Continuity of the renormalized volume under geometric limits

Franco Vargas Pallete

Continuity of the renormalized volume under geometric limits

Differential Geometry 2016-05-26 v1

Authors: Franco Vargas Pallete

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.

Keywords

renormalized volume geometric limits hyperbolic 3-manifolds geometrically finite acylindrical manifold convex core geodesic class

Cite

@article{arxiv.1605.07986,
  title  = {Continuity of the renormalized volume under geometric limits},
  author = {Franco Vargas Pallete},
  journal= {arXiv preprint arXiv:1605.07986},
  year   = {2016}
}