The interrelationship of integrable equations, differential geometry and the geometry of their associated surfaces

TL;DR

This paper surveys the deep connections between integrable equations, differential geometry, and surface theory, highlighting recent advances and introducing new results using the moving frame and differential forms.

Contribution

It presents new insights into the relationship between integrable equations and surface geometry, including generalizations of classical surface generation methods.

Findings

Elucidation of the role of the moving frame in surface theory

Introduction of structure equations via differential forms

New results on generalized Weierstrass-Enneper methods

Abstract

A survey of some recent and important results which have to do with integrable equations and their relationship with the theory of surfaces is given. Some new results are also presented. The concept of the moving frame is examined, and it is used in several subjects, which are discussed. Structure equations are introduced in terms of differential forms. Forms are shown to be very useful in relating geometry, equations and surfaces, which appear in many sections. The topics of the chapters are different and separate, but joined together by common themes and ideas. Several subjects which are not easy to access are elaborated, such as Maurer-Cartan cocycles and recent results with regard to generalizations of the Weierstrass-Enneper method for generating constant mean curvature surfaces in three and higher dimensional Euclidean spaces.