LECTURE 1. SYMPLECTIC FIXED POINTS AND MORSE THEORY 157converges to zero uniformly as s -t ±oo.^3 Hence OtU - Xt(u) converges to zero,
uniformly int, as t -t oo. Hence it follows from Exercise 1.22 below that, for every
€ > 0, there exists a T > 0 such that if Isl > T then u(s, t) E UxEP(H) Bc(x(t)).
This implies (8) for some x± E P(H). Thus we have proved that (i) implies (ii).
The proof that (ii) implies (iii) will be deferred to Section 2.7. D
Exercise 1.22. Suppose that every 1-periodic solution x E P(H) is nondegenerate.
Prove that for every € > 0 there exists a /5 > 0 such that, for every smooth loop
y: JR./Z -t M,
fo1
l7i(t) - Xt(y(t))l^2 dt < /5 sup supd(x(t),y(t)) < €.
xEP(H) tHint: Argue by contradiction and use the Arzela-Ascoli theorem to show that
every sequence Yv : JR./Z -t M with
V-->00 lim llYv - Xt(Yv)llL2(s1) =^0has a subsequence which converges uniformly to a periodic solution of (1). D
Exercise 1.23. Let u : JR. x JR./Z -t M be a connecting orbit for the gradient flow
of the symplectic action, i.e. a solution of (7) and (8). Assume exponential decay
as in Proposition 1.21. Prove that the energy of u is given by
E(u) = aH(x-,u-) - aH(x+,u+),
where u ± : B -t M are smooth functions such that u ± ( e^2 7rit) = x± ( t), and u + is
chosen to agree with the connected sum of u-and u, i.e. u+ = u-#u. D
1.6. Moduli spaces
Denote by
M(x-, x+) = M(x-, x+; H, J)
the space of all solutions of (7) and (8). For a generic Hamiltonian Ht these spaces
are finite dimensional manifolds. However, unless T = 0 in (4), the dimension of
M(x-, x+) depends on the component of the moduli space.
Theorem 1.24. There exists a subset Hreg = Hreg(J) c C^00 (M x JR./Z) of the
second category in the sense of Baire (i.e. a countable intersection of open and
dense sets) such that the 1-periodic solutions of (1) are all nondegenerate, and
the moduli space M(x- , x+; H , J) is a finite dimensional smooth manifold for all
x± E P(H) and all HE Hreg·
Moreover, if (4) holds then there is a function T/H : P(H) -t JR. such that for
each u E M(x-, x+; H , J) the dimension of the moduli space is given by
(10) dimu M(x-, x+; H, J) = μ(u; H) = TJH(x-) - TJH(x+) + 2TE(u)
locally near u.
satisfies the previous inequality also satisfies the following mean value inequality
f e < -2..._ ===> e(O) ~ ~ f e + Ar2.
} Br(D) - l6B 'TrT j Br(O) 4Exercise: Assume the last assertion for r = 1, and prove it for general r by rescaling.
3Given E: > 0, with i;;^2 <Ii, choose T > 0 such that E(u, [T-l,oo) x S^1 ) < i;;^2. Then apply (9)
with r =./£to obtain 18 su(s, t)1^2 ~ (c + 8/7r)c for s 2: T.