How does one arrive at a surface like this? One has a look at Leonor Ferrer and Francisco Martín‘s paper Minimal surfaces with helicoidal ends, where they dare not to solve a period problem in order to create a very useful deformation, which we will mimic here:
Consider the contour on the left, with straight line segments in the front. Solving the free boundary value problem yields the surface on the right. Ferrer and Martín’s deformation slides the double vertical segments in opposite directions, like so:
The solution to the free boundary problem can be reflected and rotated to produce the translational fundamental cell of a genus 9 embedded triply periodic minimal surface that has the same characteristics as the singly periodic genus one helicoid:
Existence and embeddedness proofs will follow, as well as further surfaces related to this one. This is joint work with Carlos Cardona and GPT 6.1 Codex.
