One of the toy examples that illustrates how easy it is to make minimal surfaces defined on punctured spheres is the wavy catenoid. In its simplest form it fuses a catenoid and an Enneper end together, like so:
I learned from Shoichi Fujimori that one can add a handle to these:
This would make a beautiful mincing knife… Numerically, it was easy to add more handles:
I dubbed them angel surfaces, partially because of their appearance, partially because while we think they exist, we don’t have a proof.
They are interesting for two reasons: First, they are extreme cases of two-ended finite total curvature surfaces: The degree of the Gauss map of such surfaces must be at least g+2, where g is the genus of the surface. Here, we have equality.
Secondly, they come in 1-parameter families, providing us with an interesting deformation between Enneper surfaces of higher genus.
Above is a genus 2 example close to the Chen-Gackstatter surface. Below is a genus 2 example close to a genus 2 Enneper surface, first described by Nedir do Espírito-Santo.
In other words, we get a deformation from a genus 1 to a genus 2 surface.