The Hexagonal or H-surface is one of the four triply periodic minimal surfaces discussed by Hermann Amandus Schwarz in his Preisschrift from 1867. It comes in a natural 1-parameter family with no easily distinguishable representative.
Like his other examples, this surface has genus 3 when divided by its translational symmetries. It is in may ways analogous to his P-surface. For instance, it also solves a Plateau problem for two equilateral triangles in parallel planes.
Here the triangles are translates of each other, while for the P-surface they are rotated against each other by 180 degrees. The 1-parameter family that maintains all symmetries has two extreme cases: In one limit, one obtains parallel planes joined by catenoidal neck, in the other translation invariant Scherk surfaces with 6 ends.
There is, however, a dramatic difference: While the P-surface belongs to an explicit, 5-dimensional family of embedded triply periodic minimal surfaces where the 8 branched values of the Gauss map come in pairs of antipodal points (the Meeks family), this is not the case for the H-family, where these branched values are at the vertices of a triangular prism and its triangle centers. It has only recently been shown that this surface can be further deformed into surfaces that also belong to the Meeks family.
Particularly attractive is the translation structure associated to G dh above. If you identify all parallel edges, you get a genus 3 surface with a single cone point of cone angle 10π.
H.A. Schwarz: Bestimmung einer speciellen Minimalfläche, Eine von der Königlichen Akademie der Wissenschaften zu Berlin am 4. Juli 1867 gokrönte Preisschrift. Nebst einem Nachtrage und einem Anhange.
- H. Chen’s tD family
- The Lidinoid