Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Minimal Timelike Surfaces in the Lorentz-Minkowski 3-space and Their Canonical Parameters

Ognian Kassabov, Velichka Milousheva

Abstract

We study minimal timelike surfaces in $\mathbb R^3_1$ using a special Weierstrass-type formula in terms of holomorphic functions defined in the algebra of the double (split-complex) numbers. We present a method of obtaining an equation of a minimal timelike surface in terms of canonical parameters, which play a role similar to the role of the natural parameters of curves in $\mathbb R^3$. Having one holomorphic function that generates a minimal timelike surface, we find all holomorphic functions that generate the same surface. In this way we give a correspondence between a minimal timelike surface and a class of holomorphic functions. As an application, we prove that the Enneper surfaces are the only minimal timelike surfaces in $\mathbb R^3_1$ with polynomial parametrization of degree 3 in isothermal parameters.

Minimal Timelike Surfaces in the Lorentz-Minkowski 3-space and Their Canonical Parameters

Abstract

We study minimal timelike surfaces in using a special Weierstrass-type formula in terms of holomorphic functions defined in the algebra of the double (split-complex) numbers. We present a method of obtaining an equation of a minimal timelike surface in terms of canonical parameters, which play a role similar to the role of the natural parameters of curves in . Having one holomorphic function that generates a minimal timelike surface, we find all holomorphic functions that generate the same surface. In this way we give a correspondence between a minimal timelike surface and a class of holomorphic functions. As an application, we prove that the Enneper surfaces are the only minimal timelike surfaces in with polynomial parametrization of degree 3 in isothermal parameters.
Paper Structure (4 sections, 5 theorems, 52 equations, 1 figure)

This paper contains 4 sections, 5 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Transformation of the isothermal parameters to canonical ones
  4. Holomorphic functions generating a minimal timelike surface

Key Result

Theorem 3.1

Let the minimal timelike surface $S$ be defined by the real or "imaginary" part of (eq:3.1). Any solution to differential equation (eq:3.3) defines a transformation of the isothermal parameters of $S$ to canonical ones. Moreover, the function $\tilde{g}(w)$ that defines $S$ via formula (eq:2.2) is g

Figures (1)

  • Figure 1: Enneper surfaces

Theorems & Definitions (6)

  • Theorem 3.1
  • Corollary 3.2
  • Proposition 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Remark 4.1