Smooth critical points of planar harmonic mappings

TL;DR

This paper investigates smooth critical points of planar harmonic mappings, establishing relationships between local invariants and connecting them to classical models using Milnor fibration theory, with applications to real and complex analytic germs.

Contribution

It introduces new relationships between local invariants of smooth critical points and links them to Milnor fibration theory, extending classical models to harmonic mappings.

Findings

Established a relation between complex multiplicity, local order, and Puiseux pairs.

Developed an iterative algorithm for computing invariants.

Compared harmonic and real analytic cases through examples.

Abstract

In a work in 1992, Lyzzaik studies local properties of light harmonic mappings. More precisely, he classifies their critical points and accordingly studies their topological and geometrical behaviours. We will focus our study on smooth critical points of light harmonic maps. We will establish several relationships between miscellaneous local invariants, and show how to connect them to Lyzzaik's models. With a crucial use of Milnor fibration theory, we get a fundamental and yet quite unexpected relation between three of the numerical invariants, namely the complex multiplicity, the local order of the map and the Puiseux pair of the critical value curve. We also derive similar results for a real and complex analytic planar germ at a regular point of its Jacobian level-0 curve. Inspired by Whitney's work on cusps and folds, we develop an iterative algorithm computing the invariants. Examples are presented in order to compare the harmonic situation to the real analytic one.