| We study the properties of the nonlinear Cauchy-Riemann operator and the solutions of the corresponding nonlinear Cauchy-Riemann equation: pseudoholomorphic curves in almost complex manifolds. Pseudoholomorphic curve is a natural generalization of a holomorphic map in the situation when the target space is endowed with an almost complex structure. M. Gromov has pioneered the study of the moduli space of pseudoholomorphic curves in symplectic manifolds. His work led to numerous beautiful results in symplectic geometry.; We investigate the properties of pseudoholomorphic curves in almost complex and symplectic manifolds with minimal assumptions on smoothness of the corresponding structures (almost complex structure and symplectic form).; For a symplectic manifold we prove the regularity of the weak solutions of the equation for pseudoholomorphic curves. In fact, we prove partial regularity results for more general classes of equations.; We derive conditional elliptic estimates for an associated nonlinear Cauchy-Riemann operator. Unlike the linear case, these estimates only hold provided that there are no nontrivial solutions of a certain model problem. We also study local properties of pseudoholomorphic curves. In particular, we address the question of their boundary behavior.; Finally, we introduce the notions and study the properties of an almost complex pseudodistance and J-hyperbolic manifolds, which generalizes the work S. Kobayashi in the complex case. |
