#### Volume 21, issue 1 (2021)

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The dual Bonahon–Schläfli formula

### Filippo Mazzoli

Algebraic & Geometric Topology 21 (2021) 279–315
##### Abstract

Given a differentiable deformation of geometrically finite hyperbolic $3$–manifolds ${\left({M}_{t}\right)}_{t}$, the Bonahon–Schläfli formula (J. Differential Geom. 50 (1998) 25–58) expresses the derivative of the volume of the convex cores ${\left(C{M}_{t}\right)}_{t}$ in terms of the variation of the geometry of their boundaries, as the classical Schläfli formula (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 ${ℍ}^{3}$ and the de Sitter space ${dS}^{3}$. 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.

##### Keywords
Schläfli formula, dual volume, dual Bonahon–Schläfli formula, Kleinian groups, convex cocompact, hyperbolic geometry
##### Mathematical Subject Classification 2010
Primary: 53C65, 57M50
Secondary: 30F40, 52A15, 57N10
##### Publication
Received: 10 July 2019
Revised: 19 February 2020
Accepted: 26 March 2020
Published: 25 February 2021
##### Authors
 Filippo Mazzoli Department of Mathematics University of Virginia Charlottesville, VA United States https://filippomazzoli.github.io/