80 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICSgeometrically distinct solutions u,,, where
<I> : ( T - {j' T + 8) x D --+ JR x M : ( (}' z) --+ u,, ( z)
defines an embedding. We study now this maximal family in detail.Let Fn T E [O, To) be the maximal Bishop family. We say that the family
admits uniform gradient bounds, if there exists a constant c > 0, such that every
FT admits a holomorphic parametrisation u with
l(Y'u)(z)I :::; C for all z ED.
Proposition 3.14. The Bishop family FT does not admit uniform gradient bounds.
Proof. Arguing indirectly assume we have uniform gradient bounds. Then take a
sequence (uk) parametrizing FTk with Tk --+ To and uniformly bounded gradients.
By assumption
l(Y'uk)(z)I:::; C for all z ED.
We may assume without loss of generality that uk --+ u 0 in C^00 (recall that gradient
bounds imply C^00 -bounds). The map u 0 has to be a solution of (14). Writeuo = (b, uo). One computes easily
(-1),.b)ds /\ dt = -In( uo)s l^2 ds /\ dt on D.
The righthand side cannot be identically 0. In view of the boundary conditionbl8D = 0 Hopf's strong maximum principle implies that g~ (z) > 0 for every
z E 8D, where n is the outward pointing normal to the disk. From the differential
equation this implies immediately that ( uo) l8D is transversal to the characteristic
foliation. Since the winding number on the boundary is 1 we see that u 0 has thefollowing property. For c: > 0 sufficiently small
uo : A,, : = { z I l - c: :::; I z I :::; 1} --+ JR x M
is an embedding andu 01 (uo(A,J =Ac:.
We call uo an embedding at the boundary (in the sense ofMcDuff), see [70]. So the
embedding at the boundary Uo is the C^00 -limit of embeddings, which implies by the
results on the positivity of intersections, that it has to be an embedding too. Thisagain is nontrivial and we refer the reader to [1, 70, 73]. If the boundary u 0 (8D)
touches the boundary of V we obtain a contradiction. Indeed we have a tangency,where they meet, contradicting the fact that 8V is a leaf for the characteristic
foliation, which always is transversal to the boundary of a holomorphic disk. Hence
uo(8D) lies in the interior of V and therefore hits£ at the point C(T*). ClearlyT < oo. This of course implies that To = T. Applying the implicit function
theorem to F* = uo(D), we obtain a contradiction to the maximality of the Bishop
family. This proves the proposition. DBy the previous discussion we know that we do not have uniform gradient
bounds. Next we show that we always have gradient bounds near the boundary.
Consider the maximal family FT, TE [O, To). Take counter clock-wise two other
leaves £ 2 and £ 3 of the characteristic foliation on V. Put £ 1 = £. Consider forTE (0,To) the holomorphic parametrisation UT of FT> which satisfies uT(l) E £1,
uT(i) E £2 and uT(-l) E £3. Define 9c:(T) by
9c:(T) = maxzEA, IY'uT(z)I.
Clearly 9c: remains bounded on every compact interval in [O, To). We have