On a conjecture of Almgren: area-minimizing submanifolds with fractal singularities

TL;DR

This paper constructs area-minimizing submanifolds with fractal singularities on compact Riemannian manifolds, confirming Almgren's conjecture and allowing prescribed stratification of singular sets across various geometric categories.

Contribution

It provides the first explicit construction of fractal singularities in area-minimizing submanifolds, settling Almgren's conjecture with sharp dimension results.

Findings

Existence of fractal singular sets in area-minimizing submanifolds

Ability to prescribe stratification of singularities

Results valid for integral currents, mod v currents, and varifolds

Abstract

We construct area-minimizing submanifolds with fractal singular sets on compact Riemannian manifolds. Thus, we settle a conjecture by Almgren and our answer is sharp dimensionwise. Furthermore, we can prescribe arbitrarily the strata in the Almgren stratification of the singular sets of our area-minimizing submanifolds, and our results hold in the category of integral currents, mod v currents and stable stationary varifolds.