Zygmund vector fields, Hilbert transform and Fourier coefficients in shear coordinates

Details Zygmund vector fields, Hilbert transform and Fourier coefficients in shear coordinates Zygmund vector fields, Hilbert transform and Fourier coefficients in shear coordinates

2011-03-31

arxiv.org/abs/1103.6249v1

Mathematics

Geometric Topology

Scientific paper

We parametrize the space $\mathcal{Z}$ of Zygmund vector fields on the unit circle in terms of infinitesimal shear functions on the Farey tesselation. Then we express the Hilbert transform and the Fourier coefficients of the Zygmund vector fields in terms of the above parametrization by infinitesimal shear functions. Finally, we compute the Weil-Petersson metric on the Teichm\"uller space of a punctured surface in terms of shears.

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