## Closing the Gaps

In 1982, Chi Cheng Chen and Fritz Gackstatter published a paper that described the surface below.

Like some of the classical examples of minimal surfaces, this surface is complete and has finite total curvature. A famous theorem of Osserman from 1964 asserts that any such surface can be defined on a punctured Riemann surface. In the classical examples, this had always been a sphere, but here we have a torus with one puncture.  There were some earlier examples, but this one, while not embedded, was surprisingly simple. From far away, it looks just like the Enneper surface.

How does one make such an example? One problem is illustrated above: While Osserman’s theorem also guarantees that the derivative of a conformal parametrization has a meromorphic extension to the compact surface, the integration of these so-called Weierstrass data might leave gaps.

To close the gap, we use the help of symmetries: Two vertical planes cut the surface into four congruent pieces, each represented by the upper half plane. The Weierstrass forms $\phi_1$ and  $\phi_2$ then turn out to be Schwarz-Christoffel integrands. The corresponding integrals map the upper half plane to (infinite) Euclidean polygons, shown above. The left extends to cover a bit more than a quarter plane, the right a bit less than a three quarter plane.

Incidentally, we can see the torus by fitting four copies of the right polygon together. We obtain the plane with a square missing. Identifying opposite edges of the missing square creates a torus with one puncture.

Now the condition that makes the gaps disappear is just that the two polygons fit together, which can be achieved by scaling. It’s really that simple. Similarly one can have more symmetric versions by just changing the angles in the polygons. Below is an example with sevenfold symmetry.

## 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:

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.

## Catenoids meet Enneper (Enneper III)

Sometimes, the Enneper surface will just show up. For instance, when classifying complete minimal surfaces of small total Gauss curvature, it is unavoidable. Together with the catenoid it hold the record of having only total curvature -4𝜋. Next comes -8𝜋, and for this you will encounter critters like these that have look like an Enneper surface with two catenoids poking out.

There are many others, and I view them not so much as objects to be classified and put away but rather as play grounds where one can learn what design goals are compatible with the constraint of being a minimal surface.

For instance, adding a base to the surface above is possible but pulls the two top “lobes” of Enneper and with them the two inward pointing catenoids apart:

But still, the Enneper surface comes in handy. The k-Noids, which traditionally are minimal surfaces just with catenoidal ends, have to be well balanced: The catenoids pull and push in the direction of their axes, and get boring after a while. The Enneper surface is much stronger then any number of catenoids and will win any tug-of-war.

## Praising the Underrated (Enneper I)

In 1760 Joseph Lagrange writes, after establishing the minimal surface equation of a graph and observing that planar graphs do indeed satisfy his equation, that “la solution générale doit ètre telle, que le périmètre de la surface puisse ètre détermine a volonté” — the general solution ought to be such that the perimeter of the surface can be prescribed arbitrarily.

For a hundred years, little progress was made to support Lagrange’s optimism. Few examples of minimal surfaces were found, and most of them with considerable effort. Then it the second half of the 18th century, it took the combined efforts of Pierre Ossian Bonnet, Karl Theodor Wilhelm Weierstraß, Alfred Enneper, and Hermann Amandus Schwarz to unravel a connection between complex analysis and minimal surfaces that would become the Weierstrass representation and revolutionize the theory.

One piece in this story is Enneper’s minimal surface. Enneper was not so much after minimal surfaces but after examples of surfaces where all curvature lines are planar. This was immensely popular back then, and the long and technical papers are mostly forgotten.

Above is an attempt to visualize the planes that intersect the Enneper surface in its curvature lines.

Visually easier to digest are the ruled surfaces that are generated by the surface normals along the curvature lines, because here the ruled surfaces and the Enneper surface meet orthogonally. While not planar, they are still flat, and invite therefore a paper model construction (that one can do for the curvature liens of any surface):

Print and cut out the five snakes. The orange centers are the curvature lines. Also cut all segments that go half through a snake, and fold along all segments that go all the way through a snake, by about 90 degrees, always in the same way. Then assemble by sliding the snakes into each other along the cuts, like so:

The three long snakes close up in space and need some tape to help them with that. Here is a retraced version of the same model which might help.