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There are various ways that lead to consider Willmore surfaces:
merging the study of minimal surfaces and conformal invariance, considering nonlinear models for elastic plates, or as an example of a 4th-order PDE problem in geometric analysis. I will give a brief general description of this topic, introduce some of the challenges it offers to an analyst, and finally present some aspects close to my research in this area, namely boundary value problems and associated gradient flow.
http://cvgmt.sns.it/seminar/859/