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Invariant $λ$-translators in Lorentz-Minkowski space

Antonio Bueno, Irene Ortiz

Abstract

Given $λ\in\mathbb{R}$ and $\textbf{v}\in\mathbb{L}^3$, a $λ$-translator with velocity $\textbf{v}$ is an immersed surface in $\mathbb{L}^3$ whose mean curvature satisfies $H=\langle N,\textbf{v}\rangle+λ$, where $N$ is a unit normal vector field. When $λ=0$, we fall into the class of translating solitons of the mean curvature flow. In this paper we study $λ$-translators in $\mathbb{L}^3$ that are invariant under a 1-parameter group of translations and rotations. The former are cylindrical surfaces and explicit parametrizations are found, distinguishing on the causality of both the ruling direction and the $λ$-translators. In the case of rotational $λ$-translators we distinguish between spacelike and timelike rotations and exhibit the qualitative properties of rotational $λ$-translators by analyzing the non-linear autonomous system fulfilled by the coordinate functions of the generating curves.

Invariant $λ$-translators in Lorentz-Minkowski space

Abstract

Given and , a -translator with velocity is an immersed surface in whose mean curvature satisfies , where is a unit normal vector field. When , we fall into the class of translating solitons of the mean curvature flow. In this paper we study -translators in that are invariant under a 1-parameter group of translations and rotations. The former are cylindrical surfaces and explicit parametrizations are found, distinguishing on the causality of both the ruling direction and the -translators. In the case of rotational -translators we distinguish between spacelike and timelike rotations and exhibit the qualitative properties of rotational -translators by analyzing the non-linear autonomous system fulfilled by the coordinate functions of the generating curves.
Paper Structure (11 sections, 5 theorems, 28 equations, 7 figures)

This paper contains 11 sections, 5 theorems, 28 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cylindrical $\lambda$-translators in $\mathbb{L}^3$
  4. Rotational $\lambda$-translators
  5. Existence of radial $\lambda$-translators
  6. Rotational $\lambda$-translators about a timelike axis
  7. Spacelike $\lambda$-translators
  8. Timelike $\lambda$-translators
  9. Rotational $\lambda$-translators about a spacelike axis
  10. The spacelike case
  11. The timelike case

Key Result

Theorem 3.1

A spacelike cylindrical $\lambda$-translator along a spacelike direction can be parametrized, up to a change of coordinates, as follows:

Figures (7)

  • Figure 4: Left: the base curve of a cylindrical timelike $\lambda$-translator for $\lambda=2$. Right: for $\lambda=1$. In both cases, $v_3=1$.
  • Figure 5: Left: the phase plane of a timelike rotational $\lambda$-translator about a timelike axis, and the possible orbits for $\lambda<1$. Right: the profile curves of the rotational $\lambda$-translators corresponding to each orbit. Here, $\lambda=1/2$.
  • Figure 6: Left: the phase plane of a timelike rotational $\lambda$-translator about a timelike axis, and the possible orbits for $\lambda=1$. Right: the profile curves of the rotational $\lambda$-translators corresponding to each orbit.
  • Figure 7: Left: the phase plane of a spacelike rotational $\lambda$-translator about a timelike axis, and the possible orbits for $\lambda>1$. Right: the profile curves of the rotational $\lambda$-translators corresponding to each orbit. Here, $\lambda=2$.
  • Figure 8: Left: the phase plane of a timelike rotational $\lambda$-translator about a timelike axis, and the possible orbits. Right: the profile curves of the rotational $\lambda$-translators corresponding to each orbit. Here, $\lambda=1/2$.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Definition 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Proposition 4.1
  • Theorem 4.2
  • Corollary 4.3