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Discrete Weierstrass-type representations

Mason Pember, Denis Polly, Masashi Yasumoto

Abstract

Discrete Weierstrass-type representations yield a construction method in discrete differential geometry for certain classes of discrete surfaces. We show that the known discrete Weierstrass-type representations of certain surface classes can be viewed as applications of the $Ω$-dual transform to lightlike Gauss maps in Laguerre geometry. From this construction, further Weierstrass-type representations arise. As an application of the techniques we develop, we show that all discrete linear Weingarten surfaces of Bryant or Bianchi type locally arise via Weierstrass-type representations from discrete holomorhpic maps.

Discrete Weierstrass-type representations

Abstract

Discrete Weierstrass-type representations yield a construction method in discrete differential geometry for certain classes of discrete surfaces. We show that the known discrete Weierstrass-type representations of certain surface classes can be viewed as applications of the -dual transform to lightlike Gauss maps in Laguerre geometry. From this construction, further Weierstrass-type representations arise. As an application of the techniques we develop, we show that all discrete linear Weingarten surfaces of Bryant or Bianchi type locally arise via Weierstrass-type representations from discrete holomorhpic maps.

Paper Structure

This paper contains 10 sections, 13 theorems, 106 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Discrete Legendre immersions in Laguerre geometry
  4. Discrete L-isothermic surfaces
  5. The Calapso transform
  6. Discrete Weierstrass type represenations
  7. Discrete surfaces in hyperplanes of R31
  8. Discrete surfaces in quadrics in R31
  9. The Lawson correspondence
  10. Discrete linear Weingarten surfaces of Bryant and Bianchi type

Key Result

Proposition 3.8

A discrete Legendre immersion $L$ with discrete lightlike Gauss map $G$ is L-isothermic if and only if there exist discrete surfaces $g\in \Gamma G$ and $x\in\Gamma L$ such that

Figures (4)

  • Figure 1: Thre discrete holomorphic functions and the respective discrete zero mean curvature surfaces in Euclidean, Lorentz and isotropic space.
  • Figure 2: Two examples of discrete cmc $1$ surfaces in $\mathbb{H}^3$ shown in the Poincaré ball model
  • Figure 3: A discrete cmc $1$ surface in $\mathbb{H}^3$ and its dual surface in the Poincaré ball model. The discrete holomorphic functions can be regarded of the hyperbolic and secondary Gauss maps of the surfaces.
  • Figure 4: Examples of discrete BrLW/BiLW surfaces in $\mathbb{H}^3$ or $\mathbb{S}^{2,1}$ which form a pair of discrete surface and hyperbolic Gauss map (see Remark \ref{['rem:BryantBiachniDual']}). All examples are created using the representation \ref{['eq:BryantRep']} with the discrete power function as data. We show surfaces in hyperbolic space in the Poincaré ball model and surfaces in de Sitter space in the hollow ball model.

Theorems & Definitions (31)

  • Example 2.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.4: burstall2018
  • Definition 3.5
  • Definition 3.6
  • Remark 3.7
  • Proposition 3.8
  • Lemma 3.9
  • proof
  • ...and 21 more