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 |
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Format: | Artikel |
Sprache: | eng |
Online-Zugang: | Volltext |
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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. |
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ISSN: | 0002-9947 1088-6850 |
DOI: | 10.1090/tran/9243 |