LECTURE 3. THE WEINSTEIN CONJECTURE IN THE OVERTWISTED CASE 83Lemma 3.16. Let w: D \ { 1} ---+ IR x M be a smooth map satisfying the Cauchy-Riemann equation on interior(D) and the boundary condition w(8D\ {1}) C D\ { e},
where the image of the boundary stays away from e. If E(w) < oo, then w extends
to a smooth map on D.In view of (26) the extended map, still denoted by w: D ---+ IR x M is notconstant. Therefore, the image of 8D under w is transversal to the characteristic
foliation of F and hence hits all leaves. This proves the claim and completes theproof in the case that 0 ::; p < oo in (18).
Next we assume that p = oo in (18) and shall derive another contradiction.
Assume that zk ---+ zo E 8D for a singular point z 0 of the sequence uki such that(16), (17) and (18) holds true with p = oo. We claim that the sequence Uk admits
at most finitely many singular points at the boundary 8D. Indeed, if z 0 E 8D
is such a singular point we carry out the same analysis as before and find, usingp = oo a rescaled sequence Vk defined this time on domains Dk C C such that
the boundary IR = 8H+ is always outside of Dk. Then vk ---+ v in Ci'~c for a finite
energy plane v: C ---+ IR x M as defined in the introduction. In particular, v is not
constant. For the details of this analysis we refer to [40]. The finite energy plane
v takes away (in the limit) an amount of d.A-energy which is bounded from below
by / > 0, where / is, as above, the infimum of the periods of contractible periodic
solutions, of the Reeb vectorfield X , see exercise 2.10. Now observe that we have
an a priori bound for the d.A-energy of Uk.
Exercise 3.17. The energy of a disk in the Bishop family admits the following
bound:
E(u) ::; ~ j l(d.AID)I.
Here the right hand side can be identified with the maximal positive d.A-area of D.
Observe that J d.AID = 0 by Stokes' theorem.
As a consequence of the previous discussion the number of singular points on
8D must be finite. The same argument can be used to show that also the number of
interior singular points is finite. Denote by r c D the finite collection of all singular
points of the sequence of Uk (or better of a suitable subseuence). Away from r we
have gradient bounds, and hence C^00 -bounds and we may, therefore, assume, after
taking a subsequence, that uk ---+ u in Ci'~c ( D \ f). The map u: D \ r ---+ IR x M
satisfies
(27)
us+ f(u)ut = 0 on int(D) \ r
u(8D\r)cF
E(u) < oo.
As before, we can apply Gromov's removable singularity theorem, in order to remove
the singularities on 8D. Sou is the restriction of a smooth map, again denoted by
u, which is defined on D minus a finite number of interior points.
Let z 1 , z 2 , ... , Zn be the singular points on 8D. For i = 1, ... , n we fix points
z"f E 8D close enough to zi , such that zi belongs to the interior of the oriented
segment Si = [zi, ztJ C 8D, and such that these segments are pairwise disjoint.
For every segment S i we choose a short curve / i connecting zt with zi within D
such that/in 8D = {zi, zt}. The image of the curve /i under the smooth map u