Construction of Willmore Two-Spheres Via Harmonic Maps Into 
                     
                        
                           
                              
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$ {\boldsymbol{SO^+(1,n+3)/(SO^+(1,1)}} \times {\boldsymbol{SO(n+2))}} $

doi:10.1201/9781315151830-4

Chapter

$Construction of Willmore Two-Spheres Via Harmonic Maps Into S O + ( 1 , n + 3 ) / ( S O + ( 1 , 1 ) × S O ( n + 2 ) ) $ {\boldsymbol{SO^+(1,n+3)/(SO^+(1,1)}} \times {\boldsymbol{SO(n+2))}} $$

Chapter

Construction of Willmore Two-Spheres Via Harmonic Maps Into S O + ( 1 , n + 3 ) / ( S O + ( 1 , 1 ) × S O ( n + 2 ) ) $ {\boldsymbol{SO^+(1,n+3)/(SO^+(1,1)}} \times {\boldsymbol{SO(n+2))}} $

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 Hélein, 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.