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 oΔ 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?