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Sub-Riemannian geometry of the coefficients of univalent functions

Irina Markina·Dmitri Prokhorov·Alexander Vasil'ev
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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 $\mathcal M_3$.

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Irina Markina, Dmitri Prokhorov, Alexander Vasil'ev. 2006-08-22. Sub-Riemannian geometry of the coefficients of univalent functions. https://arxiv.org/abs/math/0608532

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