1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 1. SYMPLECTIC FIXED POINTS AND MORSE THEORY 155

Exercise 1.19. Prove that the I-form iI! His closed. Hint: Consider a 2-parameter

family ofloops IR^2 --+ £M: (s1, s2)--+ Xs 1 ,s 2 and denote 6 = 8x/8s1, 6 = 8x/8s2.

Then diI! H (x; 6, 6) = Os, iI! H (x; Os 2 X) - Os 2 iI! H (x; Os, x). D

The I-form WH is not exact. However, it is the differential of a circle valued

function aH : £M --+ IR/Z. This function is defined by

aH(x) = -l u*w -1

1

Ht(x(t)) dt

for x E £M, where u : B = {z EC : lzl:::; I} --+ M is a smooth map such that

u(e^2 7rit) = x(t) fort ER Such maps u exist whenever xis a contractible loop. The

assumption J 52 v*w E Z for every smooth map v : S^2 --+ M guarantees that aH

takes values in IR/Z. Sometimes we shall denote by aH(x, u) the symplectic action

of the pair (x, u) which is well defined as a real number.

Exercise 1.20. Prove that the differential of aH is the I-form iI! H on £M. Hint:

Consider a path IR--+ £M: sf-+ Xs where Xs(t) = xo ass:::; -I and define Us by

Us(e^2 7r(r+it)) = Xs+r(t) for r:::; 0 and t ER 0

Floer's idea is to carry out Morse theory for the symplectic action functional

in analogy to the Morse-Smale-Witten complex in finite dimensional Morse theory.

1.5. Connecting orbits

Let us now fix a time dependent Hamiltonian Ht = Ht+ 1 : M --+ IR such that

the I-periodic solutions x : IR/Z --+ M of (I) are all nondegenerate. We wish to

study the gradient flow lines of the action functional aH : £M --+ IR/Z. For this

we must choose a metric on the loop space. Such a metric can be obtained from

a I-periodic family of almost complex structures lt = lt+l E :J(M,w) with cor-

responding metrics (~, rJ)t = w(~, l trJ). The resulting inner product on the tangent

space Tx£M = C^00 (IR/Z,x*TM) is given by

(~, 'r/) = 1

1

(~(t), rJ(t))t dt.

Since the differential of aH is the I-form iI! H it follows that the gradient of aH with

respect to this metric is given by

gradaH(x)(t) = lt(x(t))±(t) - \?Ht(x(t))

where the gradient of Ht is taken with respect to the metric(., ·)ton M. A gradient

flow line of aH is a smooth I-parameter family ofloops IR--+ £M: sf-+ u(s, ·)which

satisfies 8u/8s +grad aH(u(s, · )) = 0. In view of the above formula for grad aH this
becomes the partial differential equation
au au
(7)
08

+ lt(u) at - \1 Ht(u) = 0

for smooth maps u: IR^2 --+ M which satisfy the periodicity condition u(s, t +I) =

u(s, t). Note that, in the case where J , H , and u are independent oft, this is the

upward gradient flow of H = Ht. In the case where u(s, t) = x(t) is independent

of s, this reduces to the Hamiltonian equations (I), and in the case Ht = 0 and

lt = J this is the equation for J-holomorphic curves.