Sub-Riemannian geometry of the coefficients of univalent functions

2006/08/22 by Irina Markina, Markina, Irina, Dmitri Prokhorov +3
Mathematics · #17B66 #30C50 #53C17 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #math.CV #math.DG #msc:17B66 #msc:30C50 #msc:53C17

paper · pdf · doi:10.48550/arxiv.math/0608532

19 pages

arxiv created 2006/08/22 · arxiv updated 2009/12/01

Abstract

We consider coefficient bodies \mathcal M_n for univalent functions. Based on the Löwner-Kufarev parametric representation we get a partially integrable Hamiltonian system in which the first integrals are Kirillov's operators for a representation of the Virasoro algebra. Then \mathcal M_n are defined as sub-Riemannian manifolds. Given a Lie-Poisson bracket they form a grading of subspaces with the first subspace as a bracket-generating distribution of complex dimension two. With this sub-Riemannian structure we construct a new Hamiltonian system and calculate regular geodesics which turn to be horizontal. Lagrangian formulation is also given in the particular case \mathcal M₃.

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