Work on triply periodic minimal surfaces

Question

I have seen in some engineering departments that they manufacture models of periodic minimal forms (characterised by equal and opposite curvature at every points on the surface). In pure mathematics, they are known as triply periodic minimal surfaces.

If I understand rightly, these have been observed experimentally in crystallography and polymer chemistry but I assume they must have been studied in differential geometry as well. The Wikipedia page mentions the classification of these surfaces as an open problem: has there been any recent progress on this? The physics literature also mentions the possibility of constructing minimal surfaces with the properties of a quasicrystal (ie. minimal surfaces with a quasicrystalline order). Again, has there been any further geometric work on this construction?

Answer 1

This is an active research topic. I'm currently working on the front line towards a classification of TPMSs of genus 3 (TPMSg3s). My collaborators include Weber and Traizet. I also know a Japanese team working on the moduli space.

Recent progress include surprising discoveries of new examples:

• https://arxiv.org/abs/1804.01442 (j/w Weber, to appear in Trans. AMS)
• https://arxiv.org/abs/1807.10631 (j/w Weber, to appear in Trans. AMS)

and rigorous proof of some deformations of the gyroid.

• https://arxiv.org/abs/1901.04006

These examples are important because they reveal concrete singularities in the moduli space of TPMSg3s. So their existence basically proves that the classification is very complicated.

However, an ultimate classification of TPMSg3s is not completely hopeless. We are making big progress on the boundary of the moduli space, and we expect to see more new examples in the near future. I estimate about 5 years of hard works before we can finally evaluate the feasibility of the classification. I'm optimistic.

We also constructed uncountably many examples of non-periodic minimal surfaces that of great interest for material scientists and crystallographers.

• https://arxiv.org/abs/1908.06276 (j/w Traizet, to appear in SIAM J. Math. Anal.)

Some of them can be regarded as "quasi-periodic", but not "quasi-crystallographic". I have been thinking about quasi-crystallographic minimal surfaces, but currently have no progress at all.

My involvement in the topic is mostly motivated by physics, and I'm actively collaborating with many experimental teams.

Answer 2

Matthias Weber has been active in this area (classifying TPMSs) recently. For example. I suggest you contact him.