The number of minimal surfaces bounded by Enneper's wire

TL;DR

This paper investigates the number of minimal surfaces bounded by Enneper's wire, revealing exactly three such surfaces for certain radii, and solves three open problems from Nitche's 1989 list.

Contribution

It establishes the exact count of minimal surfaces bounded by Enneper's wire for specific radii and characterizes their properties, addressing open questions in minimal surface theory.

Findings

Exactly three minimal surfaces bounded by Enneper's wire for R between 1 and √3.

Two additional minimal surfaces are absolute area minima with specific symmetries.

These surfaces depend continuously on R and have positive second variation of area.

Abstract

Enneper's wire, the image of the circle of radius $R$ under Enneper's surface, bounds exactly three minimal surfaces for $R$ between 1 and $\sqrt 3$, and these three surfaces depend continuously on $R$. The other two surfaces (besides Enneper's surface) are absolute minima of area among disk-type surfaces bounded by Enneper's wire. These surfaces each have a unique horizontal tangent plane, whose height can be computed from $R$, and they are invariant under reflections in the planes $x_1=0$ and $x_2 = 0$. These two surfaces have positive second variation of area, and depend continuously on $R$. This result solves three open problems from the list in Nitche's 1989 book. Enneper's wire is the only Jordan curve $\Gamma$ bounding more than one minimal surface for which a specific bound on the number of minimal surfaces bounded by $\Gamma$ is known.