1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

156 D. SALAMON, FLOER HOMOLOGY

The construction of the Floer homology groups relies on a careful analysis of
the gradient flow lines of the symplectic action, i.e. of the solutions of (7). The
energy of such a solution is defined by

We shall only consider solutions with finite energy. The key observation is that a
solution u of (7) has finite energy if and only if it converges to periodic solutions
of (1) as s __, ±oo, provided that all periodic solutions are nondegenerate (see
Figure 5).

x

u

Figure 5. A gradient flow line of the symplectic action

Proposition 1.21. Let u : IR x IR/Z __, M be a solution of (7). Then the following

are equivalent.

(8)

(i) E(u) < oo.

(ii) There exist periodic solutions x± E P(H) such that

lim u(s, t) = x±(t).

s-+±oo

and lims-+±oo 05 u(s, t) = 0, where both limits are uniform in the t-variable.

(iii) There exist constants 8 > 0 and c > 0 such that

l8su(s, t)i:::; ce-oJsJ

for alls, t E IR.

Proof. That (iii) implies (i) is obvious. We prove that (i) implies (ii). The proof

relies on the a-priori estimate

(9)

for solutions of (1). Here n > 0 and c > 0 are constants independent of u. A

proof of this estimate can be found in [45].^2 If E(u) < oo then (9) shows that o 5 u

(^2) The proof of (9) is based o n an inequality of the form
~e ~-A-Be^2
for the energy density e(s,t) = 1a.u(s,t)l2, where~= &^2 /&s^2 + &^2 /&t^2 denotes the standard
Laplacian. This inequality holds for all solut'.ons of (7) with constants A > 0 and B > 0 depending

on w, J , and H. It is worth pointing out that H = 0 =* A = 0. Now every function e which