The dual Bonahon-Schl\"afli formula

TL;DR

This paper provides a self-contained proof of the dual Bonahon-Schl"afli formula, relating the variation of dual volume of hyperbolic 3-manifolds' convex cores to boundary geometry, extending classical volume variation formulas.

Contribution

It offers a new, self-contained proof of the dual Bonahon-Schl"afli formula, avoiding reliance on Bonahon's original work, and explores dual volume variations in hyperbolic 3-manifolds.

Findings

Derived a self-contained proof of the dual Bonahon-Schl"afli formula.

Connected dual volume variation to boundary geometry in hyperbolic manifolds.

Extended classical volume variation principles to dual volumes in hyperbolic and de Sitter spaces.

Abstract

Given a differentiable deformation of geometrically finite hyperbolic $3$-manifolds $(M_t)_t$, the Bonahon-Schl\"afli formula expresses the derivative of the volume of the convex cores $(C M_t)_t$ in terms of the variation of the geometry of its boundary, as the classical Schl\"afli formula 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 $\mathbb{H}^3$ and the de Sitter space $\mathrm{dS}^3$. The corresponding dual Bonahon-Schl\"afli formula has been originally deduced from Bonahon's work by Krasnov and Schlenker. Here, making use of the differential Schl\"afli formula and the properties of the dual volume, we give a (almost) self-contained proof of the dual Bonahon-Schl\"afli formula, without making use of Bonahon's original result.