LECTURE 5
Sketch Proof of the N onsqueezing Theorem
Suppose that ¢ : B^2 n(l) --+ Z(r) is a symplectic embedding. Its image lies in
some compact subset B^2 (r) x K of Z(r) that can be considered as a subset of the
compact manifold (8^2 x T^2 n-^2 , 0), where f2 is the sum u E9 K.Wo of a symplecticform u on 82 with total area 7rr^2 + E and a suitable multiple of the standard form
wo on r^2 n-^2. Let Jo be the usual almost complex structure on R^2 n and let J be an
n-compatible almost complex structure on 82 x r^2 n-^2 that restricts to ¢*(Jo) on
the image of the ball. (It is easy to construct such J using the methods of proof of
Proposition 3 .9.) As we will see below, the theory of J-holomorphic curves ensures
that there is at least one J-holomorphic curve through each point of 82 x r^2 n-^2
in the class A = [8^2 x pt]. Let C be such a curve through the image ¢(0) of the
origin, and let 8 be the component of the inverse image ¢-^1 (C) that goes through
the origin. Then 8 is a proper^1 J 0 -holomorphic curve in the ball B^2 n(l) through 0.
Since J 0 is the usual complex structure, this means that 8 is a 9 0 -minimal surface
(where 90 is the usual metric on R^2 n.) But it is well-known that the proper surface
of smallest area through the center of a ball of radius 1 is a fl.at disc with area 7r.
Hence
7r :::; 9o-area of 8 = r Wo = 1 f2 < 1 f2 = r f2 = 7rT^2 + E.
Js q,(S) c Js2xpt
Since this is true for all E > 0 we must have r ~ l.
What we have used here from the theory of J-holomorphic curves is the exis-tence of a curve in class A = [8^2 x pt] through an arbitrary point in 82 x r^2 n-^2.
It is easy to check that when J equals the product almost complex structure Jsplit
there is exactly one such curve through every point. For in this case the two pro-
jections are holomorphic so that every Jsplwholomorphic curve in 82 x R^2 n-^2 is
the product of curves in each factor. But the curve in r^2 n -^2 represents the zero
homology class and so must be constant. Now, the basic theory of J-holomorphic
curves is really a deformation theory: if you know that curves exist for one J you
(^1) This means that the intersection of S with any compact subset of the ball is compact. Thus it
goes all the way out t o the boundary.
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