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**Problèmes elliptiques, surfaces de Riemann et structures symplectiques [d’après M. Gromov]. (Elliptic problems, Riemann surfaces and symplectic structures (according to M. Gromov)).**
*(French)*
Zbl 0644.53031

Sémin. Bourbaki, 38ème année, Vol. 1985/86, Exp. No. 657, Astérisque 145/146, 111-136 (1987).

[For the entire collection see Zbl 0601.00002.]

The paper is a review of results of M. Gromov from the 70’s and 80’s in global and integral symplectic geometry. This is a very advanced domain in symplectic geometry which is closely connected with algebraic geometry, modern functional analysis, and a very special consideration in topology and geometry of manifolds.

The first series of the reviewed results refer to stability of symplectic and contact structures. It includes the Eliashberg-Gromov theorem stating that the space of symplectomorphisms of any manifolds is \(C^ 0\)-closed and the results of Gromov-Lees about the homotopic equivalence of some Lagrangian immersions. Very nice are the so-called topological uncertainty relations of Gromov which preclude e.g. any symplectic embedding of Euclidean balls in some products of two balls such that one of them has a small radius. They imply the non-existence of any symplectomorphism between a ball and a half-ball having the same volume.

The main part of the paper is devoted to positive elliptic relations which define some submanifolds of total spaces of bundles of Grassmannians over symplectic manifolds. Their general definition and many results about them are due to Gromov. For example he observed a special property of such submanifolds if the corresponding symplectic manifold is \(S^ 2\times S^ 2\). Some of Gromov’s results may be briefly formulated similar to the one that a Hamiltonian isotropy of a symplectic manifold admits at least two fixed points. It is a generalization of the classical Poincaré problem about fixed points of maps of rings. At the end of the paper the author outlines the idea of the Gromov construction of exotic symplectic and contact structures in \(R^{2n}\) and a special result for \(R^ 4\).

The paper is a review of results of M. Gromov from the 70’s and 80’s in global and integral symplectic geometry. This is a very advanced domain in symplectic geometry which is closely connected with algebraic geometry, modern functional analysis, and a very special consideration in topology and geometry of manifolds.

The first series of the reviewed results refer to stability of symplectic and contact structures. It includes the Eliashberg-Gromov theorem stating that the space of symplectomorphisms of any manifolds is \(C^ 0\)-closed and the results of Gromov-Lees about the homotopic equivalence of some Lagrangian immersions. Very nice are the so-called topological uncertainty relations of Gromov which preclude e.g. any symplectic embedding of Euclidean balls in some products of two balls such that one of them has a small radius. They imply the non-existence of any symplectomorphism between a ball and a half-ball having the same volume.

The main part of the paper is devoted to positive elliptic relations which define some submanifolds of total spaces of bundles of Grassmannians over symplectic manifolds. Their general definition and many results about them are due to Gromov. For example he observed a special property of such submanifolds if the corresponding symplectic manifold is \(S^ 2\times S^ 2\). Some of Gromov’s results may be briefly formulated similar to the one that a Hamiltonian isotropy of a symplectic manifold admits at least two fixed points. It is a generalization of the classical Poincaré problem about fixed points of maps of rings. At the end of the paper the author outlines the idea of the Gromov construction of exotic symplectic and contact structures in \(R^{2n}\) and a special result for \(R^ 4\).

Reviewer: J.Czyz

### MSC:

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |

35J99 | Elliptic equations and elliptic systems |