1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

78 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS

Then J(O, z) = z and therefore D 2 f(O, z ) =Id for all z E D. We find now by the
implicit function theorem a unique smooth map

g: (-c:',c:') x D-----+ D (c:':::; c:)

with

f(T,g(T,z)) = Z

i.e.

Hence we must have g(T,. ) = crT.

Parametrize £ 1 by £ 1 (7) := (1,T). We find a local diffeomorphism a defined on

some open neighbourhood of 0 E IR, mapping 0 to 0, such that

Hence we have proved the following

Proposition 3.12. Given smooth embeddings u 1 and u 2 as in proposition 3 .11

there exist c: > 0 , a smooth diffeomorphism a : ( -c:, c:) __, a( ( -c:, c:)) satisfying

a(O) = 0, and a smooth map (-c:,c:) 3 T f-----+ crT, where crT is a biholomorphic self

map of the disk, with cro = I do, such that

for all (T,z ) E (-c:,c:) x D.

So in other words there is up to parametrisation a unique disk family T f-----+ DT,

with Do = D x {O}, determined by the first order elliptic boundary value problem.

Here DT = u(T, D). Combining the previous two propositions with theorem 3.2 we

get:

Corollary 3.13. Let (W, J) be an almost complex four-manifold and let F C W

be a totally real submanifold. Moreover let u 0 be an embedded J-holomorphic disk

with boundary values in F and index k = 0 in the local normal farm. Then there

exists a smooth embedding u: (-c:,c:) x D __, W such that with u(T)(z) = u(T,z)

we have

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

0JU(T) = 0,

u(O)(z ) = uo(z).

Moreover the associated disk family T f-----+ u(T)(D) is unique up to the parametri-

sation.