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Spinor representation in isotropic 3-space via Laguerre geometry

Joseph Cho, Dami Lee, Wonjoo Lee, Seong-Deog Yang

TL;DR

This work develops a Laguerre-geometric framework for isotropic 3-space $\mathbb{I}^3$, unifying it with Euclidean geometry through Minkowski space $\mathbb{L}^4$ and a Hermitian matrix model. It introduces spin transformations and a Dirac-type compatibility equation to obtain a spinor representation of conformal spacelike surfaces in $\mathbb{I}^3$, leading to Weierstrass-type representations for minimal surfaces and Kenmotsu-type representations for nonzero cmc surfaces. The resulting spinor data yield explicit constructions of cmc surfaces, including spheres, cylinders, Delaunay-type, and singly periodic dihedral-symmetric examples, illustrating the practical use of the framework. Overall, the paper provides a coherent, metric-free perspective on isotropic surface theory that parallels Euclidean methods and enables concrete surface constructions.

Abstract

We give a detailed description of the geometry of isotropic space, in parallel to those of Euclidean space within the realm of Laguerre geometry. After developing basic surface theory in isotropic space, we define spin transformations, directly leading to the spinor representation of conformal surfaces in isotropic space. As an application, we obtain the Weierstrass-type representation for zero mean curvature surfaces, and the Kenmotsu-type representation for constant mean curvature surfaces, allowing us to construct many explicit examples.

Spinor representation in isotropic 3-space via Laguerre geometry

TL;DR

This work develops a Laguerre-geometric framework for isotropic 3-space , unifying it with Euclidean geometry through Minkowski space and a Hermitian matrix model. It introduces spin transformations and a Dirac-type compatibility equation to obtain a spinor representation of conformal spacelike surfaces in , leading to Weierstrass-type representations for minimal surfaces and Kenmotsu-type representations for nonzero cmc surfaces. The resulting spinor data yield explicit constructions of cmc surfaces, including spheres, cylinders, Delaunay-type, and singly periodic dihedral-symmetric examples, illustrating the practical use of the framework. Overall, the paper provides a coherent, metric-free perspective on isotropic surface theory that parallels Euclidean methods and enables concrete surface constructions.

Abstract

We give a detailed description of the geometry of isotropic space, in parallel to those of Euclidean space within the realm of Laguerre geometry. After developing basic surface theory in isotropic space, we define spin transformations, directly leading to the spinor representation of conformal surfaces in isotropic space. As an application, we obtain the Weierstrass-type representation for zero mean curvature surfaces, and the Kenmotsu-type representation for constant mean curvature surfaces, allowing us to construct many explicit examples.
Paper Structure (20 sections, 10 theorems, 155 equations, 6 figures)

This paper contains 20 sections, 10 theorems, 155 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Geometry of isotropic 3-space
  3. Laguerre geometry of Euclidean 3-space
  4. Spheres and planes
  5. Contact elements and normals
  6. Surfaces and their contact elements
  7. Isometries fixing the origin
  8. Summary
  9. Laguerre geometry of isotropic 3-space
  10. Spheres in isotropic space
  11. Planes in isotropic space
  12. Contact elements and normals
  13. Surfaces and their contact elements
  14. Isometries in isotropic space -- Hermitian matrix model
  15. Spin transformations and spinor representation
  16. ...and 5 more sections

Key Result

Proposition 2.7

Let $P$ be a spacelike plane in isotropic $3$-space given via for some $\alpha, \beta, \gamma, q \in \mathbb{R}$. Then the unit normal $\nu$ to $P$ is given by

Figures (6)

  • Figure 1: Spheres and planes of Euclidean space in Laguerre geometry.
  • Figure 2: Spheres and planes with different orientations.
  • Figure 3: Contact elements of Euclidean space in Laguerre geometry.
  • Figure 4: Spheres in isotropic geometry via isotropy projection
  • Figure 5: Spacelike planes and lightlike planes of isotropic space via isotropic hyperplanes in Laguerre geometry.
  • ...and 1 more figures

Theorems & Definitions (28)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Lemma 2.9
  • proof
  • ...and 18 more