NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

42 CASIM ABBAS
are contact forms on W (h) whenever δ>0 is sufficiently small. Pick (x,τ) ∈ W(h),
and let {u, v, w} be a basis of T(x,τ)W (h) with π∗u = π∗v = 0. Then
(λ1 ∧ dλ1)(x,τ)(u, v, w)
= δ 2 (˜α ∧ d ˜α)(x,τ)(u, v, w) − δ [dτ(π∗w) d ˜α(x,τ)(u, v)]
= 0
for sufficiently small δ>0,anddλ1 is a volume form on W . Now we have to continue
the contact forms λ1 beyond ∂W(h) ≈ ∂W × S1 onto ∂W × D2 . At this point it is
convenient to change coordinates. We identify C × S1 with ∂W × (D2 1+ε \D2 1 ), where
D2 ρ is the 2-disk of radius ρ. Using polar coordinates (r, φ) on D2 1+ε with 0 ≤ φ ≤ 2π
and 0 0 and γ ′
1 (r) < 0 if r>0,
hence the curves γ = γδ have to turn counterclockwise in the first quadrant starting
at the point (γ1(0), 0) and later connecting with (−δ(1 − ε0), 1). In the case where
δ>0, the Reeb vector fields are given by
Xδ(θ,r,φ) = γ ′ 2 (r)
µ(r)
∂
∂θ − γ ′ 1 (r)
µ(r)
∂
∂φ ,