The Angel Surfaces

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:G=1

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.G=2

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.Espiro Santo

In other words, we get a deformation from a genus 1 to a genus 2 surface.