Construction of embedded periodic surfaces in $\mathbb{R}^n$

TL;DR

This paper constructs new embedded minimal surfaces in higher-dimensional Euclidean spaces that are periodic and characterized by specific Jordan curves, expanding the understanding of minimal surface embeddings in codimension n-2.

Contribution

It introduces a method to construct embedded n-periodic minimal surfaces in , particularly for codimension -2, using Jordan curves and Schwarz reflection techniques.

Findings

Constructed embedded n-periodic minimal surfaces in 

Characterized Jordan curves leading to embedded surfaces

Identified exactly five such curves for n=4

Abstract

We construct embedded minimal surfaces which are $n$-periodic in $\mathbb{R}^n$. They are new for codimension $n-2\ge 2$. We start with a Jordan curve of edges of the $n$-dimensional cube. It bounds a Plateau minimal disk which Schwarz reflection extends to a complete minimal surface. Studying the group of Schwarz reflections, we can characterize those Jordan curves for which the complete surface is embedded. For example, for $n=4$ exactly five such Jordan curves generate embedded surfaces. Our results apply to surface classes other than minimal as well, for instance polygonal surfaces.