For a translation invariant surfaces with 2k annular “Scherk” ends, one expects 2k-3 parameters that control the angles between the ends. Here is a 1-parameter subfamily (with some symmetry) of 6-ended surfaces of genus 1, deforming the most symmetric toroidal Karcher-Scherk surfaces.

We keep two ends horizontal (using the figure as a reference), and reflectional symmetries at the vertical coordinate planes. Both limits are then quite interesting:

It appears that above the catenoidal necks between the top ends will bubble off and the limit will be a 6-ended Scherk surface with parallel top ends. That surface, however, does not exist. More likely is that that the bottom piece will also become wider and decompose into two singly periodic Scherk surfaces.

The other limit shows three planes joined by a pattern of catenoidal nodes. This is the prototype of a very general construction of Kevin Li who in his thesis regenerates singly periodic surfaces of arbitrary genus from such nodal planes by bending the planes.

Resources

K. Li: Singly-Periodic Minimal Surfaces with Scherk Ends Near Parallel Planes, PhD Thesis, Bloomington 2012

Mathematica Notebook

PoVRay Sources