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Parametrizations of minimal timelike surfaces in the four-dimensional pseudo-Euclidean space of index two

Katsuhiro Moriya

Abstract

We construct representation formulas for local null curves in the four-dimensional pseudo-Euclidean space of index two and derive corresponding parametrizations for local minimal timelike surfaces without integration. As a special case of the representation formula, we construct a representation formula for local null curves in the three-dimensional pseudo-Euclidean space of index one that involves integration. Our results provide examples of minimal timelike surfaces.

Parametrizations of minimal timelike surfaces in the four-dimensional pseudo-Euclidean space of index two

Abstract

We construct representation formulas for local null curves in the four-dimensional pseudo-Euclidean space of index two and derive corresponding parametrizations for local minimal timelike surfaces without integration. As a special case of the representation formula, we construct a representation formula for local null curves in the three-dimensional pseudo-Euclidean space of index one that involves integration. Our results provide examples of minimal timelike surfaces.
Paper Structure (4 sections, 6 theorems, 42 equations, 6 figures)

This paper contains 4 sections, 6 theorems, 42 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Null curves and minimal timelike surfaces
  3. Proof of theorems
  4. Examples

Key Result

Theorem 1

Let $\langle\enskip,\enskip\rangle$ be the inner product of $\mathbb{E}^4_2$ and $z(x)$ and $w(y)$ be two null curves defined on open intervals $I_1$ and $I_2$ respectively in $\mathbb{E}^4_2$. If $\langle z(x), w(y)\rangle \neq 0$ for $(x, y) \in I_1 \times I_2$, then defines a Lorentzian minimal surface in $\mathbb{E}^4_2$. Conversely, locally every Lorentzian minimal surface in $\mathbb{E}^4

Figures (6)

  • Figure 1: $\alpha_4$
  • Figure 2: $\tilde{\alpha}_4$
  • Figure 3: $f_4$
  • Figure 4: $\alpha_5$
  • Figure 5: $\tilde{\alpha}_5$
  • ...and 1 more figures

Theorems & Definitions (17)

  • Theorem 1: Chen zbMATH05568272
  • Theorem 2
  • Corollary 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Definition 1
  • Definition 2
  • Lemma 3
  • proof
  • proof : Proof of Theorem \ref{['thm:repnc']}
  • ...and 7 more