Willmore surfaces in spheres via loop groups: a survey

TL;DR

This survey reviews the use of the DPW method to study Willmore surfaces in spheres, covering their global properties, constructions, characterizations, deformations, and related harmonic map theories.

Contribution

It provides a comprehensive overview of the DPW method's application to Willmore surfaces and harmonic maps, including new results on duality and finite uniton type maps.

Findings

Development of the DPW framework for Willmore surfaces

Construction methods for Willmore 2-spheres

Duality theorem for harmonic maps into symmetric spaces

Abstract

In the past decades, the authors made some systematic research on global and local properties of Willmore surfaces in terms of the DPW method. In this note we give a survey, mainly including the basic framework of the DPW method for the global geometry of Willmore surfaces via the conformal Gauss map, applications on constructions of Willmore $2$-spheres, characterizations of minimal surfaces, Willmore deformations of Willmore surfaces and Bjoerling problems for Willmore surfaces. Moreover, we also obtained some results on harmonic maps via DPW, including a duality theorem for harmonic maps into an inner non-compact symmetric space and its dual inner compact symmetric space, and harmonic maps of finite uniton type.