Approaching the isoperimetric problem in $H^m_{\mathbb{C}}$ via the hyperbolic log-convex density conjecture

TL;DR

This paper proves that in hyperbolic spaces with certain smooth, radial, log-convex densities, geodesic balls are the optimal shapes for enclosing volume with minimal perimeter, extending known results from Euclidean spaces.

Contribution

It establishes the isoperimetric property of geodesic balls in hyperbolic spaces with log-convex densities, generalizing previous Euclidean results to hyperbolic and symmetric spaces.

Findings

Geodesic balls are isoperimetric in hyperbolic spaces with log-convex densities.

Extension of Chambers' Euclidean results to hyperbolic and symmetric spaces.

Application to rank one symmetric spaces of non-compact type.

Abstract

We prove that geodesic balls centered at some base point are isoperimetric in the real hyperbolic space $H_{\mathbb R}^n$ endowed with a smooth, radial, strictly log-convex density on the volume and perimeter. This is an analogue of the result by G. R. Chambers for log-convex densities on $\mathbb R^n$. As an application we prove that in any rank one symmetric space of non-compact type, geodesic balls are isoperimetric in a class of sets enjoying a suitable notion of radial symmetry.