On the persistence of pseudo-holomorphic curves on an almost complex torus (with an appendix by J�rgen P�schel)

Abstract

In recent years pseudo-holomorphic curves on almost complex manifolds have played an important role in connection with the developments in symplectic geometry, notably on account of Gromov's work [8] in 1985. In particular, the existence of compact pseudo-holomorphic curves of the type of S 2 or the closed disc have been established on particular manifolds. In this paper we study, unrelated from symplectic geometry, non-compact pseudo-holomorphic curves on an almost complex torus (T2n,J). These curves carry a complex structure and therefore represent Riemann surfaces. In our case they are of the type of ~ or a cylinder ~* = IE\(0). In particular we are interested in the question of the persistence of these curves under perturbation of the almost complex structure. This question has similar features as that of the persistence of invariant tori in classical mechanics, as initiated by Kolmogorov [2,15,18]. However, in contrast with that theory we are dealing with nonlinear elliptic systems of partial differential equations of Cauchy-Riemann type. Our results provide global solutions for such systems.