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Minmax Hierarchies, Minimal Fibrations and a PDE based Proof of the Willmore Conjecture
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We introduce a general scheme that permits to generate successive min-max problems for producing critical points of higher and higher indices to Palais-Smale Functionals in normal Banach manifolds equipped with complete Finsler structures. We call the resulting tree of minmax problems a minmax hierarchy. We give several examples and in particular we explain how to implement this scheme in the framework of the viscosity method introduced by the author some years ago in order to give a new proof of the Willmore conjecture after the famous result by Marques and Neves.
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Minimizing the Gauss map area of surfaces in $\mathbb{S}^3$
Closed oriented genus-g surfaces in S^3 have Gauss map area at least 4π(1+g); equality holds only for round spheres, and minimizing embedded sequences bubble into one positive and g negative spherical cycles.
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