Discrete constant mean curvature cylinders and isothermic tori

TL;DR

The work develops a structure-preserving discretisation framework for isothermic surfaces by tying the monodromy of Darboux transforms to discrete polarised curves through flat connections. This reduction enables explicit, closed-form discrete parametrisations of isothermic cylinders, cmc cylinders, and tori, including bubbletons and cmc bubbletons, with resonance conditions governing closure. The authors also demonstrate consistency with the smooth theory via continuum limits, strengthening the link between discrete integrable systems and classical differential geometry. The approach provides concrete discrete models with potential impact on geometric modeling and the study of global properties of isothermic surfaces.

Abstract

We consider the monodromy problem of Darboux transforms of discrete isothermic surfaces using the integrable theory of discrete polarised curves. Then we provide, for the first time, closed-form discrete parametrisations of discrete isothermic cylinders, discrete constant mean curvature cylinders, and discrete isothermic tori.

Figures (9)

• Figure 1: Discrete constant mean curvature cylinder (on the left) and discrete isothermic torus (on the right) obtained as Darboux transformations of discrete isothermic surfaces.
• Figure 2: An elementary quadrilateral with auxiliary points and their labels.
• Figure 3: Discrete isothermic bubbletons drawn using explicit parametrisation \ref{['eqn:expPar']} with $k = 2$, $\rho = 1$, and $c_2 = -10$ with various number of subdivisions (on the left: $M = 40$, $N = 5$, on the right: $M = 80$, $N = 10$).
• Figure 4: Discrete isothermic bubbletons drawn using explicit parametrisation \ref{['eqn:expPar']} with $k = 3$, $\rho = 1$, $M = 80$, and $N = 8$ with varying initial conditions (on the left: $c_2 = -4$, on the right: $c_2 = -8$).
• Figure 5: Discrete cmc bubbletons drawn using explicit parametrisation \ref{['eqn:expPar']} where $k = 2$, $\rho = 1$ with initial condition as in \ref{['eqn:cmcInit']} at various number of subdivisions (on the top left: $M = 40$, $N = 5$; on the top right: $M = 40$, $N = 10$; on the bottom left: $M = 160$, $N = 5$; on the bottom right: $M = 160$, $N = 15$).

Theorems & Definitions (14)

• Definition 2.1
• Remark 2.2
• Remark 2.4
• Definition 2.5
• Definition 2.8: hertrich-jeromin_discrete_1999
• Remark 2.11
• Definition 2.12
• Proposition 3.1
• Remark 3.2
• Proposition 3.3