Fine structure of singularities in area-minimizing currents mod(𝑞)

Abstract We study fine structural properties related to the interior regularity of 𝑚-dimensional area-minimizing currents mod ⁡ ( q ) \operatorname{mod}(q) in arbitrary codimension. We show (i) the set of points where at least one tangent cone is translation invariant along m − 1 m-1 directions is locally a connected C 1 , β C^{1,\beta} submanifold, and moreover such points have unique tangent cones; (ii) the remaining part of the singular set is countably ( m − 2 ) (m-2) -rectifiable, with a unique flat tangent cone (possibly with multiplicity) at H m − 2 \mathcal{H}^{m-2} -a.e. flat singular point. These results are consequences of fine excess decay theorems as well as almost monotonicity of a universal frequency function.