Pseudoholomorphic Discs Attached to Pseudoconcave Domains

Abstract

We discuss almost complex perturbations of linear discs. We give precise estimates for the (1, α) norm of these deformations and for the dependence on parameters. In particular, we show how families of such discs give rise to local foliations of the space. Also, if Ω is a domain whose boundary is endowed with at least one negative eigenvector w ₁ at 0 for the standard structure of ℂⁿ, then small discs with velocity w ₁ which are analytic for a C ¹-perturbation of the structure, have boundary which is contained in ? in a neighborhood of 0. In particular, if the almost complex structure is real analytic, almost holomorphic functions extend along the corresponding foliation of discs.