The infimum of the dual volume of convex co-compact hyperbolic $3$-manifolds

TL;DR

This paper establishes a relationship between the infimum of the dual volume and the Riemannian volume of convex cores in hyperbolic 3-manifolds, providing bounds related to bending lamination length.

Contribution

It proves that the infimum of the dual volume equals that of the Riemannian volume for convex cores under quasi-isometric deformations and derives a sharp linear lower bound for quasi-Fuchsian manifold volumes.

Findings

Infimum of dual volume equals infimum of Riemannian volume for convex cores.

Derived a linear lower bound for convex core volume based on bending lamination length.

Established optimal multiplicative constant in the volume bound.

Abstract

We show that the infimum of the dual volume of the convex core of a convex co-compact hyperbolic $3$-manifold with incompressible boundary coincides with the infimum of the Riemannian volume of its convex core, as we vary the geometry by quasi-isometric deformations. We deduce a linear lower bound of the volume of the convex core of a quasi-Fuchsian manifold in terms of the length of its bending measured lamination, with optimal multiplicative constant.