Convexity of the renormalized volume of hyperbolic $3$-manifolds

TL;DR

This paper proves that the renormalized volume functional of certain hyperbolic 3-manifolds is strictly convex at the hyperbolic metric with totally geodesic boundary, using the positivity of its Hessian.

Contribution

It establishes the strict convexity of the renormalized volume functional at the hyperbolic metric with totally geodesic boundary, providing new insights into the geometry of hyperbolic 3-manifolds.

Findings

Hessian of the renormalized volume is a strictly positive bilinear form.

The hyperbolic metric with totally geodesic boundary minimizes the volume of the convex core.

The volume functional is strictly convex at its minimum point.

Abstract

The Hessian of the renormalized volume of geometrically finite hyperbolic $3$-manifolds without rank-$1$ cusps, computed at the hyperbolic metric $g$ with totally geodesic boundary of the convex core, is shown to be a strictly positive bilinear form on the tangent space to Teichm\"uller space. The metric $g$ is known from results of Bonahon and Storm to be an absolute minimum for the volume of the convex core. We deduce the strict convexity of the functional volume of the convex core at its minimum point.