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We introduce a general scheme that permits to generate successive min-maxproblems for producing critical points of higher and higher indices toPalais-Smale Functionals in normal Banach manifolds equipped with completeFinsler structures. We call the resulting tree of minmax problems a minmaxhierarchy. We give several examples and in particular we explain how toimplement this scheme in the framework of the viscosity method introduced bythe author some years ago in order to give a new proof of the Willmoreconjecture after the famous result by Marques and Neves.

Minmax Hierarchies, Minimal Fibrations and a PDE based Proof of the Willmore Conjecture