Minmax Hierarchies and Minimal Surfaces in Manifolds

TL;DR

This paper develops a hierarchical min-max framework for generating higher-index critical points of functionals, applying it to minimal surfaces in spheres, including a characterization of the Clifford Torus.

Contribution

It introduces a minmax hierarchy scheme for constructing minimal surfaces with increasing area and Morse index, extending viscosity methods to higher codimension cases.

Findings

Established a minmax hierarchy for minimal surfaces in spheres.

Provided a characterization of the Clifford Torus via minmax methods.

Proved lower semi-continuity of Morse index in the viscosity approach.

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.