Gromov nonsqueezing theorem

Gromov nonsqueezing theorem
Name	Gromov nonsqueezing theorem
Field	Symplectic topology
Statement	A ball in a symplectic vector space cannot be symplectically embedded into a thinner cylinder whose base radius is smaller than the ball's radius.
Proven	1985
Author	Mikhail Gromov

Gromov nonsqueezing theorem The Gromov nonsqueezing theorem is a foundational result in symplectic topology stating that a symplectic embedding of a Euclidean ball into a symplectic cylinder exists only when the cylinder's radius is at least the ball's radius. It establishes a rigidity phenomenon contrasting with flexible embeddings in volume-preserving settings, and it has deep ties to theories developed by Mikhail Gromov, Simon Donaldson, Andreas Floer, and others. The theorem influenced subsequent developments linked to Yasha Eliashberg, Clifford Taubes, Edward Witten, and institutions such as IHÉS and MSRI.

Statement

Let B^{2n}(R) denote the closed ball of radius R in Euclidean space R^{2n} equipped with the standard symplectic form ω_0, and let Z^{2n}(r) denote the symplectic cylinder D^2(r) × R^{2n-2} with base disk of radius r. The theorem asserts that there is a symplectic embedding B^{2n}(R) → Z^{2n}(r) if and only if R ≤ r. This precise radius constraint contrasts with embeddings in the category of John Nash-style isometric or Henri Lebesgue-measure-preserving embeddings, and it ties to rigidity phenomena explored by Vladimir Arnold and André Weinstein.

Historical context and significance

Gromov published the theorem in 1985 in a landmark work that catalyzed modern symplectic geometry and symplectic topology. Its proof used techniques from pseudo-holomorphic curve theory inspired by interactions between Gromov and researchers at IHÉS and Steklov Institute of Mathematics. The result answered questions posed by Vladimir Arnold and reshaped perspectives at gatherings such as conferences at Princeton University and University of California, Berkeley where figures like Michael Atiyah, Raoul Bott, and Isadore Singer influenced geometric analysis. The nonsqueezing theorem established symplectic rigidity as a central theme alongside flexibility results such as the h-principle advanced by M. Gromov and influenced later breakthroughs by Dusa McDuff, Leonid Polterovich, and Yakov Eliashberg.

Sketch of proof

Gromov's original argument used the theory of pseudo-holomorphic curves in almost complex manifolds, constructing holomorphic spheres or disks to derive an obstruction to squeezing. The strategy couples a choice of compatible almost complex structure with compactness results akin to those developed by Klaus Mohnke and bubbling analysis reminiscent of work by Andreas Floer and Donaldson techniques. One forms a family of holomorphic curves intersecting a prospective image of the ball; energy and area calculations, along with monotonicity lemmas rooted in ideas from S. S. Chern and Kunihiko Kodaira, yield the radius inequality. Alternative proofs employ tools from symplectic capacity theory developed by Helmut Hofer, Eliashberg–Gromov, and Leonid Polterovich, or combinatorial methods motivated by Seiberg–Witten invariants introduced by Clifford Taubes and Edward Witten.

Applications and consequences

The nonsqueezing theorem underpins the definition and properties of symplectic capacity invariants, which were formalized through work by Helmut Hofer, Leonid Polterovich, Eliashberg, and Yakov Eliashberg. It has implications for Hamiltonian dynamics studied by Viktor Ginzburg, constraints on embedding problems examined by Dusa McDuff, and restrictions on isotopy classes considered by Paul Seidel. In mathematical physics, connections appear with mirror symmetry programs influenced by Maxim Kontsevich and with invariants from Seiberg–Witten theory and Gromov–Witten invariants developed by Edward Witten and Ravi Vakil. The theorem also informs computational aspects pursued at institutions like Princeton University and Courant Institute of Mathematical Sciences.

Extensions and generalizations

Extensions include quantitative non-squeezing results for symplectic embeddings in convex and noncompact settings investigated by Dusa McDuff and Felix Schlenk, and versions for other target manifolds studied by Yasha Eliashberg, Polterovich, and Hofer–Zehnder-type capacities. Higher-dimensional analogues and relative nonsqueezing statements relate to work by Eva Miranda, Patrick Massot, and researchers at ETH Zurich. Generalizations also connect to persistence modules in topological data analysis contexts explored by Gunnar Carlsson and to Floer-theoretic obstructions influenced by Andreas Floer and Paul Seidel.

Examples and counterexamples

A prototypical example is that B^{4}(R) cannot embed into D^2(r) × R^2 unless R ≤ r, while volume-preserving embeddings exist for R arbitrarily large when only Lebesgue measure is preserved, illustrating the contrast between symplectic and volumetric categories noted by John von Neumann in operator theory. Counterexamples to naive generalizations appear when replacing the standard symplectic form with degenerate two-forms studied by Jean-Marie Souriau or when considering topological embeddings without symplectic compatibility, as examined by Gromov and Eliashberg.

Related concepts in symplectic topology

Close relatives include symplectic capacity, pseudo-holomorphic curve, Hamiltonian diffeomorphism group, Weinstein conjecture, Floer homology, Gromov–Witten invariants, and the h-principle. The theorem intersects with ideas in contact topology developed by Yakov Eliashberg and Paul Seidel, and with global analysis topics advanced by Michael Atiyah and Isadore Singer.

Category:Symplectic topology