Harmonic Enneper Immersion in ${\mathbb{R}}^3$

Abstract

We present a method for constructing harmonic immersions in ${\mathbb{R}}^3$, known as the Enneper-type representation. We also prove that any harmonic immersion in ${\mathbb{R}}^3$ can be obtained using this approach. Furthermore, we determine the number of non-planar rotational harmonic immersions in ${\mathbb{R}}^3$ that connect two coaxial circles in parallel planes, where both circles have the same radius $r>0$ and are separated by a distance $l>0$.