Periodic k-Noids (Minimal Surfaces in the Wild II)

The k-Noids that Shireen built last winter will keep roaming the Swiss landscape, from May to August in Wülflingen. Maybe it is time to corral them. My suggestion is to build fences of catenoids. The most classical one looks like this:


It is, technically, a translation invariant minimal surface that has two ends and genus 1 in the quotient. A simple generalization and an even simpler 90 degree rotation leeds to towers with catenoidal openings.


If that isn’t safe enough, you can have them with double walls (i.e. with genus two in the quotient) like so


or so:


All these examples have many ends in the quotient. The surprise is that there also is the elusive Uninoid which only has one catenoidal end in the quotient, namely by a 180 degree screw motion:


Here things get tricky. Michelle Hackman has found more complicated versions of this in her thesis. Here is a Uninoid that is invariant under a screw motion with quarter turn.


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