Symplectic capacity ($c(X,\omega )$)

A symplectic capacity is monotone under symplectic embeddings, scales by $c(X,a\omega )=ac(X,\omega )$ for $a>0$, and has positive unit-ball capacity and finite unit-cylinder capacity. It is invariant under symplectomorphisms. A normalized symplectic capacity additionally assigns $\pi$ to both unit domains.

A normalized symplectic capacity has the displayed unit-ball and unit-cylinder values. Dilation gives $\pi R^2$ for both radius-$R$ domains. Monotonicity then implies the sharp Gromov non-squeezing theorem.

A locally uniform limit of symplectomorphisms that is a smooth diffeomorphism is symplectic. By capacity preservation under uniform limits and linear capacity rigidity, its derivative is symplectic or anti-symplectic. Repeat this argument after taking the product of every map with the identity on a symplectic plane. An anti-symplectic derivative on the original factor and a symplectic derivative on the added factor would give neither sign on the product, a contradiction. The argument uses symplectic capacities in the original and stabilized dimensions.

An invertible real linear map preserving symplectic capacities of all solid ellipsoids is symplectic or anti-symplectic. For a pair with $\omega (u,v)=1$, if $0<|\omega (T^{T}u,T^{T}v)|<1$, complete this pair and its normalized transformed pair to symplectic bases. The resulting transpose matrix sends a unit ball into a symplectic cylinder of radius less than one, contradicting the capacity axioms after iterating the resulting linear map: its first coordinate pair contracts geometrically, while the image ellipsoid retains positive capacity. Perturbation handles a zero pairing; using $T^{-1}$ handles absolute value greater than one. Thus the absolute bilinear pairings agree. Their squares agree as polynomials, and the polynomial ring being an integral domain forces one global sign.

Suppose diffeomorphisms preserving capacities of solid ellipsoids converge uniformly on compact sets to a diffeomorphism $\varphi$. For $0<a<1$, uniform closeness and degree of a continuous mapping give ${\varphi }_j(aE)\subseteq \varphi (E)\subseteq {\varphi }_j(a^{-1}E)$. The capacity axioms squeeze $c(\varphi (E))$ between $a^{2}c(E)$ and $a^{-2}c(E)$. Letting $a$ tend to one proves equality. Rescaling a smooth $\varphi$ near a point and repeating the argument also proves capacity preservation by its derivative.