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

Priyank Vasu

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 $l > 0$.

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

Abstract

We present a method for constructing harmonic immersions in , known as the Enneper-type representation. We also prove that any harmonic immersion in can be obtained using this approach. Furthermore, we determine the number of non-planar rotational harmonic immersions in that connect two coaxial circles in parallel planes, where both circles have the same radius and are separated by a distance .
Paper Structure (5 sections, 8 theorems, 35 equations)

This paper contains 5 sections, 8 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Application
  5. Acknowledgement

Key Result

Theorem 2.3

AlarconLopez2013 Let $X: \Omega \rightarrow \mathbb{R}^3$ be a harmonic immersion. Then, satisfies the following conditions: Conversely, if $\Omega$ is a simply connected domain and $\phi_j;\, j=1,2,3,$ are holomorphic functions satisfying the conditions above, then the map is a well-defined harmonic immersion (here, $z_{0}$ is an arbitrary fixed point of $\Omega$).

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.5: Kalaj
  • Definition 2.6
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • ...and 8 more