An early gluing construction of Martin Traizet allowed gluing families of singly periodic Scherk surfaces together to create triply periodic minimal surfaces. His original construction allowed for Scherk surfaces of different frequencies. But the phases had to be anchored, for instance by requiring a horizontal symmetry plane.
Examples of genus 4 include Wei’s surface (to the right) and a version with straight lines on the left. Both these surfaces pose only a 1-dimensional period problem and have simple existence proofs.
Several years ago, I found a way to numerically and minimally connect the two surfaces, and I made a short movie that suggested that it was possible to slide the two Scherk surfaces independently.
Nobody I showed it to believed that this was possible. I am reviving this because there has been recent progress. We now have an AI-assisted existence proof (based on interval arithmetic) and embeddedness proof. The embeddedness is based on deforming these surfaces to Traizet nodal limits on tori, where the slide deformation is already visible.
There are many open problems:
- The monodromy acts non-trivially on the homology of the quotient surface, but trivially on the homology of the quotient lattice torus. Simply put, the handles really move, but the lattice only wobbles. Can one adjust the parameters so that the lattice stays rigid?
- Are there other examples of non-simply connected moduli spaces of embedded minimal surfaces?
- The space of embedded triply periodic minimal surfaces of genus 4 is now well studied. Is it connected? How many (irreducible) components does it have?
This is a collaboration with Carlos Cardona and Codex 6.1. We will share all supporting documents shortly. Our goal is not to solve famous conjectures, but to open up new territories for more research. We don’t think that AI-supported proofs are the end, but rather a beginning for research that leads to new understanding.
