Differential geometry Symplectic geometry Theorem

Gromov's non-squeezing theorem

Builds on Symplectic manifold Pseudoholomorphic curve

Statement

Let $n \ge 1$ and give $\R^{2n}$ the coordinates $(x_1, y_1, \ldots, x_n, y_n)$ and the standard symplectic form $\omega_0 = \sum_{i=1}^n dx_i \wedge dy_i$ . For $r > 0$ , let $B^{2n}(r)$ be the open ball of radius $r$ centered at $0$ . For $R > 0$ , let $Z^{2n}(R) = \set{(x, y) \in \R^{2n} : x_1^2 + y_1^2 < R^2}$ be the open cylinder of radius $R$ over a disc in the $(x_1, y_1)$ -plane. If there is a symplectic embedding $\varphi \colon B^{2n}(r) \to Z^{2n}(R)$ , that is, a smooth embedding with $\varphi^* \omega_0 = \omega_0$ , then $r \le R$ .

Remarks

Gromov proved the theorem in 1985 with pseudoholomorphic curves. Assume $r > R$ and an embedding exists; compactify the cylinder to $S^2 \times T^{2n-2}$ with $S^2$ of area slightly more than $\pi R^2$ . For a suitable almost complex structure, there is a pseudoholomorphic sphere in the class of $S^2$ through the image of the center of the ball. Its area is less than $\pi R^2 + \varepsilon$ . Pulled back to the ball, it gives a proper minimal surface through the center with area less than $\pi r^2$ . This contradicts the monotonicity formula, which says that such a surface has area at least $\pi r^2$ .

The hypothesis that the disc lies in a symplectic plane is needed. The map $(x_1, y_1, x_2, y_2) \mapsto (\lambda x_1, \lambda^{-1} y_1, \lambda x_2, \lambda^{-1} y_2)$ with small $\lambda > 0$ is a linear symplectomorphism that squeezes any ball into the cylinder $\set{x_1^2 + x_2^2 < R^2}$ over the $(x_1, x_2)$ -plane, which is Lagrangian. For $n = 1$ the theorem is just the statement that area-preserving maps preserve area.

Going deeper

For a student

The theorem marks the birth of symplectic topology: it shows that symplectic maps are not just volume-preserving maps. It leads to the notion of a symplectic capacity, a size of a symplectic set that is invariant under symplectic embeddings. By Liouville's theorem, Hamiltonian flows preserve volume; non-squeezing gives a finer constraint on them, a classical analogue of the uncertainty principle.

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Pseudoholomorphic curve concept
Symplectic manifold concept
Gromov's non-squeezing theorem statement

Tag	`RDX55HQ3XT`
URI	`m2://RDX55HQ3XT`
Kind	statement, theorem
Writing	W0 stub
MSC	53D35, 57R17
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About	Symplectic manifold
Uses	Symplectic manifold, Pseudoholomorphic curve
Same as	Wikidata Q3527214 Wikipedia Non-squeezing theorem