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

In this work we present new tools for studying the variations of the Willmore functional of immersed surfaces into ℝm. 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 ℝm, away possibly from isolated branched points, which minimizes the Willmore energy among all possible Lipschitz immersions in ℝm having an L2-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 extend to smooth conformal Willmore embeddings or global isothermic embeddings of the surface in that case. Finally, as a by-product of our analysis, we establish that inside a compact subspace of the moduli space the following holds: the weak limit of Palais Smale Willmore sequences are conformal Willmore, the weak limits of Palais Smale sequences of conformal Willmore are either conformal Willmore or global isothermic and finally we observe also that weakly converging Palais Smale sequences of global isothermic immersions are global isothermic. The analysis developed along the paper - in particular these last results - opens the door to the possibility of constructing new critical saddle points of the Willmore functional without or with constraints using min-max methods