From pseudo-holomorphic functions to the associated real manifold

2016/06/27 by Giampiero Esposito, Esposito, Giampiero, Raju Roychowdhury +1
Mathematics · Physics and Astronomy · #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.CV #math.MP

paper · pdf · doi:10.48550/arxiv.1606.08144

25 pages

arxiv created 2016/06/27 · arxiv updated 2016/06/28

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.

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