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 [1].

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

A. Bobenko, T. Pavlyukevich, and B. Springborn, Hyperbolic constant mean curvature one surfaces: spinor representation and trinoids in hypergeometric functions, Math. Z. 245 (2003), no. 1, 63—91 [2023953].