The shear coordinate is a countable coordinate system to describe increasing self-maps of the unit circle, which is furthermore invariant under modular transformations. Characterizations of circle homeomorphism, quasisymmetric homeomorphisms were obtained by D.Šarić. We are interested in characterizing Weil-Petersson circle homeomorphisms using shears. This class of homeomorphisms arises from the study of Kähler geometry on the universal Teichmüller space and connects various distant fields that will be mentioned briefly.
For this, we introduce diamond shear which is the minimal combination of shears producing WP homeomorphisms. Diamond shears are closely related to log-Lambda length introduced by R. Penner. We obtain sharp results comparing the class of circle homeomorphisms with square summable diamond shears with the Weil-Petersson class and Hölder classes. We also express the Weil-Petersson metric tensor and symplectic form in terms of infinitesimal shears and diamond shears.
This talk is based on joint work with Dragomir Šarić and Catherine Wolfram. See https://arxiv.org/abs/2211.11497.