A discrete version of the Darboux transform for isothermic surfaces

TL;DR

This paper develops a discrete framework for isothermic surfaces in 4D Euclidean space, introducing transformations and characterizations for constant mean curvature surfaces, bridging discrete and smooth differential geometry.

Contribution

It defines and analyzes discrete Christoffel, Darboux, and Ribaucour transforms for isothermic nets, and characterizes discrete constant mean curvature surfaces using these transformations.

Findings

Defined discrete Ribaucour congruences.

Characterized discrete constant mean curvature surfaces.

Established properties of discrete isothermic nets.

Abstract

We study Christoffel and Darboux transforms of discrete isothermic nets in 4-dimensional Euclidean space: definitions and basic properties are derived. Analogies with the smooth case are discussed and a definition for discrete Ribaucour congruences is given. Surfaces of constant mean curvature are special among all isothermic surfaces: they can be characterized by the fact that their parallel constant mean curvature surfaces are Christoffel and Darboux transforms at the same time. This characterization is used to define discrete nets of constant mean curvature. Basic properties of discrete nets of constant mean curvature are derived.