The Log Convex Density Conjecture in Hyperbolic Space

TL;DR

This paper extends the proof of the Log Convex Density Conjecture from Euclidean space to hyperbolic space, showing that spheres about the origin minimize weighted perimeter for certain densities.

Contribution

It generalizes Chambers' results from Euclidean space to hyperbolic space, adapting the isoperimetric problem with density to a new geometric setting.

Findings

Spheres about the origin minimize weighted perimeter in hyperbolic space for certain densities.

The results adapt the Log Convex Density Conjecture to hyperbolic geometry.

The paper provides a framework for isoperimetric problems with densities in hyperbolic space.

Abstract

The isoperimetric problem with a density or weighting seeks to enclose prescribed weighted area with minimum weighted perimeter. According to Chambers' recent proof of the Log Convex Density Conjecture, for many densities on $\mathbb{R}^n$ the answer is a sphere about the origin. We generalize his results from $\mathbb{R}^n$ to $\mathbb{H}^n$ with related but different volume and perimeter densities.