The infinitesimal earthquake theorem for vector fields on the circle

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 generali...

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Veröffentlicht in:Transactions of the American Mathematical Society 2024-12
1. Verfasser: Diaf, Farid
Format: Artikel
Sprache:eng
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Zusammenfassung: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/9243