Gromov's non-squeezing theorem

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.