Nonsqueezing

Nonsqueezing, often referred to as Gromov's nonsqueezing theorem, is a fundamental result in symplectic geometry that reveals a rigidity property of symplectic embeddings. It shows that symplectic structure imposes constraints beyond what volume preservation alone would allow, making certain seemingly possible embeddings impossible.

The standard formulation considers the 2n-dimensional Euclidean space with its standard symplectic form ω0 = Σ dx_i ∧ dy_i. The closed ball B^{2n}(R) cannot be symplectically embedded into the cylinder Z^{2n}(r) = B^2(r) × R^{2n-2} if the radius r of the cylindrical factor is smaller than the ball’s radius R. Equivalently, no symplectic embedding exists from B^{2n}(R) into Z^{2n}(r) when r < R. This contrasts with the volume comparison, since the ball and the cylinder can have the same or even larger volume in some dimensions yet still obstruct an embedding due to symplectic constraints.

History and concepts: The theorem was proven by Mikhail Gromov in 1985 using the theory of pseudo-holomorphic curves. It introduced the idea of symplectic capacities—numerical invariants that are monotone under symplectic embeddings and can obstruct embeddings even when volume considerations pass. The most common capacity related to the nonsqueezing statement assigns the value πR^2 to the ball and πr^2 to the cylinder.

Impact: Nonsqueezing is a cornerstone result in symplectic topology, illustrating the intrinsic rigidity of symplectic maps and influencing the study of symplectic embeddings, Hamiltonian dynamics, and related invariants such as Gromov width.