PARTIAL INTEGRABILITY OF ALMOST COMPLEX STRUCTURES AND THE EXISTENCE OF SOLUTIONS FOR QUASILINEAR CAUCHY-RIEMANN EQUATIONS

We study the local solvability of the system of quasilinear Cauchy-Riemann equations for d unknown functions in n complex variables, which is a system of elliptic type and overdetermined if n >= 2. We consider an associated almost complex structure on Cn+d and its partial integrability and prove by using the Newlander-Nirenberg theorem and its algebraic generalizations that the existence of a pseudoholomorphic function on the zero set is equivalent to the local solvability of the original quasilinear system. We discuss an algorithm for finding pseudoholomorphic functions on the zero set and then present examples.