LECTURE 4
The N onsqueezing Theorem and Capacities
In Lecture 2, I explained various results that showed how flexible symplecto-
morphisms are and how little local structure a symplectic manifold has. Now I
want to show you the phenomenon of symplectic rigidity that is encapsulated in
Gromov's nonsqueezing theorem [G]. We will consider the cylinderZ(r) = B^2 (r) X R^2 n -^2 = {(x, y) E R^2 n: xi+ y~:::; r^2 }
with the restriction of the usual symplectic form w 0.
Theorem 4.1 (Gromov). If there is a symplectomorphism that maps the unit ballB^2 n(l) in (R^2 n,w 0 ) into the cylinder Z(r), then r ~ l.
This deceptively simple result is, as we shall see, enough to characterise sym-plectomorphisms among all diffeomorphisms. It clearly shows that symplectomor-
phisms are different from volume-preserving diffeomorphisms since it is easy to
construct a volume-preserving diffeomorphism that squeezes the unit ball into an
arbitrarily thin cylinder. We will begin discussing the proof at the end of this
lecture. For now, let's look at its implications.
The clearest way to understand the force of Theorem 4.1 is to use the Ekeland-
Hofer idea of capacity. A symplectic capacity is a function c that assigns an element
in [O, oo] to each symplectic manifold of dimension 2n and satisfies the following
axioms:
(i) (monotonicity) if there is a symplectic embedding¢ : (U,w) ~ (U',w') then
c(U,w):::; c(U',w').
(ii) (conformal invariance) c(U,>.w) = >.^2 c(U,w).
(iii) ( nontriviality)0 < c(B^2 n(l),wo) = c(Z(l),wo) < oo.
It is the last property c(Z(l),w 0 ) < oo that implies that capacity is an essen-
tially 2-dimensional invariant, for example that it cannot be a power of the total
volume. Sometimes one considers capacities that satisfy a less stringent version of
(iii): namely
(iii')
0 < c(B^2 n(l),wo), c(Z(l),wo) < oo.
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