Smooth critical points of planar harmonic mappings

Mohammed El Ammrani (1) , Michel Granger (1) , Jean-Jacques Loeb (1) , Lei Tan (1)

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

Lyzzaik studied the local properties of light harmonic mappings. More precisely, he classified their critical points and accordingly studied their topological and geometrical behaviour. One aim of our work is to shed some light on the case of smooth critical points, thanks to miscellaneous numerical invariants. Inspired by many computations, and with a crucial use of Milnor fibration theory, we get a fundamental and quite unexpected relation between three of these invariants. In the final part of the work we offer some examples providing significant differences between our harmonic setting and the real analytic one.

Mots clés: Planar harmonic mappings; critical locus; critical value set; multiplcities; normal forms. Milnor fifrers.; normal forms. Milnor fifrers

Domaines: Variables complexes [math.CV]; Géométrie différentielle [math.DG]

Mohammed El Ammrani, Michel Granger, Jean-Jacques Loeb, Lei Tan. Smooth critical points of planar harmonic mappings. 2014. ⟨hal-01023168⟩