# Volumes of quasifuchsian manifolds

## TL;DR

This paper explores the relationship between the renormalized volume of quasifuchsian hyperbolic manifolds and the dual volume of their convex core, revealing analogies, bounds, and variational properties.

## Contribution

It establishes new connections and analogies between renormalized volume and dual volume, and discusses their implications for bounding convex core volume.

## Key findings

- Renormalized volume and dual volume share similar variational formulas.
- Objects related to these volumes are within bounded distance.
- Both volumes can bound convex core volume via Weil-Petersson distance.

## Abstract

Quasifuchsian hyperbolic manifolds, or more generally convex co-compact hyperbolic manifolds, have infinite volume, but they have a well-defined ``renormalized'' volume. We outline some relations between this renormalized volume and the volume, or more precisely the ``dual volume'', of the convex core. On one hand, there are striking similarities between them, for instance in their variational formulas. On the other, object related to them tend to be within bounded distance. Those analogies and proximities lead to several questions. Both the renormalized volume and the dual volume can be used for instance to bound the volume of the convex core in terms of the Weil-Petersson distance between the conformal metrics at infinity.