Supplement on Curved flats in the space of point pairs and Isothermic surfaces: A Quaternionic Calculus

TL;DR

This paper develops a quaternionic calculus for surface pairs in the conformal 4-sphere and explores their relation to curved flats, isothermic surfaces, and constant mean curvature surfaces, providing new insights and representations.

Contribution

It introduces a novel quaternionic calculus framework and offers a new perspective on the relations between various special surfaces, including a new Bryant-type representation.

Findings

Quaternionic calculus for surface pairs in conformal 4-sphere

New relations between curved flats and isothermic surfaces

A new Bryant-type representation for CMC-1 surfaces in hyperbolic space

Abstract

A quaternionic calculus for surface pairs in the conformal 4-sphere is elaborated. This calculus is then used to discuss the relation between curved flats in the symmetric space of point pairs and Darboux and Christoffel pairs of isothermic surfaces. A new viewpoint on relations between surfaces of constant mean curvature in certain space forms is presented --- in particular, a new form of Bryant's Weierstrass type representation for surfaces of constant mean curvature 1 in hyperbolic 3-space is given.