A Schlafli-type formula for convex cores of hyperbolic 3-manifolds

TL;DR

This paper establishes a Schlafli-type formula for the variation of the volume of convex cores in hyperbolic 3-manifolds, linking it to the bending lamination's transverse measure, extending classical polyhedral volume formulas.

Contribution

It generalizes the Schlafli formula to convex cores of hyperbolic 3-manifolds, relating volume variation to the bending lamination's transverse measure.

Findings

Volume variation equals half the length of the transverse bending lamination.

The formula extends classical polyhedral volume variation to convex cores.

Provides a new tool for understanding deformation of hyperbolic 3-manifolds.

Abstract

In 3-dimensional hyperbolic geometry, the classical Schlafli formula expresses the variation of the volume of a hyperbolic polyhedron in terms of the length of its edges and of the variation of its dihedral angles. We prove a similar formula for the variation of the volume of the convex core of a geometrically finite hyperbolic 3--manifold M, as we vary the hyperbolic metric of M. In this case, the pleating locus of the boundary of the convex core is not constant any more, but we showed in an earlier paper that the variation of the bending of the boundary of the convex core is described by a geodesic lamination with a certain transverse distribution. We prove that the variation of the volume of the convex core is then equal to 1/2 the length of this transverse distribution.