Two New Embedded Triply Periodic Minimal Surfaces of Genus 4

TL;DR

This paper introduces two new families of embedded triply periodic minimal surfaces of genus 4, expanding the known catalog and demonstrating new geometric configurations with specific boundary and symmetry properties.

Contribution

The authors discover and prove the existence of two new embedded triply periodic minimal surfaces of genus 4, including their boundary structures and relation to known surfaces like the Costa surface.

Findings

Two new families of minimal surfaces added to the known list.

Surfaces can be tiled by minimal pentagons with specific boundary segments.

One family limits to the Costa surface, showing a new occurrence for triply periodic minimal surfaces.

Abstract

We add two new 1-parameter families to the short list of known embedded triply periodic minimal surfaces of genus 4 in $\mathbb{R}^3$. Both surfaces can be tiled by minimal pentagons with two straight segments and three planar symmetry curves as boundary. In one case (which has the appearance of the CLP surface of Schwarz with an added handle) the two straight segments are parallel, while they are orthogonal in the second case. The second family has as one limit the Costa surface, showing that this limit can occur for triply periodic minimal surfaces. For the existence proof we solve the 1-dimensional period problem through a combination of an asymptotic analysis of the period integrals and geometric methods. v2: corrected embeddedness proof. Minor further improvements.