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Singularities on timelike minimal surfaces in Lorentzian Heisenberg group

Shintaro Akamine, Hirotaka Kiyohara

TL;DR

This work develops a comprehensive singularity theory for timelike minimal surfaces in the Lorentzian Heisenberg group $(\operatorname{Nil}_3(\tau), g_+)$ by representing such surfaces through Lorentzian harmonic maps into the de Sitter sphere $\mathbb{S}^2_1$ and exploiting a duality with timelike CMC surfaces in $\mathbb{L}^3$. It derives explicit criteria for typical singularities—cuspidal edges, swallowtails, and cuspidal cross caps—in terms of the harmonic data $(g, \hat{\omega})$ and shows how these singularities correspond to regular points on the dual CMC surfaces via a Kenmotsu-type representation and Abresch–Rosenberg differentials. The authors also construct concrete examples, notably B-scroll type surfaces, to demonstrate swallowtails and cross caps, thereby proving the existence of timelike minimal surfaces in $\operatorname{Nil}_3(\tau)$ with these singularities. Overall, the paper advances the understanding of singularities in Lorentzian geometry on Nil$_3$ by linking harmonic map data to precise local models and dual Lorentzian–CMC geometry in $\mathbb{L}^3$.

Abstract

Timelike minimal surfaces in the three-dimensional Lorentzian Heisenberg group are shown to be constructed from Lorentzian harmonic maps into the de-Sitter two-sphere, and they naturally admit singular points. In particular, we provide criteria for cuspidal edges, swallowtails, and cuspidal cross caps, and present several explicit examples.

Singularities on timelike minimal surfaces in Lorentzian Heisenberg group

TL;DR

This work develops a comprehensive singularity theory for timelike minimal surfaces in the Lorentzian Heisenberg group by representing such surfaces through Lorentzian harmonic maps into the de Sitter sphere and exploiting a duality with timelike CMC surfaces in . It derives explicit criteria for typical singularities—cuspidal edges, swallowtails, and cuspidal cross caps—in terms of the harmonic data and shows how these singularities correspond to regular points on the dual CMC surfaces via a Kenmotsu-type representation and Abresch–Rosenberg differentials. The authors also construct concrete examples, notably B-scroll type surfaces, to demonstrate swallowtails and cross caps, thereby proving the existence of timelike minimal surfaces in with these singularities. Overall, the paper advances the understanding of singularities in Lorentzian geometry on Nil by linking harmonic map data to precise local models and dual Lorentzian–CMC geometry in .

Abstract

Timelike minimal surfaces in the three-dimensional Lorentzian Heisenberg group are shown to be constructed from Lorentzian harmonic maps into the de-Sitter two-sphere, and they naturally admit singular points. In particular, we provide criteria for cuspidal edges, swallowtails, and cuspidal cross caps, and present several explicit examples.
Paper Structure (9 sections, 16 theorems, 84 equations, 2 figures, 1 table)

This paper contains 9 sections, 16 theorems, 84 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Timelike surfaces
  4. Duality between timelike minimal surfaces in $\operatorname{Nil}_3(\tau)$ and timelike constant mean curvature surfaces in $\mathbb{L}^3$
  5. Minimal surfaces in $\operatorname{Nil}_3(\tau)$ with singularities
  6. Types of singularities
  7. Singularities in $\Sigma^\omega$
  8. Singularities in $\Sigma^g \setminus \Sigma^\omega$
  9. Examples

Key Result

Proposition 2.1

A paracomplex number $z$ has the inverse $1/z$ if and only if $z$ satisfies that $|z|^2 \neq 0$.

Figures (2)

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

Theorems & Definitions (37)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Lemma 2.3: KK
  • proof : Proof of Proposition \ref{['prop: derivative']}
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6: c.f. KK
  • ...and 27 more