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Sub-Riemannian geometry of the coefficients of univalent functions | Irina Markina
; Dmitri Prokhorov
; Alexander Vasil’ev
; | Date: |
22 Aug 2006 | Subject: | Complex Variables; Differential Geometry | Abstract: | We consider coefficient bodies $mathcal M_n$ for univalent functions. Based on the L"owner-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 $mathcal M_3$. | Source: | arXiv, math/0608532 | Services: | Forum | Review | PDF | Favorites |
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