A Free Boundary Isoperimetric Problem in the Hyperbolic Space between parallel horospheres

TL;DR

This paper studies the shape of regions with minimal boundary area for a given volume between two parallel horospheres in hyperbolic 3-space, using rotational symmetry and constant mean curvature surfaces.

Contribution

It reduces the isoperimetric problem to rotationally invariant regions and characterizes solutions via profile curves of constant mean curvature surfaces.

Findings

Identification of possible isoperimetric regions between horospheres.

Analysis of profile curves for constant mean curvature surfaces.

Reduction of the problem to rotational symmetry.

Abstract

In this work we investigate the following isoperimetric problem: to find the regions of prescribed volume with minimal boundary area between two parallel horospheres in hyperbolic 3-space (the area of the part of the boundary contained in the horospheres is not included). We reduce the problem to the study of rotationally invariant regions and obtain the possible isoperimetric solutions by studying the behaviour of the profile curves of the rotational surfaces with constant mean curvature in the hyperbolic space.