An orthorhombic deformation family of Schwarz' H surfaces

TL;DR

This paper introduces a new 2-parameter family of triply periodic minimal surfaces that includes Schwarz's H surfaces and connects them to Meeks' family, revealing continuous deformations between these classes.

Contribution

It constructs a novel 2-parameter family of embedded TPMS of genus three that encompasses H surfaces and links them to Meeks' family, expanding understanding of minimal surface deformations.

Findings

H surfaces can be deformed into Meeks surfaces within the TPMS space.

A 2-parameter family of embedded TPMS of genus three is constructed.

H surfaces are shown to be part of a continuous deformation family.

Abstract

The classical H surfaces of H. A. Schwarz form a 1-parameter family of triply periodic minimal surfaces (TPMS) that are usually described as close relatives to his more famous P surface. However, a crucial distinction between these surfaces is that the P surface belongs to a 5-dimensional smooth family of embedded TPMS of genus three discovered by W. Meeks, while the H surfaces are among the few known examples outside this family. We construct a 2-parameter family of embedded TPMS of genus three that contains the H family and meets the Meeks family. In particular, we prove that H surfaces can be deformed continuously within the space of TPMS of genus three into Meeks surfaces.