Sub-Riemannian geometry of the coefficients of univalent functions

Markina, Irina; Prokhorov, Dmitri; Vasil'ev, Alexander

Sub-Riemannian geometry of the coefficients of univalent functions

Authors

Markina, Irina

Prokhorov, Dmitri

Vasil'ev, Alexander

Description

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 $\mathcal M_3$.
19 pages