Universal Teichmuller space can be identified with quasisymmetric homeomorphisms of the unit circle fixing three points. Weil-Petersson universal Teichmuller space is, in a certain sense, the L2 subspace of the universal Teichmuller space. It is the largest subspace where one can make sense of homogenous Kahler geometry and it connects various distant fields that will be mentioned briefly.
The shear coordinate is a concrete, countable coordinate system to describe non-decreasing self-maps of the circle. Characterizations of circle homeomorphism and quasisymmetric homeomorphisms were obtained by D.Šarić. We study the shear coordinates of Weil-Petersson circle homeomorphisms and compare them to L2 spaces of shears. We also obtain sharp results comparing them to Holder classes of circle homeomorphisms.
This talk is based on joint work with Dragomir Šarić and Catherine Wolfram. See https://arxiv.org/abs/2211.11497.