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 $l > 0$.