LECTURE 3. THE WEINSTEIN CONJECTURE IN THE OVERTWISTED CASE 81Proposition 3.15. There exists an 0 < c < 1 and a constant C > 0 such that
g"(T) :SC for all TE [O, To).
Proof. Arguing indirectly we find a sequence Tk ----> To and a sequence Zk ----; zo,
._ with lzol = 1 such that
Rk := l'Vuk(zk)I----> oo,where we wrote Uk for uTk Let us show that this assumption will lead to a contra-
dition. We choose a sequence ck > 0, satisfying
(16)
Applying lemma 2.6 to the continuous functions l'Vuk(z)I we can choose new
sequences ck, Zk such that we have, in addition to (16) the following local uniform
estimates
(17)Consider now the sequence Rk dist(zk, 8D). By taking a subsequence, we may
assume that
(18) Rk dist(zk, 8D) ----> p E [O, oo].There are two cases: either p < oo or p = oo. We first treat the case p < oo and shall
derive a contradiction to the fact that every boundary curve uk(8D) CV hits each
leaf of the characteristic foliation of V precisely once. We parameterize the leaveson F by the circle 81 = { z E C I I z I = 1} in such a way that 1 corresponds to t\, i
to £ 2 and -1 to f 3. In addition, we equipp 81 with the normalized measure 2 ~dB.
We are going to construct a sequence 8k c 8D of closed segments, containing the
distinguished point zo E 8D in the interior, and shrinking in length to 0, such that
the measure mk of the set of parameter values of those leaves which are intersected
by uk(8k) converge to 1. Observe that the singular point zo, i.e. the point where the
bubbling-off takes place, can be contained only in at most two of the closed segments
[1, i], [i, -1] and [-1, 1] on 8D. We denote by S the segment not containing z 0.
Then the measure of the parameter values of leaves hit by uk(8) is , in view of the
normalization condition on the parametrisation at least ~, for every k. Since the
boundary curve is transversal to the leaves, every leaf is hit precisely once, so that
we have, for all k, the estimate(19)Our aim is to prove that mk ----> 1, so that we arrive at a contradiction to (19).It remains to construct the sequence of segments 8k c 8D such that mk ----> 1.
We denote by fl+ the closed upper halfplane fl+ = {z EC I Im(z) ;::: O} and choose
a holomorphic embedding t.p of a sufficiently large neighborhood of 0 E fl+, which
maps 0 into zo = t.p(O) and a segment around 0 of JR= ()fl+ onto a segment around
z 0 in 8D. By composing Uk with <p we may assume, without loss of generality, thatthe sequence uk o t.p is defined on the closed half disk D+ = D n fl+. Writing uk
again, instead of Uk o t.p, we have a sequence uk: n+ ---->JR x M satisfying
(uk)s + ](uk)(uk)t = 0 on int(D+)
(20) uk([-1, 1]) c F