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Framed null curves and timelike surfaces via Lorentzian harmonic maps into de-Sitter 2-space

Shintaro Akamine, Hirotaka Kiyohara

Abstract

We construct a class of Lorentzian harmonic maps into the de-Sitter $2$-space satisfying the eigenvalue equation $\Box N=2H^2N$ for the d'Alambert operator $\Box$ and a non-zero constant $H$ from framed null curves. We also investigate two classes of timelike surfaces associated with these Lorentzian harmonic maps: the first one is timelike surfaces with constant mean curvature $H$ in Lorentz-Minkowski $3$-space and the second one is timelike minimal surfaces in the three-dimensional Lorentzian Heisenberg group $\operatorname{Nil}_3(H)$. In particular, we characterize some properties of singularities on timelike minimal surfaces in $\operatorname{Nil}_3(H)$ via an invariant of framed null curves.

Framed null curves and timelike surfaces via Lorentzian harmonic maps into de-Sitter 2-space

Abstract

We construct a class of Lorentzian harmonic maps into the de-Sitter -space satisfying the eigenvalue equation for the d'Alambert operator and a non-zero constant from framed null curves. We also investigate two classes of timelike surfaces associated with these Lorentzian harmonic maps: the first one is timelike surfaces with constant mean curvature in Lorentz-Minkowski -space and the second one is timelike minimal surfaces in the three-dimensional Lorentzian Heisenberg group . In particular, we characterize some properties of singularities on timelike minimal surfaces in via an invariant of framed null curves.
Paper Structure (9 sections, 14 theorems, 73 equations, 3 figures)

This paper contains 9 sections, 14 theorems, 73 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Minimal surfaces in $\operatorname{Nil}_3(\tau)$ with singularities
  3. Singularities on fronts and frontals
  4. Criteria for singularities via the duality between $f$ and $f_L$
  5. B-scroll type surfaces in $\operatorname{Nil}_3(\tau)$
  6. Singularities on null scroll type timelike minimal surfaces in $\operatorname{Nil}_3(\tau)$
  7. Geometric meaning of $\kappa_2$ and construction of B-scroll type surfaces
  8. Invariance of singularity
  9. Examples

Key Result

Theorem 1.2

Let $H$ be a non-zero constant and $h=h(s)\colon I \to \mathbb{R}$ be a smooth function on an interval $I\subset \mathbb{R}$ with non-vanishing derivative. Then there exist Moreover, regarding on singularities of $f$ in $\operatorname{Nil}_3(H)$, the following assertions hold.

Figures (3)

  • Figure 1: A timelike constant mean curvature (CMC) $H=1$ B-scroll in $\mathbb{L}^3$ (left) and the corresponding timelike minimal B-scroll type surface in $\operatorname{Nil}_3(1)$ with cuspidal cross caps (right).
  • Figure 2: A timelike constant mean curvature (CMC) $H=1$ B-scroll in $\mathbb{L}^3$ (left) and the corresponding timelike minimal B-scroll type surface in $\operatorname{Nil}_3(1)$ with a cuspidal cross cap (right).
  • Figure 3: A timelike constant mean curvature (CMC) $H=1$ B-scroll in $\mathbb{L}^3$ (left) and the corresponding timelike minimal B-scroll type surface in $\operatorname{Nil}_3(1)$ with swallowtails (right).

Theorems & Definitions (26)

  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1
  • Remark 2.2
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • Remark 4.3
  • proof
  • Proposition 4.4: cf. ILLopez
  • ...and 16 more