Willmore surfaces in spheres: the DPW approach via the conformal Gauss map

TL;DR

This paper develops a DPW method for Willmore surfaces using conformal Gauss maps, enabling new classifications and examples of such surfaces in various geometries.

Contribution

It introduces a novel DPW approach for Willmore surfaces and constructs new examples, including a Willmore two-sphere in S^6 without duals.

Findings

Descriptions of minimal surfaces in R^{n+2}

Descriptions of isotropic surfaces in S^4

A new Willmore two-sphere in S^6 without dual surfaces

Abstract

The paper builds a DPW approach of Willmore surfaces via conformal Gauss maps. As applications, we provide descriptions of minimal surfaces in $\mathbb R^{n+2}$, isotropic surfaces in $S^4$ and homogeneous Willmore tori via the loop group method. A new example of a Willmore two-sphere in $S^6$ without dual surfaces is also presented.