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-Rie
Results & Lemmas (6)
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Proposition 4.1.
Proposition 4.1. Let Mn be the n-th coefficient body and L1,... Ln be vec- tor fields defined by (11, 16). Then the system (L1, L2) satisfies…
Proposition 4.1. Let Mn be the n-th coefficient body and L1, . . . Ln be vec- tor fields defined by (11, 16). Then the system (L1, L2) satisfies the bracket generating condition and the distribution is D = span (L1, L2).
Proposition 4.2.
Proposition 4.2. The Hausdorff(complex) dimension of the sub-Riemannian manifold Mn is equal to • ( n 2 + 1)2 −9 4 for odd n; • ( n 2 + 1)2…
Proposition 4.2. The Hausdorff(complex) dimension of the sub-Riemannian manifold Mn is equal to • ( n 2 + 1)2 −9 4 for odd n; • ( n 2 + 1)2 −2 for even n.
Proposition 4.3.
Proposition 4.3. A path γ(s) = (c1(s),..., cn(s)) in Mn is horizontal if and only if ˙c3(s) = 3c2(s)˙c1(s) + 2c1(s) ˙c2(s) −2c1(s)˙c1(s)…
Proposition 4.3. A path γ(s) = (c1(s), . . . , cn(s)) in Mn is horizontal if and only if ˙c3(s) = 3c2(s)˙c1(s) + 2c1(s) ˙c2(s) −2c1(s)˙c1(s) . . . . . . ˙cn(s) = ncn−1(s)˙c1(s) + (n −1)cn−2(s) ˙c2(s) −2c1(s)˙c1(s) . (17)
Proposition 4.4.
Proposition 4.4. Any solution of the Hamiltonian system (18) is a horizontal path.
Proposition 4.4. Any solution of the Hamiltonian system (18) is a horizontal path.
Proposition 4.5.
Proposition 4.5. Define l3 as l3 = ¯ξ3 + 2c1¯ξ4 +... + (n −2)cn−3¯ξn. Then, (i) ˙l1 = ¯l2l3 and ˙l2 = −¯l1l3. (ii) The energy of the system…
Proposition 4.5. Define l3 as l3 = ¯ξ3 + 2c1¯ξ4 + . . . + (n −2)cn−3¯ξn. Then, (i) ˙l1 = ¯l2l3 and ˙l2 = −¯l1l3. (ii) The energy of the system 1 2(|u1|2+|u2|2) is conserved along the geodesics. The Carnot-Carath´eodory length of the tangent vector is conserved along the geodesics.
Proposition 4.6.
Proposition 4.6. The solution to the Euler-Lagrange system (22) is a solution to the Hamiltonian system (18) if and only if it is a…
Proposition 4.6. The solution to the Euler-Lagrange system (22) is a solution to the Hamiltonian system (18) if and only if it is a horizontal path.