Geometry of contact transformations and domains: orderability versus squeezing

Gromov’s famous non-squeezing theorem (1985) states that the standard symplectic ball cannot be symplectically squeezed into any cylinder of smaller radius. Does there exist an analogue of this result in contact geometry? Our main finding is that the answer depends on the sizes of the domains in question: We establish contact non-squeezing on large scales, and show that it disappears on small scales. The algebraic counterpart of the (non)-squeezing problem for contact domains is the question of existence of a natural partial order on the universal cover of the contactomorphisms group of a contact manifold. In contrast to our earlier beliefs, we show that the answer to this question is very sensitive to the topology of the manifold. For instance, we prove that the standard contact sphere is non-orderable while the real projective space is known to be orderable. Our methods include a new embedding technique in contact geometry as well as a generalized Floer homology theory which contains both cylindrical contact homology and Hamiltonian Floer homology. We discuss links to a number of miscellaneous topics such as topology of free loops spaces, quantum mechanics and semigroups.

Dedicated to Dusa McDuff on the occasion of her 60th birthday

Keywords

contact manifolds, contact squeezing and orderability, Floer homology, holomorphic curves

Mathematical Subject Classification 2000

Primary: 53D10, 53D40

Secondary: 53D35, 53D50

References

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Publication

Received: 12 February 2006

Accepted: 4 September 2006

Published: 28 October 2006

Proposed: Eleny Ionel

Seconded: Tomasz Mrowka, Peter Ozsváth

Correction: 1 February 2009

Authors

Yakov Eliashberg
Department of Mathematics Stanford University Stanford, CA 94305-2125 USA

Sang Seon Kim
Departamento de Matemática Instituto Superior Técnico Av Roviso Pais 1049-001 Lisboa Portugal

Leonid Polterovich
School of Mathematical Sciences The Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University 69978 Tel Aviv Israel

Volume 10, issue 3 (2006)

Yakov Eliashberg, Sang Seon Kim and Leonid Polterovich

Abstract