Experimental Noids

Constant mean curvature 1 surfaces in hyperbolic 3-space

These experimentally computed 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 fanciful bubble.

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.

Fournoid with unequal end parameters μ=(0.05, 0.15, 0.18, 0.21), poles=(1, i, –1, –i).
Fournoid with two pairs of unequal end parameters μ=(0.15, 0.15, 0.05, 0.05), poles=(1, i, –1, –i).
Fournoid with one negative end parameter μ=(–0.15, 0.15, 0.2, 0.2), poles=(1, i, –1, –i).
Fournoid with non-coplanar end axes. Parameters μ=(–0.1, 0.1, 0.1, 0.1), poles=(1, i, –4, –i).
Fivenoid with four equal and one unequal end parameter μ=(0.1, 0.1, 0.1, 0.1, 0.2), poles=(1, i, –1, –i, –-0.707–0.707i).

References

  1. Alexander I. Bobenko, Tatyana V. Pavlyukevich and Boris A. Springborn, Hyperbolic constant mean curvature one surfaces: spinor representation and trinoids in hypergeometric functions, Math. Z. 245 (2003), no. 1, 63–91.