The infinitesimal earthquake theorem for vector fields on the circle

Diaf, Farid

doi:10.1090/tran/9243

The infinitesimal earthquake theorem for vector fields on the circle
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by Farid Diaf;

Trans. Amer. Math. Soc. 377 (2024), 8721-8767

DOI: https://doi.org/10.1090/tran/9243

Published electronically: September 17, 2024

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Abstract:

We prove that any continuous vector field on a circle is the extension in a suitable sense, of a unique infinitesimal earthquake of the hyperbolic plane. Furthermore, we obtain other extension results when the vector field is assumed only to be upper or lower semicontinuous. This leads to a generalization of Kerckhoff’s and Gardiner’s infinitesimal earthquake theorems to a broader setting, using a completely novel approach. The proof is based on the geometry of the dual of Minkowski three-space, also called Half-pipe three-geometry. In this way, we obtain a simple characterization of Zygmund vector fields on the circle in terms of width of convex hulls.

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