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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).

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If at a point z 0 ∈ D, f has a zero and g is holomorphic, then, as was pointed out in [5] the map Φ 1 fails to be an immersion at z = z 0 (see also the appendix).If g(z) has a zero at z = z 0 , and f is defined and nonzero there, then one gets an umbilic at the image of z 0 in R 3 .This allows one to use the DPW method, to construct a CMC immersion with a prescribed distribution of umbilics in the domain D.

Let G be a complex Lie group and ΛG denote the group of maps from the unit circle S 1 into G, of a suitable class. A differentiable map F from a manifold M into ΛG, is said to be of connection order ( b a ) if the Fourier expansion in the loop parameter λ of the S 1 -family of Maurer-Cartan forms for F, namely F −1 λ dF λ , is of the form b i=a α i λ i . Most integrable systems in geometry are associated to such a map. Roughly speaking, the DPW method used a Birkhoff type splitting to reduce a harmonic map into a symmetric space, which can be represented by a certain order ( 1 −1 ) map, into a pair of simpler maps of order ( −1 −1 ) and ( 1 1 ), respectively. Conversely, one could construct such a harmonic map from any pair of ( −1 −1 ) and ( 1 1 ) maps. This allowed a Weierstrass type description of harmonic maps into symmetric spaces. We extend this method to show that, for a large class of loop groups, a connection order ( b a ) map, for a < 0 < b, splits uniquely into a pair of ( −1 a ) and ( b 1 ) maps. As an application, we show that constant non-zero sectional curvature submanifolds with flat normal bundle of a sphere or hyperbolic space split into pairs of flat submanifolds, reducing the problem (at least locally) to the flat case. To extend the DPW method sufficiently to handle this problem requires a more general Iwasawa type splitting of the loop group, which we prove always holds at least locally. Mathematics Subject Classification (2000)Primary 37K10 · 37K25 · 53C42 · 53B25; Secondary 53C35 J. Dorfmeister was supported by DFG grant DO 776.

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