Last week I explained a really complicated way to get from Scherk’s doubly periodic minimal surface to the helicoid, through a family of Schwarz Diamond surfaces. As was known already to Scherk, this can be done much easier, namely by “shearing” the standard Scherk surface above. I put apostrophes because a simple Euclidean shearing isn’t enough to keep the surface minimal.
Bill Meeks and Hippolyte Lazard-Holly have shown that these are the only embedded doubly periodic minimal surfaces of genus 0 (after taking the quotient by their translational periods). Things get tricky for larger genus.
First of all one needs to distinguish whether the parallel half planes “on top” are parallel to the ones “at the bottom” or not. Today we stick with the case that they are not parallel, and are in fact orthogonal. Then there is just one such surface of genus 1 (I am pretty sure, but I think nobody has written a proof). This was first constructed by Hermann Karcher. It’s pretty clear (and provable) that one can continue like this, creating doubly periodic surfaces with more handles, like the genus 2 example below.
It would be a nice theorem if they all would be unique. But I don’t think so. Below is a picture of a genus 3 surface where the handles are arranged differently.
Proving that this really exists won’t be easy, but interesting, because it would allow one to speculate what will happen if one can shear this surface like the original Scherk surface.