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1 is a sequence of almost complex structures on M converging in Coo toJ. Then /-L n := 4(/-L + /-L 0 (In x I n)) is a Hermitian metric on (M,Jn) converging in Coo to /-L as n ~ 00. 3. For each sufficiently large n, and thus for all n, we may assume that £o(n) = £0' c(n) = c andCML(n) =CML , independently of n. This can easily be seen from the proofs in the last chapter. If we are interested in getting uniform estimates for higher derivatives of In-holomorphic maps one difficulty arises. Namely, the space [S,M] of I-jets depends on I n as a point set.

Hence IITfll, with respect to the Poincare metric on D \ {O}, is also bounded since n is locally an isometry. Since the area of (P) \ {z E IHII Im(z) ~ r } is finite for r > 0, this implies that f has finite area on some neighbourhood of 0 and we are in the 0 situation of Case 1. This finishes the proof of (a). Observe that the following result was obtained above. 2. A f-holomorphic map f: S \ {a} ~ (M,l) with relatively compact image satisfying assumption (ii) of the theorem on the removal of singularities also satisfies assumption (i) and thus has finite area in some neighbourhood of a.

Area ~ (length of boundary)2 (see Appendix A) which is "similar" to the one claimed in the lemma. II. 1. Let R = injrad(M, Il) be the injectivity radius of M. 1). Let expp := expp IBR(O) denote the exponential map of Mat p restricted to the R-ball BR(O) c (TpM,Jp,ll p ) where p is an arbitrary point in M fixed from now on. We identify canonically the tangent spaces to TpM with TpM itself. The standard symplectic form roo of (TpM,Jp,ll p ) is given by We pull back roo to B R (p) c M with exp pI and get an exact symplectic form on BR(p).

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