Geometry and Topology Seminar
Homologically area-minimizing currents are minimizers of area among homologous competitors. By Almgren-De Lellis-Spadaro's big regularity theorem, such currents are chains over smooth submanifolds whose closure adds a set of codimension at least two. Prominent examples include holomorphic subvarieties and special Lagrangians. For instance, a classification of area-minimizers for every homology class in a complex torus would immediately give a true or false for the Hodge conjecture of this variety. Thus, understanding the behavior of area-minimizing currents would have immense applications to many facets of differential geometry. In this talk, we will review some recent progress on the behavior of area-minimizing surfaces in general manifolds. We start with the simplest type of singularities, i.e., the self-intersection of area-minimizing immersions. We illustrate that both non-smoothable rigid singularities and malleable singularities with arbitrary, even fractal, deformations exist in abundance. This settles a conjecture of Almgren in the 1980s about the existence of area-minimizing currents with fractal singular sets. Then we go to more general types of singularities and show that the bordism rings give fairly general obstructions to the smoothing of singularities. Combining with the previous case, we show that area-minimizing currents in every homology class in general dimensions and codimensions can have non-smoothable singularities, thus essentially settling a conjecture of White in the 1980s about the generic smoothness of area-minimizing currents. We will also discuss some cases where we can fully determine the moduli space of area-minimizing currents.