From pseudo-holomorphic functions to the associated real manifold

TL;DR

This paper develops a global theory of pseudo-holomorphic functions using differential inequalities, linking them to real manifolds and classical curve geometry, providing new insights into their structure and associated equations.

Contribution

It introduces a novel approach to describe pseudo-holomorphic functions via differential inequalities and relates them to real manifolds and classical differential geometry.

Findings

Pseudo-holomorphic functions define a 1-dimensional real manifold.

The associated envelopes lead to nonlinear PDEs that eliminate parameters.

Classical curve geometry offers new perspectives on these equations.

Abstract

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.