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arXiv · 2404.19736

On the derivatives of the Liouville currents

Xinlong Dong·Dragomir Šarić·Zhe Wang
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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.

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Xinlong Dong, Dragomir Šarić, Zhe Wang. 2024-04-30. On the derivatives of the Liouville currents. https://arxiv.org/abs/2404.19736

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