Mathematics Seminar Series: Geometric Analysis

Title: A min-max gap characterization of minimal foliations on a torus and applications

Abstract: Aubry–Mather theory concerns minimal orbits of convex Hamiltonian systems. Morse, Bangert, and Auer introduced a higher-dimensional analogue for minimal hypersurfaces. In this talk, I will describe their theory in the setting of area-minimizing hypersurfaces on a torus. After passing to the universal cover, one can associate to each area-minimizing hypersurface an asymptotic invariant, called its homological direction. Hypersurfaces with the same homological direction form a lamination. A central problem is to determine when such a lamination is actually a foliation. I will discuss a recent work in which Almgren–Pitts min-max theory is used to characterize minimal foliations on tori. I will also explain how to find minimal hypersurfaces that are not area minimizing and that lie inside the gaps of a lamination.