1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 3. THE WEINSTEIN CONJECTURE IN THE OVERTWISTED CASE 77

Now we have

0 = dd I -

7

H(T, a(T)) t =O = DH(0,0)(1, a'(O)) = D8J(u 0 )a'(O)

which implies a'(O) = 0, hence

Du(O,z)(u,(3) = ((3 + uho(z ),uko(z)) =/. (0,0) for u ,(3 =f:. 0 , z E 8D.

Since u(O, z ) = (z, 0) we infer by the previous discussion that u is an embedding

for c:' sufficiently small.

Hence we have proved the following:

Proposition 3.11. Under the assumptions described before, i.e. k = 0, there exists

a smooth embedding u: (-c:,c:) x D---+ C^2 such that with u(T)(z) = u(T,z) we have

u(T)(z ) E F for all z E 8D,

BJU(T) = 0,

u(O)(z ) = (z , 0).

In the following we will study the uniqueness properties of this map. It turns

out that the map is unique up to some trivial modifications. Define the following

three lines on F = 8 D x JR:

£1 := {1} x JR,

£2 := {i} x JR,

£3 := {-i} x R

Let u be as described in the previous theorem. There exists for given T a unique
biholomorphic selfmap of the disk D , say u 7 , such that W 7 = u(T, *) o u 7 satisfies

Wr(l) E £1, Wr(i) E £2, Wr(-i) E £3.

We would like to show that u 7 (z ) depends smoothly on T. Consider the map

f : ( -c:, c:) x D ---+ C

(T, z)---+ pr 1 (u(T, z )).

Figure 10. This figure shows the one-dime nsiona l family of mutually disjoint

embedded disks guaranteed by Proposition 3. 11.