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Special cases of the discretization principle via permutability

Joseph Cho, Mason Pember, Wayne Rossman

Abstract

We show how permutability of transforms of smooth surfaces with particular characteristics leads to discrete surfaces with discrete analogues of the same characteristics.

Special cases of the discretization principle via permutability

Abstract

We show how permutability of transforms of smooth surfaces with particular characteristics leads to discrete surfaces with discrete analogues of the same characteristics.
Paper Structure (14 sections, 10 theorems, 18 equations, 1 figure)

This paper contains 14 sections, 10 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Permutability as a bridge between smooth and discrete surfaces
  3. Smooth surfaces
  4. Isothermic surfaces and Darboux transformations
  5. Integrable reduction of isothermicity via polynomial conserved quantities
  6. Discrete surfaces
  7. Main result
  8. Applications to various surface classes
  9. Integrable reductions for isothermic surfaces to constant mean curvature surfaces
  10. $\Omega$-surfaces and their integrable reductions
  11. Isothermic surfaces
  12. Guichard surfaces
  13. $L$-isothermic surfaces
  14. Linear Weingarten surfaces

Key Result

Theorem 2.3

Suppose that $f^+$ is an isothermic surface. If $\hat{f}^+$ is a Darboux transformation of $f^+$ with finite spectral parameter $\mu$, while $f^-$ is an isotropic Darboux transformation of $f^+$, then there uniquely exists a fourth surface $\hat{f}^-$ that is simultaneously a Darboux transformation

Figures (1)

  • Figure 1: Grid from permutability of transforms of smooth surfaces

Theorems & Definitions (19)

  • Definition 2.1: hertrich-jeromin_IntroductionMobiusDifferential_2003
  • Definition 2.2: cf. burstall_DiscreteOmeganetsGuichard_2023, clarkeIntegrabilitySubmanifoldGeometry2012
  • Theorem 2.3: cf. burstall_DiscreteOmeganetsGuichard_2023, clarkeIntegrabilitySubmanifoldGeometry2012
  • Definition 2.4: burstall_SpecialIsothermicSurfaces_2012
  • Remark 2.5
  • Definition 2.6: burstall_SpecialIsothermicSurfaces_2012
  • Definition 2.7: cf. burstall_DiscreteSurfacesConstant_2014
  • Definition 2.8: burstall_DiscreteSurfacesConstant_2014
  • Lemma 2.9
  • proof
  • ...and 9 more