The chiral gyrating H'-T surface family: construction from the dual qtz--qzd nets and existence proof using a toroidal Weierstrass method

Abstract

This paper provides a construction and existence proof for a 1-parameter family of chiral unbalanced triply-periodic minimal surfaces of genus 4. We name these {\textit{gyrating H'-T} surfaces, because they are related to Schoen's H'-T surfaces in a similar way as the Gyroid is to the Primitive surface. Their chirality is manifest in a screw symmetry of order six. The two labyrinthine domains on either side of the surface are not congruent, rather one representing the quartz net (\texttt{qtz}) and the other one the dual of the quartz net (\texttt{qzd}). The family tends to the Scherk saddle tower in one limit and to the doubly periodic Scherk surface in the other. The motivation for the construction was to construct a chiral tunable unbalanced surface family, originally as a template for photonic materials. The numeric construction is based on reverse-engineering of the tubular surface of two suitably chosen dual nets, using the \textit{Surface Evolver}} to minimize area or curvature variations. The existence is proved using Weierstrass parametrizations defined on the branched torus.