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Relation between the minimal Lorentz surfaces in $\mathbb R^4_2$ and $\mathbb R^3_1$

Krasimir Kanchev, Ognian Kassabov, Velichka Milousheva

Abstract

In this paper we give Weierstrass-type representation formulas for the null curves and for the minimal Lorentz surfaces in the Minkowski 3-space $\mathbb R^3_1$ using real-valued functions. Applying the Weierstrass-type representations for the null curves, we find a correspondence between the null curves in $\mathbb R^4_2$ and the pairs of null curves in $\mathbb R^3_1$. Based on this correspondence, we obtain a relation between the minimal Lorentz surfaces in $\mathbb R^4_2$ and the pairs of minimal Lorentz surfaces in $\mathbb R^3_1$.

Relation between the minimal Lorentz surfaces in $\mathbb R^4_2$ and $\mathbb R^3_1$

Abstract

In this paper we give Weierstrass-type representation formulas for the null curves and for the minimal Lorentz surfaces in the Minkowski 3-space using real-valued functions. Applying the Weierstrass-type representations for the null curves, we find a correspondence between the null curves in and the pairs of null curves in . Based on this correspondence, we obtain a relation between the minimal Lorentz surfaces in and the pairs of minimal Lorentz surfaces in .
Paper Structure (6 sections, 17 theorems, 54 equations)

This paper contains 6 sections, 17 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weierstrass-type representation for null curves in $\mathbb R^4_2$ and $\mathbb R^3_1$
  4. Weierstrass-type representation for minimal Lorentz surfaces in $\mathbb R^4_2$ and $\mathbb R^3_1$
  5. Relation between the null curves in $\mathbb R^4_2$ and $\mathbb R^3_1$
  6. Relation between the minimal Lorentz surfaces in $\mathbb R^4_2$ and $\mathbb R^3_1$

Key Result

Proposition 2.1

Kanchev2020 A minimal Lorentz surface $\mathcal{M}$ in $\mathbb R^4_2$ is of general type if and only if $K^2-\varkappa^2\neq 0$ . Moreover, $\mathcal{M}$ is of first or second type (resp. of third type) if and only if $K^2-\varkappa^2>0$ (resp. $K^2-\varkappa^2<0$ ).

Theorems & Definitions (30)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1
  • Theorem 3.1
  • Remark 3.1
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Proposition 3.4
  • Theorem 3.5
  • ...and 20 more