Higher order derivatives

Abstract

This chapter is concerned with the estimate of higher order derivatives of J-holomorphic maps. For that purpose the differentials of J-holomorphic maps are made into pseudo-holomorphic maps to which the Gromov-Schwarz lemma applies. This is used in the proof of Gromov’s theorem on the removal of singularities for J-holomorphic maps. It is a generalization of a theorem of Riemann from complex analysis, which says that a holomorphic map f : S \ {a} → S ² from a Riemann surface minus an interior point a to the Riemann sphere can be extended to a holomorphic map S → S ², provided f does not have an essential singularity at a. As another application it can be proved that the derivatives of a locally uniformly convergent sequence of pseudo-holomorphic maps also converge locally uniformly. This generalizes a theorem of Weierstraß for holomorphic functions.