Adapted Renormalized Volume for Hyperbolic 3-Manifolds with Compressible Boundary

TL;DR

This paper introduces a new version of the renormalized volume for hyperbolic 3-manifolds with compressible boundary, ensuring boundedness and continuity properties analogous to the classical case, with applications to convex core volume bounds.

Contribution

It defines an adapted renormalized volume for manifolds with compressible boundary, extending classical properties and providing new geometric bounds and continuity results.

Findings

The adapted renormalized volume is bounded from below.

It has a uniformly bounded differential and Weil-Petersson gradient.

It extends continuously to the boundary strata of the Teichmüller space.

Abstract

The renormalized volume is a smooth function associating to every convex co-compact hyperbolic $3$-manifold $M$ a real number. When the boundary of $M$ is incompressible, the renormalized volume is always positive, otherwise there are sequences of convex co-compact structures on $M$ whose renormalized volumes diverge to minus infinity. We define here a new version of the renormalized volume which adapts to the compressible boundary case, satisfying properties analogous to those of the classical one in the incompressible setting. In particular, the adapted renormalized volume is bounded from below, its differential has uniformly bounded supremum norm, and its gradient has uniformly bounded Weil-Petersson norm. Moreover, it stays at uniformly bounded distance from the convex core volume function. As a corollary, we obtain a bound on the convex core volume of handlebodies in terms of the Weil-Petersson distance from a certain subset of the Teichm\"uller space, where the convex core volume is bounded by a known constant. Furthermore, the adapted renormalized volume extends continuously, as a function on the Teichm\"uller space of $\partial\bar M$, to the strata in the boundary of its Weil-Petersson completion corresponding to compressible multicurves. We provide a geometric interpretation of the limit quantity by defining a renormalized volume, and its adapted version, for convex co-compact hyperbolic $3$-manifolds with a finite set of marked points in the boundary.