NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

32 CASIM ABBAS
that
Here the energy E(ũ) is defined by
⎧
⎪⎨
π ◦ Tu◦ j = J ◦ π ◦ Tu on ˙S
(u
⎪⎩
∗λ) ◦ j = da + γ on ˙S
dγ = d(γ ◦ j) = 0 on S
E(ũ) < ∞.
E(ũ) = sup
ϕ∈
˙S
ũ ∗ d(ϕλ),
(1.1)
where consists of all smooth maps ϕ : R → [0, 1] with ϕ ′ (s) ≥ 0 for all s ∈ R.
Note that equation (1.1) reduces to the usual pseudoholomorphic curve equation
in the symplectization R × M if we set γ = 0. The following proposition, which is a
modification of a result by Hofer [17], shows that solutions to problem (1.1) approach
cylinders over periodic orbits of the Reeb vector field.
PROPOSITION 1.5
Let (M,λ) be a closed 3-dimensional manifold equipped with a contact form λ.Then
the associated Reeb vector field has periodic orbits if and only if the associated PDE
problem (1.1) has a nonconstant solution.
Proof
Let (S,j,Ɣ,ũ, γ ) be a nonconstant solution of (1.1). If Ɣ = ∅, then the results in [17]
imply that, near a puncture, the solution is asymptotic to a periodic orbit (see also [3]
for a complete proof). Here we use the fact that γ is exact near the punctures. The
aim now is to show that, in the absence of punctures, the map a is constant while the
image of u lies on a periodic Reeb orbit. Assume that Ɣ =∅.Since
we find, after applying d, that
u ∗ λ =−da ◦ j − γ ◦ j,
ja =−d(da ◦ j) = u ∗ dλ.
In view of the equation π ◦ Tu◦ j = J ◦ π ◦ Tu, we see that u ∗ dλ is a nonnegative
integrand. Applying Stokes’s theorem, we obtain
S u∗ dλ = 0, implying that
π ◦ Tu≡ 0.