Sub-Riemannian geometry of the coefficients of univalent functions

We consider coefficient bodies M-n for univalent functions. Based on the Lowner-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 M, are defined as sub-Riemannian manifolds. Given a Lie-Poisson bracket they form a grading of subspaces with the first subspace as a bracketgenerating 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 M-3. (c) 2006 Elsevier Inc. All rights reserved.