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In this paper we make a detailed and self-contained study of theconformalGauss map. Then, starting from the seminal work of R. Bryant and thenotion of conformal Gauss map, we recover many fundamental properties ofWillmore surfaces. We also get new results like some characterizations ofminimal and constant meancurvature (CMC) surfaces in term of their conformalGauss map behavior.

Conformal Gauss Map Geometry and Application to Willmore Surfaces in Model Spaces