Excess decay for minimizing hypercurrents mod $2Q$

We consider codimension $1$ area-minimizing $m$-dimensional currents $T$ mod an even integer $p=2Q$ in a $C^2$ Riemannian submanifold $\mathrm{\Sigma }$ of the Euclidean space. We prove a suitable excess-decay estimate towards the unique tangent cone at every point $q\in \mathrm{s}\mathrm{p}\mathrm{t}(T)\setminus {\mathrm{s}\mathrm{p}\mathrm{t}}^p(\mathrm{\partial }T)$ where at least one such tangent cone is $Q$ copies of a single plane. While an analogous decay statement was proved in arXiv:2111.11202 as a corollary of a more general theory for stable varifolds, in our statement we strive for the optimal dependence of the estimates upon the second fundamental form of $\mathrm{\Sigma }$. This technical improvement is in fact needed in arXiv:2201.10204 to prove that the singular set of $T$ can be decomposed into a $C^{1,\alpha }$ $(m-1)$-dimensional submanifold and an additional closed remaining set of Hausdorff dimension at most $m-2$.