Given a differentiable deformation of geometrically finite hyperbolic
–manifolds
,
the
Bonahon–Schläfli formula (J. Differential Geom. 50 (1998)
25–58) expresses the derivative of the volume of the convex cores
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
and the de
Sitter space
.
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.
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