialists busy for decades. From the 2000s he also wrote on random groups, including groups that cannot be coarsely embedded in Hilbert space, on sofic groups and on topological combinatorics, and he has written extensively on mathematical structures in biology, language and learning.

Honours and influence

Gromov received the Prize of the Moscow Mathematical Society (1971), the Oswald Veblen Prize in Geometry (1981), the Prix Élie Cartan (1984), the Prix de l’Union des Assurances de Paris (1989), the Wolf Prize in Mathematics (1993), the AMS Steele Prize for Seminal Contribution to Research and the Lobachevsky Medal (1997), the Balzan Prize (1999), the Kyoto Prize in Basic Sciences (2002), the Nemmers Prize in Mathematics (2004), the János Bolyai International Mathematical Prize (2005) and the Abel Prize (2009). He is a member of the French Académie des sciences (1997), a foreign associate of the US National Academy of Sciences and a member of the American Academy of Arts and Sciences (both 1989), and a Foreign Member of the Royal Society (2011). The University of Geneva awarded him an honorary doctorate in 1992.

Influence

• Geometric group theory: hyperbolic groups, growth, quasi-isometry invariants and asymptotic cones made groups into geometric objects; the field’s methods fed into the proof of the virtual Haken conjecture.
• Riemannian and metric geometry: Gromov–Hausdorff convergence, comparison geometry, collapse and Alexandrov spaces are standard; Perelman’s proof of the geometrization conjecture uses this framework.
• Symplectic topology: holomorphic curves underlie Gromov–Witten theory, Floer homology and Fukaya categories.
• Analysis and probability: concentration of measure and convex integration are now standard tools far from their origins.

Connections

People linked to Mikhail Gromov across this collection. A name without a link has no page here yet. See Mikhail Gromov in the network of all 337 lives →

Teachers and advisors

• Vladimir Rokhlin 1969

Students

• Abdelghani Zeghib 1985
• Christophe Bavard 1984
• Denis Auroux 1999
• François Labourie 1987
• Mikhail Katz 1984
• Pierre Pansu 1982
• Yann Ollivier 2003

Influenced by

• John Milnor
• John Nash
• Stephen Smale

Colleagues and collaborators

• Dennis Sullivan 1977-1997
• Jacques Tits
• William Thurston
• Grigori Perelman 1992
• H. Blaine Lawson 1980-1983
• Ilya Piatetski-Shapiro 1987
• Jeff Cheeger
• Richard Schoen 1992
• Vitali Milman 1983
• Yakov Eliashberg

Also mentioned in this life

• Michael Atiyah
• Stefan Banach
• János Bolyai
• Georg Cantor
• Augustin-Louis Cauchy
• Arthur Cayley
• Pafnuty Chebyshev
• Richard Courant
• Paul Dirac
• Élie Cartan
• Leonhard Euler
• Carl Friedrich Gauss
• Jacques Hadamard
• William Rowan Hamilton
• Felix Hausdorff
• Werner Heisenberg
• David Hilbert
• Kazimierz Kuratowski
• Sophus Lie
• Nikolai Lobachevsky
• Paul Lévy
• Gregorio Ricci-Curbastro
• Bernhard Riemann
• Edward Witten

Fields

• geometric group theory 4 lives

On CoursesHub

• Differential Geometry

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Mikhael Leonidovich Gromov, MacTutor History of Mathematics.
2. Mikhael Gromov (mathematician), Wikipedia.
3. The Abel Prize, Mikhail Leonidovich Gromov, Abel Prize 2009.
4. Mikhael Leonidovich Gromov, Mathematics Genealogy Project.
5. Simons Foundation, Science Lives: Mikhail Gromov (video interview).

Mathematics

1. M. Gromov, Stable mappings of foliations into manifolds, Izv. Akad. Nauk SSSR 33 (1969).
2. M. Gromov, Almost flat manifolds, J. Differential Geom. 13 (1978).
3. M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. IHÉS 53 (1981).
4. M. Gromov, Structures métriques pour les variétés riemanniennes, ed. J. Lafontaine and P. Pansu, Cedic/Fernand Nathan, 1981; English edition Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhäuser, 1999.
5. M. Gromov, H. B. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publ. Math. IHÉS 58 (1983).
6. M. Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983).
7. M. Gromov, V. Milman, A topological application of the isoperimetric inequality, Amer. J. Math. 105 (1983).
8. M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985).
9. M. Gromov, Partial Differential Relations, Springer, 1986.
10. M. Gromov, Hyperbolic groups, in Essays in Group Theory, MSRI Publ. 8, Springer, 1987.
11. M. Gromov, Asymptotic invariants of infinite groups, in Geometric Group Theory vol. 2, LMS Lecture Notes 182, 1993.
12. Y. Eliashberg, N. Mishachev, Introduction to the h-Principle, AMS, 2002.
13. M. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer, 1999.
14. D. McDuff, D. Salamon, J-holomorphic Curves and Symplectic Topology, AMS, 2004.
15. M. Katz, Systolic Geometry and Topology, AMS, 2007.