The renormalized volume and the volume of the convex core of quasifuchsian manifolds

TL;DR

This paper establishes a relationship between the renormalized volume and the convex core volume of quasifuchsian manifolds, providing bounds related to Teichmüller space geometry and implications for curvature of holomorphic disks.

Contribution

It proves the equivalence of renormalized volume and convex core volume up to a constant and derives bounds based on Weil-Petersson distance, linking hyperbolic geometry and Teichmüller theory.

Findings

Renormalized volume equals convex core volume up to an additive constant.

Provides an upper bound on renormalized volume via Weil-Petersson distance.

Shows large holomorphic disks in Teichmüller space have significant negative curvature.

Abstract

We show that the renormalized volume of a quasifuchsian hyperbolic 3-manifold is equal, up to an additive constant, to the volume of its convex core. We also provide a precise upper bound on the renormalized volume in terms of the Weil-Petersson distance between the conformal structures at infinity. As a consequence we show that holomorphic disks in Teichm\"uller space which are large enough must have "enough" negative curvature.