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Discrete Lorentz surfaces and s-embeddings I: isothermic surfaces

Niklas Christoph Affolter, Felix Dellinger, Christian Müller, Denis Polly, Nina Smeenk

Abstract

S-embeddings were introduced by Chelkak as a tool to study the conformal invariance of the thermodynamic limit of the Ising model. Moreover, Chelkak, Laslier and Russkikh introduced a lift of s-embeddings to Lorentz space, and showed that in the limit the lift converges to a maximal surface. They posed the question whether there are s-embeddings that lift to maximal surfaces already at the discrete level, before taking the limit. This paper is the first in a two paper series, in which we answer that question in the positive. In this paper we introduce a correspondence between s-embeddings (incircular nets) and congruences of touching Lorentz spheres. This geometric interpretation of s-embeddings enables us to apply the tools of discrete differential geometry. We identify a subclass of s-embeddings -- isothermic s-embeddings -- that lift to (discrete) S-isothermic surfaces, which were introduced by Bobenko and Pinkall. S-isothermic surfaces are the key component that will allow us to obtain discrete maximal surfaces in the follow-up paper. Moreover, we show here that the Ising weights of an isothermic s-embedding are in a subvariety.

Discrete Lorentz surfaces and s-embeddings I: isothermic surfaces

Abstract

S-embeddings were introduced by Chelkak as a tool to study the conformal invariance of the thermodynamic limit of the Ising model. Moreover, Chelkak, Laslier and Russkikh introduced a lift of s-embeddings to Lorentz space, and showed that in the limit the lift converges to a maximal surface. They posed the question whether there are s-embeddings that lift to maximal surfaces already at the discrete level, before taking the limit. This paper is the first in a two paper series, in which we answer that question in the positive. In this paper we introduce a correspondence between s-embeddings (incircular nets) and congruences of touching Lorentz spheres. This geometric interpretation of s-embeddings enables us to apply the tools of discrete differential geometry. We identify a subclass of s-embeddings -- isothermic s-embeddings -- that lift to (discrete) S-isothermic surfaces, which were introduced by Bobenko and Pinkall. S-isothermic surfaces are the key component that will allow us to obtain discrete maximal surfaces in the follow-up paper. Moreover, we show here that the Ising weights of an isothermic s-embedding are in a subvariety.

Paper Structure

This paper contains 29 sections, 18 theorems, 63 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Plan of the paper
  3. Main results
  4. Lorentz sphere geometry
  5. Lorentz geometry
  6. Lorentz Möbius geometry
  7. Timelike Lorentz Laguerre geometry
  8. Oriented planes, spheres and isotropic lines
  9. Cyclographic model
  10. Timelike Lorentz Laguerre transformations
  11. Timelike Lorentz Lie geometry
  12. Contact congruences and circle patterns
  13. Conical nets and circle patterns
  14. Lorentz lift
  15. Cyclographic lift and origami-map
  16. ...and 14 more sections

Key Result

Theorem 2.4

Every contact congruence $c$ defines a circle pattern $p$ via the intersection see also Figure fig:cccp. The conical net ${\odot{p}}$ is the orthogonal projection of the center net ${\odot{c}}$.

Figures (13)

  • Figure 1: An overview of the relations between our main geometric objects (see Section 2).
  • Figure 2: Left: the local combinatorics of a contact congruence. At each vertex there is a timelike sphere and at each face an isotropic line. Adjacent spheres share a tangent plane that contains two isotropic lines. Right: the corresponding geometric picture in Lorentz space ${\mathcal{\mathbf{L^3}}}$.
  • Figure 3: The intersection of a contact congruence with a plane is a circle pattern.
  • Figure 4: A circle packing $p$ (with $p_{\bullet}$ in blue and $p_{\circ}$ in black) and the corresponding incircular net ${\odot{p_{\bullet}}}$ (green), with incircles (brown).
  • Figure 5: The null-spheres $c_{\bullet}$ (left) and time-like spheres $c_{\circ}$ (right) of a null congruence $c$ as well as the isotropic lines $c_{{}}$ (red).
  • ...and 8 more figures

Theorems & Definitions (66)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Theorem 2.9
  • Definition 2.10
  • ...and 56 more