Construction of Willmore two-spheres via harmonic maps into $SO^+(1,n+3)/(SO^+(1,1)\times SO(n+2))$

TL;DR

This paper characterizes totally isotropic Willmore two-spheres and their adjoint transforms using harmonic maps into a specific symmetric space, providing explicit construction methods and examples.

Contribution

It offers a new description of totally isotropic Willmore two-spheres via normalized potentials derived from harmonic maps, including concrete construction techniques.

Findings

Normalized potentials of Willmore two-spheres are explicitly described.

Harmonic maps involved are of finite uniton type.

Concrete examples of Willmore two-spheres and their transforms are constructed.

Abstract

This paper aims to provide a description of totally isotropic Willmore two-spheres and their adjoint transforms. We first recall the isotropic harmonic maps which are introduced by H\'elein, Xia-Shen and Ma for the study of Willmore surfaces. Then we derive a description of the normalized potential (some Lie algebra valued meromorphic 1-forms) of totally isotropic Willmore two-spheres in terms of the isotropic harmonic maps. In particular, the corresponding isotropic harmonic maps are of finite uniton type. The proof also contains a concrete way to construct examples of totally isotropic Willmore two-spheres and their adjoint transforms. As illustrations, two kinds of examples are obtained this way.