A min-max gap characterization of minimal foliations on the torus

TL;DR

This paper extends Mather's energy to higher dimensions using min-max theory to analyze minimal foliations on tori, revealing conditions for their existence and properties.

Contribution

It introduces a higher-dimensional analogue of Mather's variational barrier theory using min-max energy, providing new criteria for minimal foliation existence.

Findings

Generic metrics with gaps in minimal laminations contain non-area-minimizing minimal hypersurfaces.

Established criteria for the existence of minimal hypersurface foliations on tori.

Derived a recurrence property for totally irrational minimal foliations.

Abstract

We extend an energy introduced by Mather to the setting of Almgren-Pitts min-max theory and obtain a parametric, higher-dimensional analogue of Mather's variational barrier theory for twist maps and geodesics on tori. We use this energy to establish several criteria for the existence of foliations of the $n$-torus by minimal hypersurfaces. We show that for a generic metric, whenever a lamination by area-minimizing hypersurfaces of the $n$-torus contains a gap, there exists a minimal hypersurface inside the gap that is not area-minimizing. As an application, we derive a recurrence property for totally irrational minimal foliations.