From pseudo-holomorphic functions to the associated real manifold

This paper studies first the differential inequalities that make it possible to build a global theory of pseudo-holomorphic functions in the case of one or several complex variables. In the case of one complex dimension, we prove that the differential inequalities describing pseudo-holomorphic functions can be used to define a one-real-dimensional manifold (by the vanishing of a function with nonzero gradient), which is here a 1-parameter family of plane curves. On studying the associated envelopes, such a parameter can be eliminated by solving two nonlinear partial differential equations. The classical differential geometry of curves can be therefore exploited to get a novel perspective on the equations describing the global theory of pseudo-holomorphic functions.