Sub-Riemannian geometry of the coefficients of univalent functions
Authors:
Irina Markina
Affiliation:
a Department of Mathematics, University of Bergen, Johannes Brunsgate 12, Bergen 5008, Norway b Department of Mathematics and Mechanics, Saratov State University, Saratov 410026, Russia
Abstract:
We consider coefficient bodies Mn 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 Mn 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 to calculate regular geodesics which turn to be horizontal. Lagrangian formulation is also given in the particular case M3.