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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).

(2) Laboratoire d'étude des transferts en hydrologie et environnement (LTHE)-Niamey (3) Service d'aéronomie-Paris (4) Laboratoire d'océanographie et du climat (Locean)-Paris (5) Laboratoire de météorologie dynamique (LMD)-Paris (6) Laboratoire d'études en géophysique et océanographie spatiales (Legos)-Brest (7) Centre d'études spatiales de la biosphère (Cesbio)-Toulouse (8) Centre de recherches de climatologie (CRC)-Dijon (9) Amma Project Office-IPSL-Paris (10) Medias-France-Toulouse (11) Laboratoire d'aérologie-Toulouse (12) Centre d'étude des environnements terrestre et planétaires (CETP)-Vélizy (12) Laboratoire interuniversitaire des systèmes atmosphériques (Lisa)-Créteil (14) Hydrosciences Montpellier (HSM)-Cotonou (1) Résumé Le projet international Amma, d'ini-tiative française, a pour objectif d'améliorer la connaissance et la compréhension de la mousson d'Afrique de l'Ouest et de sa variabi-lité, de l'échelle journalière à l'échelle interannuelle. Le projet est motivé par la forte variabilité des précipita-tions associées à ce système de mous-son, et par ses conséquences sur la sécurité alimentaire, les ressources en eau et la santé. Abstract Amma, a multidisciplinary study of the West African monsoon

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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.