94 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS
denote the local components in the model M. Since v(A) c w-UM, a(oA) c 8D,
where v = (a, b) we deduce that a(A) c D , perhaps after replacing A by some
smaller set. This shows that (35) holds. D
Summing up we have proved the following for our pseudoholomorphic curve
equation in the symplectisation:
Assume (M, .A) is a three-manifold equipped with a contact form A. Let J
be an admissible complex multiplication for the associated contact structure ~ _..,
M. Denote by J the associated translation invariant almost complex structure on
JR x M. Let F C M be a totally real surface. Consider the nonlinear boundary
value problem
(42) Us+ ](u)iit 0 on fJu(oD) c F = {O} x F.
Then we have the following theorem
Theorem 4.3. Let uo and v be embedded solutions of (42) such that k = 2. Let
(Dr)rE(-c:,c:) be the local disk family near Do= uo(D). Assume that v(D) n Do =f.
0 and v(D) n Dr = 0 for all T E (-c:, 0). Then there exists a biholomorphic
transformation cp: D _.., D with v = u 0 o cp. In particular v(D) = D 0.
Clearly the above theorem was the missing link in proving the Weinstein con-
jecture in the case tha t 7r 2 (M) =f. 0 and the contact structure is tight.