Sub-Riemannian geometry of the coefficients of univalent functions (Q883503)

The authors consider coefficient bodies \(\mathcal M_n=\{(c_1,\dots,c_n): f\in\tilde{\mathbf S}\}\) where \(\tilde{\mathbf S}\) is the class of holomorphic univalent functions \(f(z)=z(1+\sum_{n=1}^{\infty}c_nz^n)\) which are smooth on the boundary \(S^1\) of the unit disk \(U\). Based on the Loewner-Kufarev parametric representation they get a Hamiltonian system for the coefficients \(c_k(t)\) determined by a solution \(w(z,t)=e^{-t}z(1+\sum_{n=1}^{\infty}c_n(t)z^n)\) to the Loewner-Kufarev equation \[ \frac{dw}{dt}=-wp(w,t),\;\;\;w(z,0)=z,\;\;\;t\geq0, \] \(p(z,t)=1+p_1(t)z+\dots\) is holomorphic in \(U\), \(\Re p(z,t)>0\). This Hamiltonian system appears to be partially integrable, and the first integrals are Kirillov's operators for a representation of the Virasoro algebra. If \(L_1,\dots L_n\) are the first integrals of the Hamiltonian system and \(H\) is the Hamiltonian function, then \([L_j,H]=0\), \([L_j,L_k]=(j-k)L_{k+j}\), when \(k+j\leq n\), or 0 otherwise. 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 the authors construct a new Hamiltonian system to calculate regular geodesics which turn to be horizontal. Lagrangian formalism is also given in the particular case \(\mathcal M_3\).