The dual Bonahon–Schläfli formula

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.