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Zero mean curvature surfaces in isotropic space with planar curvature lines

Joseph Cho, Masaya Hara

Abstract

We give a comprehensive account of zero mean curvature surfaces in isotropic 3-space with planar curvature lines. After giving a complete classification all such surfaces, we show that they belong to a 1-parameter family of surfaces. We then investigate their relationship to Thomsen-type surfaces in isotropic 3-space, those zero mean curvature surfaces in isotropic 3-space that are also affine minimal.

Zero mean curvature surfaces in isotropic space with planar curvature lines

Abstract

We give a comprehensive account of zero mean curvature surfaces in isotropic 3-space with planar curvature lines. After giving a complete classification all such surfaces, we show that they belong to a 1-parameter family of surfaces. We then investigate their relationship to Thomsen-type surfaces in isotropic 3-space, those zero mean curvature surfaces in isotropic 3-space that are also affine minimal.

Paper Structure

This paper contains 14 sections, 12 theorems, 130 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Isotropic 3-space
  4. Surface theory
  5. Classification of zero mean curvature surfaces with planar curvature lines
  6. Analytic classification
  7. Recovering the Weierstrass data
  8. Geometry of zero mean curvature surfaces with planar curvature lines
  9. Trivial Enneper-type surface
  10. Axial directions
  11. Deformation of zero mean curvature surfaces with planar curvature lines
  12. Deformation to the trivial Enneper surface
  13. Deformation to the catenoid
  14. Zero mean curvature surfaces that are also affine minimal

Key Result

Lemma 3.1

For an umbilic-free zero mean curvature surface in isotropic space, the following are equivalent:

Figures (6)

  • Figure 1: Classification diagram of zero mean curvature surfaces with planar curvature lines
  • Figure 2: Zero mean curvature surfaces with planar curvature lines in isotropic space: trivial Enneper surface and catenoid on the top row; Enneper surface and Bonnet-type surface on the bottom row.
  • Figure 3: Path on classification diagram taken to find a deformation to the trivial Enneper surface.
  • Figure 4: Continuous deformation of zero mean curvature surfaces keeping planar curvature lines
  • Figure 5: Zero mean curvature surfaces that are also affine minimal in isotropic space: trivial Enneper surface and helicoid on the top row; Enneper surface and Thomsen-type surface on the bottom row.
  • ...and 1 more figures

Theorems & Definitions (22)

  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • Remark 3.6
  • ...and 12 more