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

TL;DR

This paper introduces a new method for constructing harmonic immersions in three-dimensional space, proving its universality and analyzing specific rotational cases connecting coaxial circles.

Contribution

It develops an Enneper-type representation for harmonic immersions in 3, showing that all such immersions can be generated by this approach.

Findings

Any harmonic immersion in 3 can be obtained using the proposed method.

The paper determines the number of non-planar rotational harmonic immersions connecting two coaxial circles.

The method provides a complete characterization of harmonic immersions in 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 $l > 0$.