A Nonlinear Cauchy-Riemann System in Several Complex Variables

Abstract

In this paper the nonlinear Cauchy-Riemann system $$ \frac{{\partial w}}{{\partial {z_j}}} = \overline {{f_j}\left( {z,w\left( z \right)} \right),} j = 1, \ldots ,n, $$ (0.1) is studied in a domain Ω of C n . In order that at each point of Ω there is a set of local solutions of (0.1) it is necessary that there is defined in Ω an involutive differential system, which induces in Ω a complex codimension one holomorphic foliation F. Then on each leaf of the foliation there is a system of linear differential equations, which the local solutions of (0.1) fulfill, and the leaf space Ω/F of the foliation has a structure of a Riemann surface ∑, over which the solutions of the linear system form a holomorphic line bundle L with every solution of (0.1) on Ω inducing a cross section of the projection of the bundle L over ∑. Based on these geometric framework, theorems expressing function theoretic properties of the solutions of (0.1), such as Identity Theorem, Extension Theorem, and Factorization Theorem, are proved.