## Death of Proof (The Pleasures of Failure I)

Die Wahrheit ist dem Menschen zumutbar.

Occasionally, after confronting students with evidence of fact (Euler Polyhedron Theorem is a great example), I ask them whether they want to see a proof or prefer to accept the statement as a miracle. The overwhelming majority is always happy with the miracle. Such are the times. Below is such an evidence of fact: A minimal surface with 3 ends and of genus 2.

Should we doubt its existence? In 1993, John Horgan published an article in Scientific American questioning whether proofs were about to become obsolete, in times where shear length and difficulty made validation next to impossible, and numerical experiments supplied by computers could be an acceptable substitute. For many reasons, large parts of the mathematical community were outraged.

Above is another example of that surface, for a different parameter value, but something seems off. There appears to be a little crack. Maybe I didn’t  compute accurately enough? Changing the parameter a bit more widens the gap.

The question whether this surface does actually exist hinges on the possibility to truly close that gap, for at least one parameter value. It appears that we have done so in the top image. But the parameter value there is 1.01, pretty close to 1, where the surface will clearly break down. A more accurate computation shows that there still is a gap at 1.01, which we can’t see, or don’t want to see. But maybe 1.001 will do?

David Hoffman and Hermann Karcher analyzed this surface in 1993, the same year as Horgan’s article, and it became known as the Horgan surface. One can indeed prove that the gap ca not be closed, so, despite all the evidence, this minimal surface does not exist.

## Three Planes

When you take two non-parallel planes, they will intersect in a line. The singly periodic Scherk surfaces are the only minimal way to “desingularize” this, in the sense that they are the only known minimal surfaces asymptotic to these two planes. To show this is one of the many famous open problems about minimal surfaces.

The situation gets vastly more complicated with three planes. Nobody has yet succeeded in constructing a minimal surface that is asymptotic to the three coordinate planes. That is another open problem. A case where we do know something is that of three (or more) vertical planes. Martin Traizet has shown in 1994 that in case the planes are reasonably general one can wiggle them a little bit and desingularize them by gluing in singly periodic Scherk surfaces. The concrete and very symmetric example above was known before that.

The only requirement on the Scherk surfaces is that they have the same translational period and share a horizontal reflectional symmetry plane to ground them. But nothing prevents us from shifting one of the Scherk surfaces by a half-period, like up above. To make the image, I assumed another reflectional symmetry at a vertical plane (roughly parallel to the screen). This still left me with a 1-parameter family, whose existence is truly only guaranteed near the limit that looks like three Scherk surfaces (with one of them shifted). But nothing keeps us from looking at the other surfaces in this family.

Above I have turned it around so that one can appreciate the handles better. What emerges becomes clear when one pushed the parameter further:

A singly periodic Costa surface! There is a similar one constructed by Bastista and Martín where the Costa-necks are rotated by 45 degrees. It then loses its reflectional symmetries but gains straight lines.

## 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 Naming of Things

Studying minimal surfaces has become a bit like hunting for rare new species of plants or animals. Having a newly discovered specimen named after a person might be considered an honor to some, but a confusion to others, because it typically says more about the name giver than the actual specimen.

Let’s, for instance, consider the minimal surface above, which is part of a triply periodic surface. It has genus 5 (after identifying opposite sides by translations), and no name yet. In my book it comes with the code name (1,0|1,1). To explain this code, look at how the symmetry planes cut the piece above into eight pieces, and look just at the fron-top-left. There are, on the piece, four points where the surface normal is vertical: Two of them are on the left side, with normal pointing up and down, and two more in the middle of the picture above with normal pointing up at both points. So the 1 stands for up, the 0 for down, and the vertical bar separates the two boundary components.

You can see this also in the frieze pattern associated to this surface via the Weierstrass representation. The uppr contour repeats left-left-right-right turns, while the bottom just alternates left-right-left-right. Replace left by 1, right by 0, to get 110011001100… and 1010101010…, respectively, and take only the first two digits of each sequence, to get the name code. Below is a larger copy of a deformed version with the same code.

That seemed a clever thing to do, because there are at least four more such codes for genus 5 surfaces (one of them has been named Schoen’s Unnamed Surface 12 before…). Unfortunately, codes can be deceiving, because there is also this surface:

It follows the same up/down pattern of the surface normal and hence gets the same code, but the top edge bends differently. Neither the frieze pattern

nor larger chunks of deformed versions

allow to codify the difference. The mystery can be resolved by looking at what are called the divisors of the Gauss map, and use Abel’s theorem to distinguish them. But not today anymore.

## 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.

## Inside or Outside?

The last minimal surface that made it into Alan Schoen’s NASA report is the F-RD surface. It has genus 6 and looks fairly simple.

A fundamental decision one has to make these days is to choose the side one wants to live on. If, for instance, we decide on the orange side, we will have the impression to live in a network of tetrahedrally or cubically shaped rooms with connecting tunnels at the vertices of each. Not too bad, but, as things stand, we will never know what life on the other side looks like.

Luckily, our imagination is still free, and we can think about the other, green side. What we can hopefully see from the pictures above and below is that the rooms of the green world are all cubical, with tunnels towards the edges of each cube. Alternatively, we can also think of the rooms as rhombic dodecahedra, with tunnels towards the faces. That’s where F-RD got its name from: Faces – Rhombic Dodecahedron.

Incidentally, the conjugate of the F-RD surface is again one of those discussed by Berthold Steßmann, with the polygonal contours having been classified by Arthur Moritz Schoenfließ

A simple deformation of F-RD maintains the reflectional symmetries of a box over a square, but allows to change the height of the box. It turns out that there are two ways to squeeze the box together.

In both cases we get horizontal planes joined by catenoidal necks, but differently placed in each case.

## Alan Schoen’s I6-Surface

After Alan Schoen was fired from NASA at the end of 1969, he moved back to California and continued to experiment with soap film. In October 1970, he used two identical wireframes bent into figure 8 curves consisting of two squares meeting at a vertex. When he dipped them into soapy water at a small distance from each other and pulled them out, he could poke the flat disks between the two figure 8s and create a minimal surface that looks like the top half in the picture above. It extends triply periodically to a surface of genus 5.

Several pages of notes with descriptions of successful experiments made it to Ken Brakke, who used his marvelous Surface Evolver to make 3D models of the surface. It was named I6, because it happened to be the 6th surface on page I of the notes. Hermann Karcher later called it Figure 8 surface. When you move the two figure 8s close to each other, you will get a surface that looks like a periodic arrangement of single periodic Scherk surfaces:

Note that these Scherk surfaces are vertically shifted in a subtle pattern. More interestingly, there is a second, unstable surface you won’t get as a soap film:

What you see here are Translation Invariant Costa Surfaces (or Callahan-Hoffman-Meeks surfaces) we looked at last time. So Alan Schoen’s I6 surface can be considered as a triply periodic version of the Costa surface, which Celso José da Costa discovered  about 10 years later.

Of course you can poke more handles into I6, as you can with the translation invariant Costa surface. Below is an example of genus 7: