A free boundary isoperimetric problem in hyperbolic 3-space between parallel horospheres

Rosa Maria Barreiro Chaves, Márcio F. da Silva and Renato H. L. Pedrosa

Vol. 244 (2010), No. 1, 1–20

DOI: 10.2140/pjm.2010.244.1

Abstract

We investigate the isoperimetric problem of finding the regions of prescribed volume with minimal boundary area between two parallel horospheres in hyperbolic 3-space (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 behavior of the profile curves of the rotational surfaces with constant mean curvature in hyperbolic 3-space. We also classify all the connected compact rotational surfaces M of constant mean curvature that are contained in the region between two horospheres, have boundary ∂M either empty or lying on the horospheres, and meet the horospheres perpendicularly along their boundary.

Keywords

constant mean curvature surfaces, hyperbolic space, isoperimetric problem

Mathematical Subject Classification 2000

Primary: 53A10, 49Q10

Milestones

Received: 24 October 2008

Revised: 23 July 2009

Accepted: 29 July 2009

Published: 17 November 2009