Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Constant mean curvature surfaces from ring patterns: Geometry from combinatorics

Alexander I. Bobenko, Tim Hoffmann, Nina Smeenk

Abstract

We define discrete constant mean curvature (cmc) surfaces in the three-dimensional Euclidean and Lorentz spaces in terms of sphere packings with orthogonally intersecting circles. These discrete cmc surfaces can be constructed from orthogonal ring patterns in the two-sphere and the hyperbolic plane. We present a variational principle that allows us to solve boundary value problems and to construct discrete analogues of some classical cmc surfaces. The data used for the construction is purely combinatorial - the combinatorics of the curvature line pattern. In the limit of orthogonal circle patterns we recover the theory of discrete minimal surfaces associated to Koebe polyhedra all edges of which touch a sphere. These are generalized to two-sphere Koebe nets, i.e., nets with planar quadrilateral faces and edges that alternately touch two concentric spheres.

Constant mean curvature surfaces from ring patterns: Geometry from combinatorics

Abstract

We define discrete constant mean curvature (cmc) surfaces in the three-dimensional Euclidean and Lorentz spaces in terms of sphere packings with orthogonally intersecting circles. These discrete cmc surfaces can be constructed from orthogonal ring patterns in the two-sphere and the hyperbolic plane. We present a variational principle that allows us to solve boundary value problems and to construct discrete analogues of some classical cmc surfaces. The data used for the construction is purely combinatorial - the combinatorics of the curvature line pattern. In the limit of orthogonal circle patterns we recover the theory of discrete minimal surfaces associated to Koebe polyhedra all edges of which touch a sphere. These are generalized to two-sphere Koebe nets, i.e., nets with planar quadrilateral faces and edges that alternately touch two concentric spheres.

Paper Structure

This paper contains 18 sections, 22 theorems, 127 equations, 28 figures.

Table of Contents

  1. Introduction
  2. Discrete S-isothermic surfaces in ${\mathbb R}^{3}$
  3. Discrete cmc surfaces in ${\mathbb R}^{3}$
  4. The Gauss map of S$_1$-cmc surfaces
  5. Discrete curvatures
  6. Two-sphere Koebe nets and orthogonal ring patterns in the sphere
  7. Analytic description of orthogonal ring patterns in the sphere
  8. Constructing cmc surfaces in ${\mathbb R}^{3}$
  9. Minimal surfaces in the limit
  10. Examples of cmc surfaces in ${\mathbb R}^{3}$
  11. Doubly periodic S$_1$-cmc surfaces
  12. Triply periodic S$_1$-cmc surfaces
  13. Discrete S-isothermic surfaces in ${\mathbb R}^{2, 1}$
  14. Discrete cmc surfaces in ${\mathbb R}^{2, 1}$
  15. Spacelike two-sphere Koebe nets and hyperbolic orthogonal ring patterns
  16. ...and 3 more sections

Key Result

Corollary 2.2

Let $s_v, s_{v_i}, s_{v_{ij}}$ and $s_{v_j}$ be four the spheres of an S-isothermic quadrilateral. Then Further they satisfy one of the three conditions shown in Figure Fig:Rt_Q_congruences, namely

Figures (28)

  • Figure 1: Triply periodic cmc surfaces in ${\mathbb R}^{3}$: Schwarz's P surface (first line) and Schoen's I-WP surface (second line). The left column presents the smooth cmc surfaces from bobenko2021constant obtained by the loop group method from the theory of integrable systems. The right column are the corresponding discrete cmc surfaces built by touching disks. Their spherical orthogonal ring patterns are shown in the middle.
  • Figure 2: A quad graph ${\mathcal{G}}$ with interior vertices of even valency. The edges are divided into 'horizontal' $i$--edges (black) and 'vertical' $j$--edges (pink), such that in each quadrilateral the edges alternately change color.
  • Figure 3: Three types of S-isothermic quadrilaterals with vertex spheres $(i)$ possessing a common orthogonal circle, $(ii)$ intersecting in two points, or $(iii)$ intersecting in one point. The first image shows the special case of type $(i)$ where vertex spheres touch.
  • Figure 4: An S$_1$-isothermic surface. Each quad possesses an incircle that intersects adjacent spheres orthogonally and passes through their touching points. The one large vertex sphere corresponds to an umbilic point, an inner vertex of $\mathcal{G}$ with valency greater than four. The figure shows a piece of the doubly periodic S$_1$-isothermic cmc surface $\psi(U_{2, 2})$ presented in Section \ref{['sec:Rt_examples']}.
  • Figure 5: A two-sphere Koebe net. Its edges alternately touch the larger, transparent gray sphere $S_+^2$ and the smaller, white sphere $S_-^2$.
  • ...and 23 more figures

Theorems & Definitions (45)

  • Definition 2.1
  • Corollary 2.2
  • proof
  • Corollary 2.4
  • Corollary 2.5
  • Definition 3.1
  • Corollary 3.2
  • Theorem 3.3
  • Definition 3.4
  • Definition 3.5
  • ...and 35 more