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2021 The dual Bonahon–Schläfli formula
Filippo Mazzoli
Algebr. Geom. Topol. 21(1): 279-315 (2021). DOI: 10.2140/agt.2021.21.279

Abstract

Given a differentiable deformation of geometrically finite hyperbolic 3–manifolds (Mt)t, the Bonahon–Schläfli formula (J. Differential Geom. 50 (1998) 25–58) expresses the derivative of the volume of the convex cores (CMt)t in terms of the variation of the geometry of their boundaries, as the classical Schläfliformula (Q. J. Pure Appl. Math. 168 (1858) 269–301) does for the volume of hyperbolic polyhedra. Here we study the analogous problem for the dual volume, a notion that arises from the polarity relation between the hyperbolic space 3 and the de Sitter space dS3. The corresponding dual Bonahon–Schläfli formula has been originally deduced from Bonahon’s work by Krasnov and Schlenker (Duke Math. J. 150 (2009) 331–356). Applying the differential Schläfli formula (Electron. Res. Announc. Amer. Math. Soc. 5 (1999) 18–23) and the properties of the dual volume, we give a (almost) self-contained proof of the dual Bonahon–Schläfli formula, without making use of Bonahon’s results.

Citation

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Filippo Mazzoli. "The dual Bonahon–Schläfli formula." Algebr. Geom. Topol. 21 (1) 279 - 315, 2021. https://doi.org/10.2140/agt.2021.21.279

Information

Received: 10 July 2019; Revised: 19 February 2020; Accepted: 26 March 2020; Published: 2021
First available in Project Euclid: 16 March 2021

Digital Object Identifier: 10.2140/agt.2021.21.279

Subjects:
Primary: 53C65, 57M50
Secondary: 30F40, 52A15, 57N10

Rights: Copyright © 2021 Mathematical Sciences Publishers

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