Gromov's style of geometry often features a "coarse" or "soft" viewpoint, analyzing asymptotic or large-scale properties. He is also interested in
mathematical biology, the structure of the brain and the thinking process, and the way scientific ideas evolve. His well-known book
Partial Differential Relations collects most of his work on these problems. Later, he applied his methods to
complex geometry, proving certain instances of the
Oka principle on deformation of
continuous maps to
holomorphic maps. His work initiated a renewed study of the Oka–Grauert theory, which had been introduced in the 1950s. Gromov and
Vitali Milman gave a formulation of the
concentration of measure phenomena. They defined a "Lévy family" as a sequence of normalized metric measure spaces in which any asymptotically nonvanishing sequence of sets can be metrically thickened to include almost every point. This closely mimics the phenomena of the
law of large numbers, and in fact the law of large numbers can be put into the framework of Lévy families. Gromov and Milman developed the basic theory of Lévy families and identified a number of examples, most importantly coming from sequences of
Riemannian manifolds in which the lower bound of the
Ricci curvature or the first eigenvalue of the
Laplace–Beltrami operator diverge to infinity. They also highlighted a feature of Lévy families in which any sequence of continuous functions must be asymptotically almost constant. These considerations have been taken further by other authors, such as
Michel Talagrand. Since the seminal 1964 publication of
James Eells and
Joseph Sampson on
harmonic maps, various rigidity phenomena had been deduced from the combination of an existence theorem for harmonic mappings together with a vanishing theorem asserting that (certain) harmonic mappings must be totally geodesic or holomorphic. Gromov had the insight that the extension of this program to the setting of mappings into
metric spaces would imply new results on
discrete groups, following
Margulis superrigidity.
Richard Schoen carried out the analytical work to extend the harmonic map theory to the metric space setting; this was subsequently done more systematically by Nicholas Korevaar and Schoen, establishing extensions of most of the standard
Sobolev space theory. A sample application of Gromov and Schoen's methods is the fact that
lattices in the isometry group of the
quaternionic hyperbolic space are
arithmetic.
Riemannian geometry In 1978, Gromov introduced the notion of
almost flat manifolds. The famous
quarter-pinched sphere theorem in
Riemannian geometry says that if a complete Riemannian manifold has
sectional curvatures which are all sufficiently close to a given positive constant, then must be finitely covered by a sphere. In contrast, it can be seen by scaling that every
closed Riemannian manifold has Riemannian metrics whose sectional curvatures are arbitrarily close to zero. Gromov showed that if the scaling possibility is broken by only considering Riemannian manifolds of a fixed diameter, then a closed manifold admitting such a Riemannian metric, with sectional curvatures sufficiently close to zero, must be finitely covered by a
nilmanifold. The proof works by replaying the proofs of the
Bieberbach theorem and
Margulis lemma. Gromov's proof was given a careful exposition by
Peter Buser and Hermann Karcher. In 1979,
Richard Schoen and
Shing-Tung Yau showed that the class of
smooth manifolds which admit Riemannian metrics of positive
scalar curvature is topologically rich. In particular, they showed that this class is closed under the operation of
connected sum and of
surgery in codimension at least three. Their proof used elementary methods of
partial differential equations, in particular to do with the
Green's function. Gromov and
Blaine Lawson gave another proof of Schoen and Yau's results, making use of elementary geometric constructions. They also showed how purely topological results such as
Stephen Smale's
h-cobordism theorem could then be applied to draw conclusions such as the fact that every
closed and
simply-connected smooth manifold of dimension 5, 6, or 7 has a Riemannian metric of positive scalar curvature. They further introduced the new class of
enlargeable manifolds, distinguished by a condition in
homotopy theory. They showed that Riemannian metrics of positive scalar curvature
cannot exist on such manifolds. A particular consequence is that the
torus cannot support any Riemannian metric of positive scalar curvature, which had been a major conjecture previously resolved by Schoen and Yau in low dimensions. In 1981, Gromov identified topological restrictions, based upon
Betti numbers, on manifolds which admit Riemannian metrics of
nonnegative sectional curvature. The principal idea of his work was to combine
Karsten Grove and Katsuhiro Shiohama's Morse theory for the Riemannian distance function, with control of the distance function obtained from the
Toponogov comparison theorem, together with the
Bishop–Gromov inequality on volume of geodesic balls. This resulted in topologically controlled covers of the manifold by geodesic balls, to which
spectral sequence arguments could be applied to control the topology of the underlying manifold. The topology of lower bounds on sectional curvature is still not fully understood, and Gromov's work remains as a primary result. As an application of
Hodge theory,
Peter Li and Yau were able to apply their gradient estimates to find similar Betti number estimates which are weaker than Gromov's but allow the manifold to have convex boundary. In
Jeff Cheeger's fundamental compactness theory for Riemannian manifolds, a key step in constructing coordinates on the limiting space is an
injectivity radius estimate for
closed manifolds. Cheeger, Gromov, and
Michael Taylor localized Cheeger's estimate, showing how to use
Bishop−Gromov volume comparison to control the injectivity radius in absolute terms by curvature bounds and volumes of geodesic balls. Their estimate has been used in a number of places where the construction of coordinates is an important problem. A particularly well-known instance of this is to show that
Grigori Perelman's "noncollapsing theorem" for
Ricci flow, which controls volume, is sufficient to allow applications of
Richard Hamilton's compactness theory. Cheeger, Gromov, and Taylor applied their injectivity radius estimate to prove
Gaussian control of the
heat kernel, although these estimates were later improved by Li and Yau as an application of their gradient estimates.
Gromov−Hausdorff convergence and geometric group theory In 1981, Gromov introduced the
Gromov–Hausdorff metric, which endows the set of all
metric spaces with the structure of a metric space. More generally, one can define the Gromov-Hausdorff distance between two metric spaces, relative to the choice of a point in each space. Although this does not give a metric on the space of all metric spaces, it is sufficient in order to define "Gromov-Hausdorff convergence" of a sequence of pointed metric spaces to a limit. Gromov formulated an important compactness theorem in this setting, giving a condition under which a sequence of pointed and "proper" metric spaces must have a subsequence which converges. This was later reformulated by Gromov and others into the more flexible notion of an
ultralimit. Gromov's compactness theorem had a deep impact on the field of
geometric group theory. He applied it to understand the asymptotic geometry of the
word metric of a
group of polynomial growth, by taking the limit of well-chosen rescalings of the metric. By tracking the limits of isometries of the word metric, he was able to show that the limiting metric space has unexpected continuities, and in particular that its isometry group is a
Lie group. As a consequence he was able to settle the
Milnor-Wolf conjecture as posed in the 1960s, which asserts that any such group is
virtually nilpotent. Using ultralimits, similar asymptotic structures can be studied for more general metric spaces. Important developments on this topic were given by
Bruce Kleiner, Bernhard Leeb, and
Pierre Pansu, among others. Another consequence is
Gromov's compactness theorem, stating that the set of compact
Riemannian manifolds with
Ricci curvature ≥
c and
diameter ≤
D is
relatively compact in the Gromov–Hausdorff metric. The possible limit points of sequences of such manifolds are
Alexandrov spaces of curvature ≥
c, a class of
metric spaces studied in detail by
Burago, Gromov and
Perelman in 1992. Along with
Eliyahu Rips, Gromov introduced the notion of
hyperbolic groups.
Symplectic geometry Gromov's theory of
pseudoholomorphic curves is one of the foundations of the modern study of
symplectic geometry. Although he was not the first to consider pseudo-holomorphic curves, he uncovered a "bubbling" phenomena paralleling
Karen Uhlenbeck's earlier work on
Yang–Mills connections, and Uhlenbeck and Jonathan Sack's work on
harmonic maps. In the time since Sacks, Uhlenbeck, and Gromov's work, such bubbling phenomena has been found in a number of other geometric contexts. The corresponding
compactness theorem encoding the bubbling allowed Gromov to arrive at a number of analytically deep conclusions on existence of pseudo-holomorphic curves. A particularly famous result of Gromov's, arrived at as a consequence of the existence theory and the monotonicity formula for
minimal surfaces, is the "
non-squeezing theorem," which provided a striking qualitative feature of symplectic geometry. Following ideas of
Edward Witten, Gromov's work is also fundamental for
Gromov-Witten theory, which is a widely studied topic reaching into
string theory,
algebraic geometry, and
symplectic geometry. From a different perspective, Gromov's work was also inspirational for much of
Andreas Floer's work.
Yakov Eliashberg and Gromov developed some of the basic theory for symplectic notions of convexity. They introduce various specific notions of convexity, all of which are concerned with the existence of one-parameter families of diffeomorphisms which contract the symplectic form. They show that convexity is an appropriate context for an
h-principle to hold for the problem of constructing certain
symplectomorphisms. They also introduced analogous notions in
contact geometry; the existence of convex contact structures was later studied by
Emmanuel Giroux. ==Prizes and honors==