Sub-Riemannian geometry of the coefficients of univalent functions ✩

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. ThenMn 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 caseM3. © 2006 Elsevier Inc. All rights reserved.