Willmore surfaces in spheres via loop groups: a survey

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.