This surface is one of several triply periodic minimal surfaces of genus 5 that have vertical symmetry planes over a square grid and diagonal horizontal lines.

It exists as a 1-parameter family, limiting in noded planes and in doubly periodic Karcher-Scherk surfaces.

The divisor of the square of the Gauss map is given below. The red crosses indicate horizontal normal symmetry lines. Given all symmetries, the period problem is 1-dimensional and can be solved using an elementary extremal length argument.

It is amusing to note that the configuration of catenoidal necks in the nodal limit is that of the Horgan surface, which does not exist as a finite total curvature surface. Even more amusing is that the existence of this surface follows from a picture proof. One just has to know how to read the picture:

Above are the flat structures of G dh and 1/G dh on the shaded rectangle above. To close the period, one needs a hexagon as shown on the left so that both horizontal edges have the same length (say 1) and so that it has a conformal 180º rotation preserving the vertices (corresponding to the normal symmetry of the surface, fixing the small crosses in the divisor picture). The latter is the case if and only if the two cycles marked Γ₁ and Γ₂ have the same extremal length. This can be achieved by choosing the long vertical edge to be of some fixed height, and by varying the length of the short vertical edge. If that edge is short, ext Γ₁ will be smaller than ext Γ₂, while when it is long so that the adjacent diagonal edge becomes short, ext Γ₁ will be larger than ext Γ₂, using elementary properties of extremal length.

##### Resources

Mathematica Notebook and CDF

PoVRay Sources