Constant curvature rotational nets and periodic Bäcklund transforms

TL;DR

The paper develops an integrable-systems framework for discrete surfaces with constant Gaussian curvature, focusing on negative CGC (cK-nets) and their Bäcklund transforms. It introduces quaternionic models, discrete gauge theory, and flat connections for circular nets, enabling explicit parametrizations and Lax-form descriptions. A key contribution is the construction of rotationally symmetric ck-nets and their associated flat connections, along with conditions that ensure periodicity and annular topology under single and double Bäcklund transforms. It also provides explicit rc-net parametrizations via trig, hyperbolic, and Jacobi-elliptic functions, analyzes singularities, and presents a linearization method for the resulting rational difference equations. Collectively, these results offer concrete, periodic discrete surfaces with constant curvature and clarifying transformations, advancing discrete differential geometry and integrable-disystem approaches to pseudospherical surfaces.

Abstract

After giving explicit parametrizations of discrete constant negative Gaussian curvature surfaces (negative CGC, i.e. discrete pseudospherical surfaces) of revolution, we construct Bäcklund transformations that again will have explicit parametrizations and are new examples of non-rotational discrete pseudospherical surfaces. In the process of doing this, for discrete CGC circular nets, we can provide rotationally invariant families of flat connections and give conditions on them so that the Bäcklund transformations preserve periodicity, that is, have annular topology.

Figures (8)

• Figure 1: Examples of periodic cK-nets (which are seen again in Figures \ref{['figure:singular']} and \ref{['figure:backlund']}).
• Figure 2: Left: vertices and oriented edges of a quadrilateral, right: identification with pairs of integers in the domain.
• Figure 3: Left: a face of a locally embedded cc-net, right: a face of a non-embedded cc-net.
• Figure 4: Upper row (left–right): The image of $\mathbf{x}$ for rc-nets that have $K\equiv 1$ with degenerate edges and singular vertices, for $\kappa=1$, $0<\kappa<1$, $\kappa>1$. Lower row (left–right): The image of $\mathbf{x}$ for rc-nets that have $K\equiv -1$ with degenerate edges and singular vertices, again for $\kappa=1$, $0< \kappa<1$, $\kappa>1$. Yellow dots indicate degenerate edges in the rotational direction, whereas red dots indicate singular vertices.
• Figure 5: Left: The image of $\mathbf{x}$ for a member of the associated family of a $K\equiv -1$ rc-net (the vertical axis is now pointing rightward). Right: The discrete helix $\gamma_{0}$ at $j=0$ in the leftward surface.

Theorems & Definitions (56)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Lemma 1.5: Gauge invariance
• Definition 1.6
• Definition 1.7
• Definition 1.8
• Lemma 1.9: BP3
• Lemma 1.10