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

Abstract

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