Title: Minmax Hierarchies and Minimal Surfaces in Manifolds,
Abstract: 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 Banach manifolds equipped with Finsler structures.
We call the resulting tree of minmax problems a minmax hierarchy. Using the
viscosity approach to the minmax theory of minimal surfaces introduced by the
author in a series of recent works, we explain how this scheme can be deformed
for producing smooth minimal surfaces of strictly increasing area in arbitrary
codimension. We implement this scheme to the case of the $3-$dimensional
sphere. In particular we are giving a min-max characterization of the Clifford
Torus and conjecture what are the next minimal surfaces to come in the $S^3$
hierarchy. Among other results we prove here the lower semi continuity of the
Morse Index in the viscosity method below an area level.