Abstract
We construct area-minimizing submanifolds with fractal singular sets on compact Riemannian manifolds. Thus, we settle Problem 5.4 in [4] 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.
