Willmore surfaces in spheres via loop groups $I$: generic cases and some examples

Abstract

In this paper we deal with the global properties of Willmore surfaces in spheres via the harmonic conformal Gauss map using loop groups.We first derive a global description of those harmonic maps which can be realized as conformal Gauss maps of some Willmore surfaces (Theorem 3.4, Theorem 3.11 and Theorem 3.18).Then we introduce the DPW procedure for these harmonic maps, and state appropriate versions of the Iwasawa decomposition and the Birkhoff decomposition Theorems. In particular, we show how the harmonic maps associated with Willmore surfaces can be constructed in terms of loop groups.The third main result, which has many implications for the case of Willmore surfaces in spheres, shows that every harmonic map into some non-compact inner symmetric space G/K induces a harmonic map into the compact dual inner symmetric space U/(U ∩ K C ). From this correspondence we obtain additional information about the global properties of harmonic maps into non-compact inner symmetric spaces.As an illustration of the theory developed in this paper we list examples (some of which were worked out in separate papers by following the theory of the present paper). In particular, we present an explicit, unbranched (isotropic) Willmore sphere in S 6 which is not S-Willmore, and thus does not have a dual Willmore surface. This example gives a negative answer to a long open problem (originally posed by Ejiri).

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

“…Lemma 2.8. [11,9] At the umbilic points of Y , the limit of µ goes to a finite number or infinity. When µ goes to infinity, [ Ŷ ] tends to [Y ], and at the limit point we have…”

“…In this subsection we will provide two kinds of examples. The first one concerns the new Willmore two-sphere derived in [9], which is the first example of Willmore two-spheres in S 6 admitting no dual surfaces. Here we will derive this surface together with one of its adjoint surface.…”

“…In the study of Willmore two-spheres, totally isotropic surfaces also play an important role [11,19,20]. Recently, Dorfmeister and Wang used the DPW method for the conformal Gauss map to study Willmore surfaces [9]. They obtained the first new Willmore two-sphere in S 6 , which admits no dual surface.…”

“…So it is very hard to use them to discuss the global geometry of Willmore surfaces. Due to this reason, in [9] Dorfmeister and Wang mainly dealt with the conformal Gauss maps of Willmore surfaces, which are anther kind of harmonic maps related to Willmore surfaces globally.…”

Willmore surfaces in spheres via loop groups III: on minimal surfaces in space forms

“…On the other hand, it also allows us to derive Willmore surfaces different from minimal surfaces in R n+2 by excluding this special type of normalized potentials. This is particularly important for the application of the main results of [14] to generic Willmore surfaces in S n+2 .…”