Variational Principles for immersed Surfaces with $L^2$-bounded Second Fundamental Form

TL;DRAbstract

In this work we present new fundamental tools for studying the variations of the Willmore functional of immersed surfaces into $R^m$. This approach gives for instance a new proof of the existence of a Willmore minimizing embedding of an arbitrary closed surface in arbitrary codimension. We explain how the same approach can solve constraint minimization problems for the Willmore functional. We show in particular that, for a given closed surface and a given conformal class for this surface, there is an immersion in $R^m$, away possibly from isolated branched points, which minimizes the Willmore energy among all possible Lipschitz immersions in $R^m$ having an $L^2-$bounded second fundamental form and realizing this conformal class. This branched immersion is either a smooth Conformal Willmore branched immersion or an isothermic branched immersion. We show that branched points do not exist whenever the minimal energy in the conformal class is less than $8π$ and that these immersions exten

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