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

We study fine structural properties related to the interior regularity of 𝑚-dimensional area-minimizing currents <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>mod</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>q</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> mod(q) in arbitrary codimension. We show (i) the set of points where at least one tangent cone is translation invariant along <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> m-1 directions is locally a connected <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>β</m:mi> </m:mrow> </m:msup> </m:math> C1,β submanifold, and moreover such points have unique tangent cones; (ii) the remaining part of the singular set is countably <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>m</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (m-2) -rectifiable, with a unique flat tangent cone (possibly with multiplicity) at <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="script">H</m:mi> <m:mrow> <m:mi>m</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> Hᵐ⁻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.