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Minimal null scrolls in the three-dimensional Heisenberg group

Hirotaka Kiyohara

Abstract

In the three-dimensional Heisenberg group equipped with a certain left invariant Lorentzian metric, timelike minimal surfaces which have the Abresch-Rosenberg differentials with vanishing multiplication of the coefficient function and its para-complex conjugate are characterized as the surfaces defined by the multiplication of null curves and affine null lines. Moreover, the constructions of these surfaces with prescribed curvatures of null curves or prescribed null lines are given.

Minimal null scrolls in the three-dimensional Heisenberg group

Abstract

In the three-dimensional Heisenberg group equipped with a certain left invariant Lorentzian metric, timelike minimal surfaces which have the Abresch-Rosenberg differentials with vanishing multiplication of the coefficient function and its para-complex conjugate are characterized as the surfaces defined by the multiplication of null curves and affine null lines. Moreover, the constructions of these surfaces with prescribed curvatures of null curves or prescribed null lines are given.
Paper Structure (15 sections, 155 equations, 5 figures)

This paper contains 15 sections, 155 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Para-complex structure
  4. Timelike surface theory in the $3$-dimensional Minkowski space
  5. $B$-scrolls
  6. $3$-dimensional Heisenberg group ${\rm Nil}_3$
  7. Integrability conditions of timelike minimal surfaces
  8. Characterization of timelike minimal surfaces with $Q \overline Q=0$
  9. Minimal surfaces in ${\rm Nil}_3$ which induce $B$-scrolls in $\mathbb L^3_{(-,+,+)}$
  10. Null scrolls and their minimality conditions
  11. Null frames for null curves in ${\rm Nil}_3$
  12. Construction of minimal null scrolls with prescribed rulings
  13. Construction from the condition $g(A, B)=2\beta$ and $\beta=0$
  14. Construction from the condition $g(A, B ) = 2\beta$ and $\beta \neq0$
  15. Construction from the condition $g(A, \widetilde{B}) =0$

Figures (5)

  • Figure 1: Horizontal umbrella: Timelike minimal surface with the Abresch-Rosenberg differential vanishing anywhere. They are included in planes.
  • Figure 2: Vertical plane: Minimal null scrolls with the minimality condition $g(A, B)=2\beta$ and $\beta=0$.
  • Figure 3: Example of minimal null scrolls constructed from ruling valued in a circle (right) and the image of its ruling (left, dash line).
  • Figure 4: Example of minimal null scrolls constructed from ruling valued in a hyperbola (right) and the image of its ruling (left, dash line).
  • Figure 5: Example of minimal null scrolls constructed from ruling valued in a hyperbola (right) and the image of its ruling (left, dash line).

Theorems & Definitions (14)

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  • ...and 4 more