We consider codimension $1$ area-minimizing $m$-dimensional currents $T$ mod an even integer $p=2Q$ in a $C^2$ Riemannian submanifold $\Sigma$ of the Euclidean space. We prove a suitable excess-decay estimate towards the unique tangent cone at every point $q\in \spt (T)\setminus \spt^p (\partial T)$ where at least one such tangent cone is $Q$ copies of a single plane. While an analogous decay statement was proved in a recent work by Minter and Wickramaseckera 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 $\Sigma$. This improvement is in fact crucial in a previous work of ours, where we 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 $m-2$.

Preprint