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

Abstract

The paper builds a DPW approach of Willmore surfaces via conformal Gauss maps. As applications, we provide descriptions of minimal surfaces in 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.

Weierstrass–kenmotsu Representation of Willmore Surfaces in Spheres

“…Therefore, let's consider the holomorphic potential of the harmonic map f (for a discussion we refer to [10,11]). Let , be the corresponding holomorphic potential on U .…”

“…The proof is as in [11] and is not related in any way to the specific properties of conformally harmonic maps that we investigate. Therefore, let us consider the holomorphic potential of the harmonic map f (for a discussion, we refer the reader to [11,12]). Let…”

“…Therefore, let us consider the holomorphic potential of the harmonic map f (for a discussion, we refer the reader to [11,12]). Let…”

A duality theorem for harmonic maps into non-compact symmetric spaces and compact symmetric spaces

“…The global existence holds for the case of two-spheres, which have essential contribution in the study of Willmore 2-spheres in spheres [15,16]. In fact the topic of this paper stems from the study of Willmore surfaces, since the conformal Gauss map of a Willmore surface is a harmonic map into some non-compact symmetric space [6,7]. To be concrete, using this duality theorem, we can classify the conformal Gauss maps of Willmore 2-spheres by classifying all compact dual harmonic maps of these harmonic maps [15].…”

“…REMARK: Following the suggestion of some anonymous referee we have divided the paper entitled "Willmore surfaces in spheres via loop groups I: generic cases and some examples "(arXiv:1301.2756) into three parts. The present paper is part III and [6,7] are part I and part II respectively.…”