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

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 ${\mathbb{H}}^3$.

arXiv comment25 pages. Minor revisions. To appear in American Journal of Mathematics