# Constant Mean Curvature Surfaces in Hyperbolic 3-Space

Nicholas Schmitt

### Description

All noids are depicted in the Poincaré ball model of three-dimensional hyperbolic space. The ends of the noids approach the boundary of hyperbolic space at infinity, represented by a glass ball or fanciful bubble.

Binoids, also known as catenoid cousins, constructed in [1][2]. Further shown are fournoids with dihedral symmetry and platnoic noids. That is constant mean curvature 1 noids in hyperbolic 3-space. Their ends are asymptotic to catenoid cousins.

For the experimental noids the closing parameters for these surfaces were computed numerically by fixing the end parameters and varying the remaining accessory parameters in the potential. A minimizing algorithm on a measure of the simultaneous unitarizability of the monodromy was applied to find unitarizable monodromy to within a numerical threshold

### References

• Alexander I. Bobenko, Tatyana V. Pavlyukevich, and Boris A. Springborn.
Hyperbolic constant mean curvature one surfaces: spinor representation and trinoids in hypergeometric functions.
preprint, 2003.
arXiv:0206021.
• Alexander I. Bobenko and Tatyana V. Pavlyukevich.
Bryant n-noids with smooth ends or symmetry.
preprint, 2005.

#### Dr. Nicholas Schmitt   +

University: TU Berlin
Website: http://page.math.tu-berlin.de/~schmitt/