Abstract:
We consider the group of orientation-preserving
diffeomorphisms of the unit circle and its central extension, the
Virasoro-Bott group, with their respective horizontal distributions,
which are Ehresmann connections with respect to a projection to the
smooth universal Teichmüller space and the universal Teichmüller curve
associated to the space of normalized univalent functions. We find
equations for the normal sub-Riemannian geodesics with respect to the
pullback of the Kählerian metrics, namely, the Velling-Kirillov metric
on the class of normalized univalent functions and the Weil-Petersson
metric on the universal Teichmüller space. The geodesic equations are
sub-Riemannian analogues of the Euler-Arnold equation and they lead to
the CLM, KdV and other known non-linear PDEs.
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