72 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICSProposition 3.4. Let 2 < p < oo and U be a sufficiently small neighbourhood of
D x {O} = uo(D) c IC^2. Then
B := { u E W^1 ·P(D, IC^2 ) I u(oD) c F ; u(D) c U}
is a Banach submanifold of W^1 ·P(D, IC^2 ) admitting an atlas that consists of one
chart only. The tangent space at u E B is given byTuB = {! E W^1 •P(D, IC^2 ) I f(z) E Tu(z)F for z E 8D}.
Before we begin with the proof we will need the following lemma:
Lemma 3.5. There is a metric on IC^2 so that F is totally geodesic.Exercise 3.6. Prove lemma 3.5.
Now we can prove proposition 3.4.Proof. Because of lemma 3.5 we may choose a metric on IC^2 so that F is totally
geodesic. Let exp : IC^2 x W -. IC^2 be the exponential map of the associated Levi-
Civita connection where W is some open neighbourhood of (0, 0) in IC^2. Consider
v := {! E W^1 ·P(D, IC^2 ) I f(z) E T(z,o)F = izlR EB zkl^2 JR for z E 8D}
First we remark that this definition makes sense since W^1 ·P(D, IC^2 ) C C^0 (D, IC^2 )
by the Sobolev embedding theorem.We can choose a small ball BCV with respect to the W^1 •P-norm around zero
so that the images of all these f E Bare contained in some ball B 0 (0) C B 0 (0) C W.
This is true because of llfllco s; cllfllw1,p (Sobolev embedding theorem). Hence
for f E B we can define the map
exp (!) : D ---+ C^2z f----+ exp ( ( z, 0), f ( z)).We note that exp IDx{(O,O)} is just the obvious embedding D x {O} <--+ IC^2. There
is an open neighbourhood W' of D x {(O, O)} c IC^2 x IC^2 so that exp lw' is an
embedding. Choosing W small enough we may assume that D x W c W'. Hence
exp (!1) = exp (h) implies Ji = h for Ji, h E B. Since exp IDxBn(O) and its
derivative are bounded, exp(!) is again of class W^1 ·P and for U small enough we
even have
{exp(!) I f EB}= B
because F is totally geodesic. So we have a map
qi : B---+ B c W^1 ·P(D, C^2 )
f f----+ exp (!).
One verifies directly that qi is smooth and
(Dqi(f) · h)(z) = D2 exp ((z, 0), f(z)) · h(z)for f E B ; h E V. We would like to show that Dqi(o) is injective and the image
splits. Then (after choosing Band U smaller) B = qi(B) is a Banach submanifold
of W^1 ·P(D,C^2 ) and qi is a global chart. We have D 2 exp((z,O),O) = Idc2, so
Dqi(o) · h = h for h E V. Defining
X z := zlR EB izkl^2 JR for z E oD