On solvability of nonlinear partial differential systems of any order in the complex plane

Abstract

We prove a general existence theorem for nonlinear partial differential systems of any order in one complex variable. A special case of first order contains a well-known theorem of Nijenhuis and Woolf concerning local existence of J-holomorphic curves on almost complex manifolds. As an application to differential geometry, we prove that non constant harmonic map always exists locally from a Riemann surface to a Riemannian manifold with a prescribed tangent vector at a given point.