of M in a case where the area function is not bounded on the components of M (as in the

case of the Heisenberg example above).

However, it is still possible that one might be able to produce such a compacti¬cation

if one can get an area bound on the curves in each component. Gromov™s insight was

that having the area bound was enough to furnish a priori estimates on the derivatives of

curves with an area bound, at least away from a ¬nite number of points.

With all of this in mind, I can now very roughly state Gromov™s Compacti¬cation

Theorem:

Theorem 5: Let M be a compact almost complex manifold with almost complex structure

J and suppose that „¦ tames J . Then every component M± of the moduli space M of

connected unparametrized holomorphic curves in M can be compacti¬ed to a space M±

by adding a set of “cusp” curves, where a cusp curve is essentially a ¬nite union of (possibly)

singular holomorphic curves in M which is obtained as a limit of a sequence of connected

curves in M± by “pinching loops” and “bubbling”.

For the precise de¬nition of “cusp curve” consult [Gr 1], [Wo], or [Pa]. The method

that Gromov uses to prove his compactness theorem is basically a generalization of the

Schwarz Lemma. This allows him to get control of the sup-norm of the ¬rst derivatives of

a holomorphic curve in M terms of the L2 -norm (i.e., area norm) at least in regions where

1

the area form stays bounded.

Unfortunately, although the ideas are intuitively compelling, the actual details are

non-trivial. However, there are, by now, several good sources, from di¬erent viewpoints,

for proofs of Gromov™s Compacti¬cation Theorem. The articles [M 4] and [Wo] listed in

the Bibliography are very readable accounts and are highly recommended. I hear that the

(unpublished) [Pa] is also an excellent account which is closer in spirit to Gromov™s original

ideas of how the proof should go. Finally, there is the quite recent [Ye], which generalizes

this compacti¬cation theorem to the case of curves with boundary.

Actually, the most fruitful applications of these ideas have been in the situation when,

for various reasons, it turns out that there cannot be any cusp curves, so that, by Gromov™s

compactness theorem, the moduli space is already compact. Here is a case where this

happens.

Proposition 1: Suppose that (M, J ) is a compact, almost complex manifold and that

„¦ is a 2-form which tames J . Suppose that there exists a non-constant holomorphic

curve f: S 2 ’ M, and suppose that there is a number A > 0 so that S 2 f — („¦) ≥ A for

all non-constant holomorphic maps f: S 2 ’ M. Then for any B < 2A, the set MB of

unparametrized holomorphic curves f(S 2 ) ‚ M which satisfy S 2 f — („¦) = B is compact.

L.9.10 162

Proof: (Idea) If the space MB were not compact, then a point of the compacti¬cation

would would correspond a union of cusp curves which would contain at least two distinct

non-constant holomorphic maps of S 2 into M. Of course, this would imply that the limiting

value of the integral of „¦ over this curve would be at least 2A > B, a contradiction.

An example of this phenomenon is when the taming form „¦ represents an integral

class in cohomology. Then the presence of any holomorphic rational curves at all implies

that there is a compact moduli space at some level.

Applications. It is reasonable to ask how the Compactness Theorem can be applied

in symplectic geometry.

To do this, what one typically does is ¬rst ¬x a symplectic manifold (M, „¦) and then

considers the space J(„¦) of almost complex structures on M which „¦ tames. We already

know from Lecture 5 that J(„¦) is not empty. We even know that the space K(„¦) ‚ J(„¦)

of „¦-compatible almost complex structures is non-empty. Moreover, it is not hard to show

that these spaces are contractible (see the Exercises).

Thus, any invariant of the almost complex structures J ∈ J(„¦) or of the almost

K¨hler structures J ∈ K(„¦) which is constant under homotopy through such structures is

a

an invariant of the underlying symplectic manifold (M, „¦).

This idea is extremely powerful. Gromov has used it to construct many new invariants

of symplectic manifolds. He has then gone on to use these invariants to detect features of

symplectic manifolds which are not presently accessible by any other means.

Here is a sample of some of the applications of Gromov™s work on holomorphic curves.

Unfortunately, I will not have time to discuss the proofs of any of these results.

(Gromov) If there is a symplectic embedding of B 2n (r) ‚ R2n into

Theorem 6:

B 2 (R) — R2n’2 , then r ¤ R.

One corollary of Theorem 6 is that any di¬eomorphism of a symplectic manifold which

is a C 0-limit of symplectomorphisms is itself a symplectomorphism.

(Gromov) If „¦ is a symplectic structure on CP2 and there exists an

Theorem 7:

embedded „¦-symplectic sphere S ‚ CP2 , then „¦ is equivalent to the standard symplectic

structure.

The next two theorems depend on the notion of asymptotic ¬‚atness: We say that

a non-compact symplectic manifold M 2n is asymptotically ¬‚at if there is a compact set

K1 ‚ M 2n and a compact set K2 ‚ R2n so that M \ K1 is symplectomorphic to R2n \ K2

(with the standard symplectic structure on R2n ).

Theorem 8: (McDuff) Suppose that M 4 is a non-compact symplectic manifold which

is asymptotically ¬‚at. Then M 4 is symplectomorphic to R4 with a ¬nite number of points

blown up.

L.9.11 163

Theorem 9: (McDuff, Floer, Eliashberg) Suppose that M 2n is asymptotically

¬‚at and contains no symplectic 2-spheres. Then M 2n is di¬eomorphic to R2n .

It is not known whether one might replace “di¬eomorphic” with “symplectomorphic”

in this theorem for n > 2.

Epilogue

I hope that this Lecture has intrigued you as to the possibilities of applying the ideas

of Gromov in modern geometry. Let me close by quoting from Gromov™s survey paper on

symplectic geometry in the Proceedings of the 1986 ICM:

Di¬erential forms (of any degree) taming partial di¬erential equations pro-

vide a major (if not the only) source of integro-di¬erential inequalities needed for

a priori estimates and vanishing theorems. These forms are de¬ned on spaces of

jets (of solutions of equations) and they are often (e.g., in Bochner-Weitzenbock

formulas) exact and invariant under pertinent (in¬nitesimal) symmetry groups.

Similarly, convex (in an appropriate sense) functions on spaces of jets are re-

sponsible for the maximum principles. A great part of hard analysis of PDE will

become redundant when the algebraic and geometric structure of taming forms

and corresponding convex functions is clari¬ed. (From the PDE point of view,

symplectic geometry appears as a taming device on the space of 0-jets of solutions

of the Cauchy-Riemann equation.)

L.9.12 164

Exercise Set 9:

The Gromov School of Symplectic Geometry

1. Use the fact that an orientable 3-manifold M 3 is parallelizable (i.e., its tangent bundle

is trivial) and Theorem 1 to show that a compact 3-manifold can always be immersed in

R4 and a 3-manifold with no compact component can always be immersed in R3 .

2. Show that Theorem 2 does, in fact imply that any connected non-compact symplectic

manifold which has an almost complex structure has a symplectic structure. (Hint: Show

that the natural projection Z 1 (X, V ) ’ V has contractible ¬bers (in fact, Z 1(X, V ) is an

a¬ne bundle over V , and then use this to show that a non-degenerate 2-form on X can

be homotoped to a closed non-degenerate 2-form on X.)

3. Show that the hypothesis in Theorem 3 that X either be non-compact or that dim(X) <

dim(Y ) is essential.

4. Show that if E is a symplectic bundle over a compact manifold M 2n whose rank is

2n + 2k for some k > 0, then there exists a symplectic splitting E = F • T where T is

a trivial symplectic bundle over M of rank k. (Hint: Use transversality to pick a non-

vanishing section of E. Now what?)

Show also that, if E is a symplectic bundle over a compact manifold M 2n , then there

exists another symplectic bundle E over M so that E • E is trivial. (Hint: Mimic the

proof for complex bundles.)

Finally, use these results to complete the proof of the Corollary to Theorem 3.

5. Show that if a symplectic manifold M is simply connected, then the symplectic blow

ˆ

up M of M along a symplectic submanifold S of M is also simply connected. (Hint: Any

ˆ ˆ

loop in M can be deformed into a loop which misses S. Now what?)

6. Prove, as stated in the text, that if M is compact and „¦ tames J , then for any

Riemannian metric g on M (not necessarily compatible with either J or „¦) there is a

constant C > 0 so that

|v § J v| ¤ C „¦(v, J v)

where |v§J v| represents the area in Tp M of the parallelogram spanned by v and J v in Tp M.

E.9.1 165

7. First Order Equations and Holomorphic Curves. The point of this problem is

to show how elliptic quasi-linear determined PDE for two functions of two unknowns can

be reformulated as a problem in holomorphic curves in an almost complex manifold.

Suppose that π: V 4 ’ X 2 is a smooth submersion from a 4-manifold onto a 2-manifold.

Suppose also that R ‚ J 1 (X, V ) is smooth submanifold of dimension 6 which has the

property that it locally represents an elliptic, quasi-linear pair of ¬rst order PDE for

sections s of π. Show that there exists a unique almost complex structure on V so that

a section s of π is a solution of R if and only if its graph in V is an (unparametrized)

holomorphic curve in V .

(Hint: The hypotheses on the relation R are equivalent to the following conditions.

For every point v ∈ V , there are coordinates x, y, f, g on a neighborhood of v in V with

the property that x and y are local coordinates on a neighborhood of π(v) and so that a

local section s of the form f = F (x, y), g = G(x, y) is a solution of R if and only if they

satisfy a pair of equations of the form

A1 fx + B1 fy + C1 gx + D1 gy + E1 = 0

A2 fx + B2 fy + C2 gx + D2 gy + E2 = 0

where the A1 , . . . , E2 are speci¬c functions of (x, y, f, g). The ellipticity condition is equiv-

alent to the assumption that

A1 ξ + B1 · C1 ξ + D1 ·

det >0

A2 ξ + B2 · C2 ξ + D2 ·

for all (ξ, ·) = (0, 0).)

Show that the problem of isometrically embedding a metric g of positive Gauss cur-

vature on a surface Σ into R3 can be turned into a problem of ¬nding a holomorphic

section of an almost complex bundle π: V ’ Σ. Do this by showing that the bundle V

whose sections are the quadratic forms which have positive g-trace and which satisfy the

algebraic condition imposed by the Gauss equation on quadratic forms which are second

fundamental forms for isometric embeddings of g is a smooth rank 2 disk bundle over Σ

and that the Codazzi equations then reduce to a pair of elliptic ¬rst order quasi-linear

PDE for sections of this bundle.

Show that, if Σ is topologically S 2, then the topological self-intersection number of

a global section of V is ’4. Conclude, using the fact that distinct holomorphic curves in

V must have positive intersection number, that (up to sign) there cannot be more that

one second fundamental form on Σ which satis¬es both the Gauss and Codazzi equations.

Thus, conclude that a closed surface of positive Gauss curvature in R3 is rigid.

This approach to isometric embedding of surfaces has been extensively studied by

Labourie [La].

E.9.2 166

8. Prove, as claimed in the text that, for the map f» : CP1 ’ CP2 given by the rule

f» (t) = [t, »t2 , »] = [1, x, y],

the pull-back of the Fubini-Study metric accumulates at the points t = 0 and t = ∞ and

goes to zero everywhere else. What would have happened if, instead we had used the map

g» (t) = [t, t2 , »2 ] = [1, x, y]?

Is there a contradiction here?

9. Verify the claim made in the text that, for a symplectic manifold (M, „¦), the spaces

K(„¦) and J(„¦) of „¦-compatible and „¦-tame almost complex structures on M are indeed

contractible. (Hint: Fix an element J0 ∈ K(„¦), with associated inner product , 0 and

show that, for any J ∈ J(„¦), we can write J = J0 (S + A) where S is symmetric and

positive de¬nite with respect to , 0 and A is anti-symmetric. Now what?)

10. The point of this exercise is to get a look at the pseudo-holomorphic curves of a non-

integrable almost complex structure. Let X 4 = C — ∆ = {(w, z) ∈ C2 |z| < 1}, and give

X 4 the almost complex structure for which the complex valued 1-forms ± = dw + z dw and

¯¯

β = dz are a basis for the (1, 0)-forms. Verify that this does indeed de¬ne a non-integrable

almost complex structure on the 4-manifold X. Show that the pseudo-holomorphic curves

in X can be described explicitly as follows: If M is a Riemann surface and φ: M 2 ’ X is

a pseudo-holomorphic mapping, then one of the following is true: Either φ— (β) = 0 and

there exists a holomorphic function h on M and a constant z0 so that

φ = h ’ z0 ¯ z0 ,

¯ h,

or else there exists a non-constant holomorphic function g on M which satis¬es |g| < 1,

a meromorphic function f on M so that f dg and fg dg are holomorphic 1-forms without

periods on M, and a constant w0 so that

p

( f dg ’ fg dg ), g(p)

φ(p) = w0 +

p0

where the integral is taken to be taken over any path from some basepoint p0 to p in M.

(Hint: It is obvious that you must take g = φ— (z), but it is not completely obvious

where f will be found. However, if g is not a constant function, then it will clearly be

holomorphic, now consider the “function”

φ— (±)

f=

1 ’ |g|2 dg

and show that it must be meromorphic, with poles at worst along the zeroes of dg.)

E.9.3 167

Bibliography

[A] V. I. Arnol™d, Mathematical Methods of Classical Mechanics, Second Edition,

Springer-Verlag, Berlin, Heidelberg, New York, 1989.

[AH] M. F. Atiyah and N. Hitchin, The Geometry and Dynamics of Magnetic