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

TL;DR

This paper introduces a parametrization of Zygmund vector fields on the circle using shear functions, and expresses key operators and metrics in terms of these shear coordinates, linking complex analysis, hyperbolic geometry, and Teichmüller theory.

Contribution

It provides a new parametrization of Zygmund vector fields via shear functions and relates the Hilbert transform, Fourier coefficients, and Weil-Petersson metric to this framework.

Findings

Parametrization of Zygmund vector fields using shear functions.

Expression of Hilbert transform and Fourier coefficients in shear coordinates.

Computation of Weil-Petersson metric in shear coordinates.

Abstract

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.