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The Willmore energy is a classical (almost) conformal invariant measuring the complexity of an immersion.  Its critical points are called Willmore surfaces and can be understood as elasticas generalizing minimal surfaces in any of the three conformal models. In this talk, we will angle our study through the tool of the Conformal Gauss Map. This application into a de Sitter space offers a natural link between harmonic surfaces in a Lorentz context and Willmore surfaces, and allows one to turn many non linear aspects of Willmore studies, into Lorentzian linear ones, with one key issues: it breaks down at umbilic points. An approach of these singularities through a Gauss-Bonnet formula will allow us to recover links between the Willmore energy and the Conformal/Renormalized volumes.