The periodic Plateau problem and its application

TL;DR

This paper proves the existence of certain periodic minimal surfaces spanning specific curves and demonstrates their applications in constructing minimal surfaces within tetrahedra and Platonic solids.

Contribution

It introduces new existence results for periodic minimal surfaces spanning noncompact curves and applies these to minimal surface constructions in polyhedral domains.

Findings

Existence of noncompact simply connected periodic minimal surfaces spanning given curves.

Construction of four embedded minimal annuli in tetrahedra with dihedral angles ≤90°.

Identification of five types of free boundary minimal surfaces in Platonic solids.

Abstract

Given a noncompact disconnected complete periodic curve $\Gamma$ with no self intersection in $\mathbb R^3$, it is proved that there exists a noncompact simply connected periodic minimal surface spanning $\Gamma$. As an application it is shown that for any tetrahedron $T$ with dihedral angles $\leq90^\circ$ there exist four embedded minimal annuli in $T$ which are perpendicular to $\partial T$ along their boundary. It is also proved that every Platonic solid of $\mathbb R^3$ contains five types of free boundary embedded minimal surfaces of genus zero.