A Tale of Two Squares

Today, we are pushing things a little. We begin with a minimal heptagon with two horizontal straight edges, and the other five edges lying in symmetry planes parallel to the vertical coordinate planes. We also want this heptagon arranged so that extension by reflections/rotations creates an embedded triply periodic surface.

Above is a first example, showing a translational fundamental piece of a triply periodic surface of genus 6 that is tiled by eight such heptagons. We see that the horizontal straight segments are replicated into edges of squares, so that all surfaces we will look at today will sit one way or another between two squares.

A way to encode the possibilities for such heptagons is through the picture above. It represents the divisor of the square of the Gauss map on the quotient torus under the order 2 rotation about a vertical axis. Less technically, the grayed rectangle represents the heptagon, with the two vertical blue edges corresponding to the horizontal straight segments. The red dots are the seven heptagon vertices, and the symbols 0 and ∞ tell whether the Gauss map points up or down.

There are 12 possibilities that lead to different surface candidates. For each of these, a 3-dimensional period problem needs to be solved, leading in the case of success to 1-parameter families. In 8 of these cases, I was able to solve this period problem numerically. No existence or non-existence proof is known in any of these cases at this point.

There are several motivations behind this exploration. For one, I would like to know what limits can occur, and how the limit surfaces are combined. This suggests possibilities for general gluing constructions that would establish the existence of these and many other surfaces, Secondly, some of the limits are surfaces that are either only very difficult to obtain (like the Callahan-Hoffman-Meeks surface of higher genus that emerges above), or have also only been established numerically (like the exotic doubly periodic Karcher-Scherk surface of genus 3 below).

Finally, there is always the chance that an exhaustive enumeration leads to examples with unexpected properties, like the surface below, that in a diagonal view reveals an appealing tunnel system that indicates that we are probably having a special pair of skeletal graphs in front of us.

Resources

Main Page for all 8 surfaces

To Be or Not To Be

Existence is an interesting concept. Even in the most rigorous of sciences, it is not free of dispute.

Above is an image of the Horgan surface, a minimal surface that does not exist, at least not with all the glorious property it appears to have. There are less convincing attempts to convey the existence of a non-existing object, like the catenoid with a handle:

In Mathematics, pictures provide evidence and enjoyment. Existence or non-existence requires proof, and provides certainty. Still, the membrane between to be and not to be can be very thin, as in the example at the top.

The pattern made by the catenoidal necks in the top image – two up and two down, at the corners of a square – does actually occur. One of the first examples I have seen is the Callahan-Hoffman-Meeks surface with an added handle as above. Its existence is not so easy to prove – does it therefore exist a little less?

The triply periodic surface above is the simplest version of the Catenoid-Square pattern I know of, and much more recent. So, in a sense, all this research is to an unconfessed extent concerned with probing the border of existence, from either side.

Meeks vs Non-Meeks

One of the most active areas of minimal surface research concerns embedded triply periodic minimal surfaces. They are periodic with respect to a 3-dimensional lattice of translations, and roughly classified by the genus of the quotient surface. The simplest case is that of genus 3, and even that is far from being understood. The earliest set of examples is due to Hermann Amandus Schwarz, and he already understood well that they come in parameter families. Below you can see the P surface, the D-surface, and members of the H and CLP-families.

So, how many are there? A casual parameter count tells that if such a surface is reasonably generic, it should come in a smooth 5-parameter family, not counting rotations, translations, and scalings. Five is a fairly large number. The space of possible lattices is also 5-dimensional, and in the best of all possible worlds, there would be precisely one surface in every lattice. Unfortunately, it is not even true that small surface deformations are in 1:1 correspondence to small lattice deformations, and things are dramatically more complicated.

Alan Schoen added to the Schwarz examples the Gyroid (left) a member of the associate family of the P surface, and Sven Lidin the Lidinoid (right), an associate member of the associate family of a particular H surface.

Key in their construction was an understanding of the Gauss map: This is a map of degree 2 to the sphere, branched over 8 points. Schwarz had already realized their significance; above is a reproduction of Plate VI from his 1871 monograph. Figures 43 and 44 show the branched values of the oP and tP deformations of the P surface, figure 45 of the CLP surfaces, figure 46 represents a special case of the rPD family, and figure 47 the H family.

Bill Meeks realized in 1970 that for any choice of 8 branched values of the sphere that is antipodally symmetric, there are two (conjugate) embedded triply periodic minimal surfaces with these branched values. This Meeks-family covers all then known examples, except for the H-surfaces, the Gyroid, and the Lidinoid.

Since then, progress has been made in small steps. Above are two new Non-Meeks surfaces (called and oH) due to Hao Chen and myself that can be deformed into different Meeks surfaces. It is not clear at this point whether they are related. Hao Chen has also recently proven that the Lidinoid and Gyroid belong to a 1-parameter family. I am sure this year will see more progress.

Here are two immediate questions that I would love to have an answer for:

  • For what triply periodic minimal surfaces are there embedded surfaces in the associate family that are not conjugates? For genus 3, the only examples so far are the Lidinoid and the Gyroid.
  • We know that the H-surfaces belong to a 5-dimensional family (work of Martin Traizet) that intersects the Meeks family (work of Hao Chen and Matthias Weber). Are the Gyroid/Lidinoid part of this family, or do they belong to a third, separate family?

Classified

Mathematicians like to classify things. Among the complete, embedded minimal surfaces of finite total curvature in Euclidean space or space forms, this has been accomplished in most of the simplest possible cases. Let’s summarize:

In Euclidean space there are only two surfaces of genus 0 in this class: The plane and the catenoid. This is a consequence of the López-Ros theorem, proven in 1991 by Francisco López and Antonio Ros.

For translation invariant surfaces (or equivalently, minimal surfaces in ℝ³ divided by a translation, the surfaces of genus 0 (in the quotient) are the general Karcher-Scherk saddle towers. These surfaces have an even number 2n of annular ends and 2n-3 free parameters with which they can flap their ends. This has been proven by Joaquín Pérez and Martin Traizet in 2007.

If you want planar ends, the lowest possible genus is 1, and Bill Meeks, Joaquín Pérez and Antonio Ros have shown in 1998 that the Riemann minimal surfaces are the only ones.

The situation is not yet resolved for the screw motion invariant surfaces. Conjecturally, these surfaces should be Hermann Karcher’s twisted saddle towers.

The case of doubly periodic surfaces of genus 0 has been settled by Bill Meeks and Hippolyte Lazard-Holly in 2001. These surfaces have non-parallel top and bottom ends.

Doubly periodic surfaces with parallel top and bottom ends can only occur in genus one and higher. Again, the genus one case has been classified: Joaquín Pérez, Magdalena Rodríguez and Martin Traizet have shown in 2005 that these are the KMR surfaces.

The main open question is that of a classification of triply periodic minimal surfaces of genus 3. To describe the state of the art will deserve several dedicated blog posts.

Likewise, I will outline in future posts the state of the art in the next difficult (open) cases.

Finally, I should mention that there are other, equally valid viewpoints for classification, using different assumptions about the topology.

Resources

F.J. López and A. Ros, On embedded complete minimal surfaces of genus zero, Journal of Differential Geometry 33 (199), 293–300

J. Pérez, M. Traizet: The Classification of Singly Periodic Minimal Surfaces with Genus Zero and Scherk-Type Ends, Transactions of the American Mathematical Society
359 (2007), 965-990.

W. H. Meeks III, J. Pérez, A. Ros: Uniqueness of the Riemann minimal examples, Invent. Math. 133 (1998),107–132

H. Lazard-Holly and W. Meeks: Classification of doubly-periodic minimal surfaces of genus zero, Invent. math. 143 (2001), 1–27.

Joaquín Pérez, M. Magdalena Rodríguez, and Martin Traizet, The classification of doubly periodic minimal tori with parallel ends, J. Differential Geom. 69 (2005), 523–577