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

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.