The Isoperimetric Problem in a Lattice of $\mathbb{H}^3$

TL;DR

This paper investigates the isoperimetric problem in hyperbolic 3-space lattices using computational simulations to identify minimal surfaces enclosing given volumes within a specific tessellation.

Contribution

It provides the first numerical solutions for the isoperimetric problem in a hyperbolic 3-space lattice, leveraging the Surface Evolver software.

Findings

Identifies unique isoperimetric regions in hyperbolic lattice cubes.

Provides numerical solutions for minimal surfaces in hyperbolic space.

Extends understanding of geometric stability in non-Euclidean spaces.

Abstract

The isoperimetric problem is one of the oldest in geometry and it consists of finding a surface of minimum area that encloses a given volume $V$. It is particularly important in physics because of its strong relation with stability, and this also involves the study of phenomena in non-Euclidean spaces. Of course, such spaces cannot be customized for lab experiments but we can resort to computational simulations, and one of the mostly used softwares for this purpose is the Surface Evolver. In this paper we use it to study the isoperimetric problem in a lattice of the three dimensional hyperbolic space. More precisely: up to isometries, there exists a unique tesselation of $\mathbb{H}^3$ by non-ideal cubes $\mathcal{C}$. Now let $\Omega$ be a connected isoperimetric region inside the non-ideal hyperbolic cube $\mathcal{C}$. Under weak assumptions on graph and symmetry we find all numerical solutions $\Sigma=\partial\Omega$ of the isoperimetric problem in $\mathcal{C}$.