Isoperimetric interpretation for the renormalized volume of convex co-compact hyperbolic 3-manifolds

TL;DR

This paper offers a new interpretation of the renormalized volume in convex co-compact hyperbolic 3-manifolds through isoperimetric profiles, proves a sharp Minkowski inequality for horospherically convex sets, and classifies stable constant mean curvature surfaces in specific hyperbolic regions.

Contribution

It introduces an isoperimetric interpretation for the renormalized volume and establishes a Minkowski inequality for horospherically convex sets in hyperbolic space.

Findings

Renormalized volume as asymptotic difference of isoperimetric profiles.

Sharp Minkowski inequality for horospherically convex sets.

Classification of stable constant mean curvature surfaces in certain hyperbolic regions.

Abstract

We reinterpret the renormalized volume as the asymptotic difference of the isoperimetric profiles for convex co-compact hyperbolic 3-manifolds. By similar techniques we also prove a sharp Minkowski inequality for horospherically convex sets in $\mathbb{H}^3$. Finally, we include the classification of stable constant mean curvature surfaces in regions bounded by two geodesic planes in $\mathbb{H}^3$ or in cyclic quotients of $\mathbb{H}^3$.