Non-squeezing theorem

The non-squeezing theorem states that a symplectic ball of radius R cannot be symplectically embedded into a cylinder whose two-dimensional cross-section has radius r<R, regardless of the cylinder’s extent in other dimensions.

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Symplectic capacity: A numerical invariant that measures the symplectic size of a subset of a symplectic manifold. Capacity makes the theorem’s restriction precise: the ball’s capacity exceeds the narrower cylinder’s.
Symplectic manifold: A smooth even-dimensional manifold equipped with a closed, nondegenerate two-form. The theorem concerns embeddings between regions of this geometric setting.
Mikhail Gromov: A mathematician known for foundational work in geometry, topology, and the theory of pseudoholomorphic curves. He proved non-squeezing in 1985 using pseudoholomorphic curves.
Darboux theorem: A theorem stating that every symplectic form has standard local coordinates near each point. Local standardization does not imply that large symplectic regions can be globally squeezed.
Symplectic camel theorem: A result describing restrictions on moving symplectic domains through a narrow passage in phase space. It extends non-squeezing’s geometric obstruction to a passage-shaped setting.
Gromov width: The supremum of radii of symplectic balls that can be embedded in a symplectic manifold. The theorem says a cylinder’s Gromov width is set by its narrow two-dimensional cross-section.
Symplectic ball: A Euclidean ball in an even-dimensional space equipped with its standard symplectic form. It is the object whose size cannot be squeezed below the cylinder’s cross-sectional capacity.
Pseudoholomorphic curve: A surface mapped into an almost complex manifold so that its differential satisfies a complex-linearity condition. Gromov’s introduction of these curves made the theorem’s proof possible.
Liouville's theorem (Hamiltonian mechanics): A theorem stating that Hamiltonian flow preserves phase-space volume. Volume preservation is weaker: it cannot by itself forbid the embeddings ruled out here.
Ekeland–Hofer capacity: A symplectic capacity defined through variational methods for Hamiltonian systems. Capacities provide further quantitative versions of the kind of obstruction non-squeezing reveals.