Continuity of the renormalized volume under geometric limits

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.

13 pages

References in corpus (2)

• On the renormalized volume of hyperbolic 3-manifolds
• Renormalized volume on the Teichmüller space of punctured surfaces

Cited by in corpus (2)

• Additive continuity of the renormalized volume under geometric limits
• The infimum of the dual volume of convex co-compact hyperbolic $3$-manifolds