Continuity of the renormalized volume under geometric limits

TL;DR

This paper proves that the renormalized volume of geometrically finite hyperbolic 3-manifolds varies continuously under geometric limits, and identifies the minimal volume at the geodesic class.

Contribution

It extends the concept of renormalized volume to a broader class of hyperbolic 3-manifolds and proves its continuity under geometric convergence.

Findings

Renormalized volume is continuous under geometric limits.

The minimal renormalized volume occurs at the geodesic class.

The geodesic class has a totally geodesic boundary of the convex core.

Abstract

We extend the concept of renormalized volume for geometrically finite hyperbolic $3$-manifolds, and show that is continuous for geometrically convergent sequences of hyperbolic structures over an acylindrical 3-manifold $M$ with geometrically finite limit. This allows us to show that the renormalized volume attains its minimum (in terms of the conformal class at $\partial M = S$) at the geodesic class, the conformal class for which the boundary of the convex core is totally geodesic.