On the derivatives of the Liouville currents
Complex Variables
2024-05-01 v1
Abstract
The Liouville map, introduced by Bonahon, assigns to each point in the Teichm\"uller space a natural Radon measure on the space of geodesics of the base surface. The Liouville map is real analytic and it even extends to a holomorphic map of a neighborhood of the Teichm\"uller space in the Quasi-Fuchsian space of an arbitrary conformally hyperbolic Riemann surface. The earthquake paths and by their extension quake-bends, introduced by Thurston, are particularly nice real-analytic and holomorphic paths in the Teichm\"uller and the Quasi-Fuchsian space, respectively. We find a geometric expression for the derivative of the Liouville map along earthquake paths.
Cite
@article{arxiv.2404.19736,
title = {On the derivatives of the Liouville currents},
author = {Xinlong Dong and Dragomir Šarić and Zhe Wang},
journal= {arXiv preprint arXiv:2404.19736},
year = {2024}
}
Comments
24 pages, 5 figures. arXiv admin note: text overlap with arXiv:2111.07809