LECTURE 4. THE WEINSTEIN CONJECTURE IN THE TIGHT CASE 93andv(8A) c 8D x JR if 8A = 8D n A=/. 0.
Moreover• v(A) n Do =f. 0
• v(A) n DT = 0 for T E (-c, 0)
- v satisfies the assumptions (34) and (35).
Then Do IA:= u(O, A)= v(A).
. We apply the previous result to the following situation. Assume (W, J) is an
almost complex four-manifold. Assume M c W is a three-dimensional orientable
submanifold of w dividing w into two disjoint parts, which we denote by w±,
i.e. M = w-n w+ and W = Mu w+ u w-. Assume that M is pseudoconvex
with respect to w-. Let F C M be a smooth two dimensional submanifold. We
consider the following boundary value problem:
w: D-+ w-UM(40) W 5 + J(w)wt = 0 on D
andw(8D) c F.
The following theorem holds:
Theorem 4.2. Let (W, J), M and F be as described above, where M is pseudo-
convex with respect to w-. Let u and v be two embedded solutions of equation
(40) with image in w-UM. Assume that the index in the normal form is k = 0,
v(D) n w-=/. 0 , u(D) n w-=/. 0 and u(D) n v(D) =/. 0. Denote by (DT )TE(-c:,c:)the local disk family near Do = u(D). If v(D) n DT = 0 for all T < 0 near 0 then
u(D) = v(D).
Proof. First note that we have v(D\8D) C w-because of the fact that M is
pseudo-convex as seen from w-and the assumption v(D) n w-=f. 0. It follows
that v(D\8D) n F = 0. In order to prove the theorem we have to verify the crucial
assumptions (34) and (35). They were formulated for the model equation in C^2 for
a proper choice of coordinates. We study the equation
u: D---+ C^2
(41) Us+ J(u)ut = 0 on D
u(8D) c 8D x JR.
Here J(z, 0) = i for all lz l :::; l. Moreover uo(z ) = (z, 0) is a solution. The solutions
of ( 41) near uo define a disk family DT. We assume that moreover an embedding
v : A -+ C^2 is given with A C D and v(8A) C 8D x JR. Moreover v(A) and
Do= D x {O} intersect at a boundary point. Since u 0 and v come from maps with
image in w-UM we have v(A \ 8A) n F = 0. This is the crucial assumption (34).
Next consider an embedded arc in W, say 'Y : [-1, 1] -+ W, satisfying 1(t) =
u 0 ((1+t)z) fort E [-1,0], where z is any point in 8D. For c E (0, 1) small we haveas a consequence of the pseudoconvexity assumption 1(0, c) E w+. Going back to
our model we may assume that we have a piece of a pseudoconvex hypersurface
Mc C^2 , containing 8D x (-c, c) for some small c > 0. Since iJ x {O} c w- we
infer from the above discussion that { ( z, 0) E c^211 < I z I < 1 + €} c w+. Here w±