Semi-discrete isothermic surfaces

TL;DR

This paper introduces a Darboux transformation for polarized space curves, develops a theory for semi-discrete isothermic surfaces, and applies it to constant mean curvature surfaces, advancing the understanding of discrete differential geometry.

Contribution

It presents a novel Darboux transformation framework for polarized curves and applies it to semi-discrete isothermic surfaces, including those with constant mean curvature.

Findings

Darboux transformation for polarized space curves is established.

Transformation properties like Bianchi permutability are analyzed.

Semi-discrete constant mean curvature surfaces are characterized.

Abstract

A Darboux transformation for polarized space curves is introduced and its properties are studied, in particular, Bianchi permutability. Semi-discrete isothermic surfaces are described as sequences of Darboux transforms of polarized curves in the conformal n-sphere and their transformation theory is studied. Semi-discrete surfaces of constant mean curvature are studied as an application of the transformation theory.