Minimal reflection surfaces in $\mathbb S^3.$ Combinatorics of curvature lines and minimal surfaces based on fundamental pentagons

TL;DR

This paper explores the construction and properties of compact minimal surfaces in the 3-sphere formed by reflections of fundamental pentagons, analyzing their curvature line combinatorics and area characteristics.

Contribution

It introduces new minimal reflection surfaces based on pentagons and studies their combinatorial and geometric properties, expanding the class of known minimal surfaces in $\

Findings

Constructed new minimal surfaces from pentagons in $\

analyzed the curvature line combinatorics of these surfaces

discussed the area properties of the constructed minimal surfaces

Abstract

We study compact minimal surfaces in the 3-sphere which are constructed by successive reflections from a minimal $n$-gon -- so-called minimal reflection surfaces. The minimal $n$-gon solves a free boundary problem in a fundamental piece of the respective reflection group. We investigate the combinatorics of the curvature lines of reflection surfaces, and construct new examples of minimal reflection surfaces based on pentagons. We end the paper by discussing the area of these minimal surfaces.