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.

How Does This Look Like At The Boundary?

A common recommendation to the layperson who is stranded among a group of mathematicians and doesn’t know what to say is to ask the question above. It will almost always trigger a lengthy and incomprehensible response.

For example, let’s look at the surface below. It constitutes a building block that can be translated around to make larger pieces of the surface. That this works has to do with the small and large horizontal squares. It is similar to Alan Schoen’s Figure 8 surface, but a bit simpler (it only has genus 4)

This surface belongs to a 5-dimensional family about which little is known. The only simple thing I can do with it is to move the squares closer or farther apart. So, how does this look at the boundary? On one hand, when the squares get close, we see little Costa surfaces emerging, as one might expect:

At the other end of infinity, things look complicated, but depending what we focus on, there is a doubly periodic Scherk surface or a doubly periodic Karcher-Scherk surface:

Below are, for the sake of their beauty, the two translation structures associated to two of the Weierstrass 1-forms defining this surface. Next week we will study a close cousin of this surface.

Costaesque (Algorithmic Geometry V)

In 1982, Celso José da Costa wrote down the equations of a minimal surface that most mathematicians at that time thought shouldn’t exist. It shares properties with the plane and catenoid that were supposed to be unique to them. Nothing could be more wrong. Since Costa, many more minimal surfaces in that same elite class have been found.

The curiously complicated way in which the Costa surfaces merges a horizontal plane with a catenoid by avoiding any intersections has become a pattern for similar constructions that is quite aptly called Costaesque.

Amusingly, the same pattern occurs in Alan Schoen’s I6 or Figure Eight surface from 1970.

It can be viewed as a triply periodic aunt of the Costa surface but was conceived as a Plateau solution for two pairs of squares in parallel planes, each of which meet a corner to form a figure eight.

This surface has a particularly simple polygonal approximation by the bent 60 degree rhombi that we have encountered before.

Let’s take 12 such bent rhombi and assemble them into an X-piece that has the two figure 8 squarical holes. A second such X-piece is rotated by 90 degrees and attached to the top to form the polygonal version of Schoen’s I6 fundamental piece.

Alternatively, one can also tile the surface with straight 60 degree rhombi so that it becomes a triply periodic zonohedron.