After Hermann Karcher added a handle to Scherk’s doubly periodic surface, it was not difficult to come up with the Enneper-Weierstraß representation of higher genus versions and to solve the period problem numerically. Below you see the surfaces of genus 2 and 4.
The period problem is remarkably similar to that of the Chen-Gackstatter surfaces of higher genus. One can see this in the similarity of the flat structures associated to G dh and 1/G dh, as shown below for genus 3 together with the surface. Again, the period condition requires that the zigzags fit and the half-infinite strips have the same width.
In fact, Matthias Weber and Michael Wolf recycled their proof for the existence of higher genus Chen-Gackstatter surfaces to establish the existence of these surface here as well.
There are three things I’d like to know about them: First, what does the limit look like when the genus goes to infinity? My guess is that one gets two copies of the ordinary doubly periodic Scherk surfaces, shifted by a diagonal half period, and with singular vertical lines.
Secondly, can they be deformed like the Scherk surface so that the ends become non-orthogonal, and if yes, do they converge to translation invariant genus g helicoids?
Thirdly, I’d like to know whether there are exotic surfaces. There is a numerical candidate above of genus 3, together with the flat structures, that shows that one might be able to cut corners at both sides of the strip.
The existence of such a surface would indicate the possibility that there are (as limits of shearing the ends) also exotic translation invariant helicoids of genus 3, and maybe, also an exotic genus 3 helicoid.
M. Weber, M. Wolf: Handle Addition for doubly-periodic Scherk Surfaces
Mathematica notebooks for genus 2, 3, 3 exotic, 4