LECTURE 3. THE WEINSTEIN CONJECTURE IN THE OVERTWISTED CASE 71andJ(z,O)=i if lzl~l
whereF := {(z,azkl^2 ) E C^2 I z E 8D; a ER},
J(z, w) := Tcp(cp-^1 (z, w)) o J(cp-^1 (z, w)) o Tcp-^1 (z, w).Here k is a suitable integer depending on F and w.It is interesting to note that the number k for a disk occuring in our Bishop
family is 0.
Exercise 3.3. Show that given a disk map ii in the previously constructed Bishop
family the constant k occuring in the corresponding local normal form is 0.
The proof of theorem 3.2 is a bit technical and we will not give details. A proof
can be found in [1]. There the reader will also find an introduction of the Maslov
index for loops of totally real planes. In this framework the integer k above finds
its natural explanation.
By the results of the previous section the investigation of the solution set of
(12) near an embedded solution uo reduces to the following problem:
u: D -----+ c2 ,
(13) 81u 8u + J ( u) 8u = 0
8s 8ton b,
u(8D) c F,
where J is an almost complex structure satisfying J(z, 0) = i for I z I~ 1 and
F = {(z, azkl^2 ) E C^2 I z E 8D; a E JR}
is totally real for J. We are interested in solutions u near the obvious solution
u 0 (z) = (z, 0) if k = 0. In order to understand the solution set of (13) we have to
view [) 1 as a smooth map between appropriate Banach manifolds.
ccFigure 9. This figure shows the situation if k = 0.