Weierstrass representations of discrete constant mean curvature surfaces in isotropic space

TL;DR

This work develops a discrete Weierstrass framework for cmc-$H$ surfaces in isotropic 3-space by leveraging the known smooth characterization that a cmc surface can be formed as a graph-sum of a zero mean curvature surface and a sphere. The authors construct the discrete analogue by starting from a discrete isothermic zero mean curvature surface $X$ encoded by a discrete holomorphic function $g$ and its Christoffel dual $h$, and then forming a discrete surface $Y$ as a graph-sum with a sphere $S$ of mean curvature $H$, ensuring $Y$ is discrete isothermic with constant mean curvature $H$ via a carefully defined lightlike Gauss map. A complete Weierstrass data set $(g,h)$ is provided, with $dh_{ij} = \frac{H}{m_{ij}\,dg_{ij}}$, giving explicit edge-forms for $dY$ and a corresponding Gauss map $N$, enabling verification of the cmc condition. The paper additionally derives a parallel cmc surface and delivers several closed-form discrete examples, including doubly channel, cylindrical, and Delaunay-type cmc surfaces in isotropic space, highlighting the method’s analytic tractability and potential for explicit parametric representations in discrete differential geometry.

Abstract

In this paper, we obtain Weierstrass representations for discrete constant mean curvature surfaces in isotropic 3-space, and use this to construct examples with discrete closed-form parametrizations.

Figures (2)

• Figure 1: Examples of discrete cmc-$1$ surfaces with circular curvature lines. The left-hand figure is created with $(M,N) = (6,2)$, the middle figure with $(M,N) = (-2, 3)$, and the right-hand figure with $(M, N) = (4,4)$.
• Figure 2: Discrete Delaunay-type surfaces with $H \equiv 1$. The left-hand figure is created with $(N,c) = (4, -\tfrac{1}{2})$, the middle figure with $(N,c) = (3, \tfrac{1}{2})$, and the right-hand figure with $(N,c) = (10, \tfrac{1}{2})$.

Theorems & Definitions (27)

• Remark 2.2
• Example 2.3
• Example 2.4: Weierstrass representations of zero mean curvature surfaces
• Example 2.5: Weierstrass representation of non-zero constant mean curvature (cmc) surface
• Definition 2.7: pottmann_DiscreteSurfacesIsotropic_2007
• Lemma 2.8: cf. cho_DiscreteIsothermicSurfaces_
• proof
• Remark 2.9
• Remark 2.10
• Definition 2.11: pember_DiscreteWeierstrasstypeRepresentations_2023 (cf. burstall_Discrete$Omega$netsGuichard_2023)