Symplectic capacity and nonsqueezing

In standard R^{2n}, let B^{2n}(r) be the open Euclidean ball of radius r and let Z^{2n}(R) be the set q₁²+p₁²<R² with all other canonical coordinates unrestricted. Gromov’s theorem says a symplectic embedding of the ball into this cylinder exists only if r≤R. Conversely, when r≤R the standard inclusion is an embedding. The rigidity is genuinely higher-dimensional for n≥2; in a single plane the radius restriction is already a consequence of area. The cross-section is a conjugate phase plane, not an arbitrary pair of position coordinates.

The theorem does not prohibit making q₁ alone very small. A canonical rescaling can compress q₁ while expanding p₁ by the reciprocal factor, preserving dq₁∧dp₁. Nor does it prohibit complicated folding that fits a domain whenever no capacity obstruction applies. It specifically rules out the stated symplectic embedding into a narrower canonical cylinder. Replacing a symplectic map by an arbitrary volume-preserving map changes the problem: extra directions can then compensate for simultaneous compression of both members of the first pair.

In the normalization used here, a symplectic capacity assigns a nonnegative extended size to suitable symplectic manifolds or domains. It is monotone under symplectic embeddings, scales by a positive factor a when the form is scaled to aω, and gives π to both the unit ball and unit canonical cylinder. Consequently a radius-r ball and cylinder have capacity πr². This has area units, unlike 2n-dimensional volume, which scales as r^{2n}. Different capacity constructions need not agree on every domain, so the word capacity does not identify one unique scalar formula for all shapes.

The Gromov width is one example: take the supremum of πr² over standard balls that symplectically embed in the domain. Its value on a cylinder is exactly πR² by nonsqueezing and inclusion. Monotonicity then immediately gives the ball-to-cylinder inequality. This reformulation organizes the theorem but does not replace its deep proof: proving the cylinder normalization is precisely where symplectic rigidity enters. Gromov’s original method uses pseudoholomorphic curves; later capacity methods provide other routes. No such analytic proof is supplied by the elementary calculations in this lesson.

If one capacity of a proposed source exceeds the corresponding target capacity, the embedding is impossible. Passing one capacity comparison generally does not establish an embedding for arbitrary shapes; other capacities or geometric obstructions may intervene. For the standard ball-to-cylinder problem, inclusion supplies the converse when the radii allow it. That special converse should not be generalized to all domain pairs. Likewise, the theorem’s two-dimensional size is not simply a minimum Euclidean width: canonical shears and coordinate rotations can change visible widths without changing symplectic capacity.

A useful comparison starts by fixing the symplectic form, canonical pair, domain conventions and radius units. Then calculate a capacity with an established normalization, and separate a proven obstruction from an unresolved comparison. Volume remains an additional invariant under symplectic maps and can obstruct embeddings into finite-volume targets, but it is weaker than the canonical-cylinder obstruction in the example here. This combination of local normal forms and global rigidity is a defining feature of symplectic topology rather than a contradiction between Darboux and Gromov.