Gromov's Non-Squeezing Theorem

Theorem6.2Gromov Non-Squeezing

If there exists a symplectic embedding $B^{2n}(r) \hookrightarrow B^2(R) \times \mathbb{R}^{2n-2}$ of the ball of radius $r$ in $\mathbb{R}^{2n}$ into the cylinder of radius $R$ , then $r \leq R$ .

This theorem, proved by Gromov in 1985, demonstrates that symplectic geometry exhibits rigidity beyond volume preservation. While volume-preserving maps can squeeze a ball into an arbitrarily thin cylinder, symplectic maps cannot. This result launched the modern era of symplectic topology.

Proof

Gromov's proof uses pseudo-holomorphic curves. Suppose $\phi: B^{2n}(r) \hookrightarrow B^2(R) \times \mathbb{R}^{2n-2}$ is a symplectic embedding. Consider the compactification to $S^2(R) \times T^{2n-2}$ and an almost complex structure $J$ tamed by $\omega$ agreeing with $\phi_*(J_0)$ on $\phi(B(r))$ .

By Gromov's compactness theorem, there exists a $J$ -holomorphic sphere $u: S^2 \to S^2(R) \times T^{2n-2}$ through any point $p \in \phi(B(r))$ representing the class $[S^2 \times \{pt\}]$ . The area of $u$ is $\pi R^2$ (by the homology class). But $u$ must pass through the image of the ball, and by the monotonicity lemma for holomorphic curves, the area within the ball is at least $\pi r^2$ . Thus $\pi r^2 \leq \pi R^2$ , giving $r \leq R$ . $\square$

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RemarkSymplectic Capacities

The non-squeezing theorem motivates the definition of symplectic capacities: numerical invariants $c(U)$ for open subsets of symplectic manifolds satisfying monotonicity, conformality, and a normalization. The Gromov width $c_G(U) = \sup\{\pi r^2 : B^{2n}(r) \hookrightarrow U\}$ is the prototypical example. Capacities provide obstructions to symplectic embeddings beyond volume.