The fine structure of the singular set of area-minimizing integral currents I: the singularity degree of flat singular points

We consider an area-minimizing integral current of dimension $m$ and codimension at least $2$ and fix an arbitrary interior singular point $q$ where at least one tangent cone is flat. For any vanishing sequence of scales around $q$ along which the rescaled currents converge to a flat cone, we define a suitable “singularity degree” of the rescalings, which is a real number bigger than or equal to $1$. We show that this number is independent of the chosen sequence and we prove several interesting properties linked to its value. Our study prepares the ground for two companion works, where we show that the singular set is $(m-2)$-rectifiable and the tangent cone is unique at $\mathcal{H}^{m-2}$-a.e. point.