*Submitted Paper*

**Inserted:** 16 aug 2023

**Last Updated:** 9 nov 2023

**Year:** 2023

**Abstract:**

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 $q\in {\textrm{spt}} (T)\setminus {\textrm{spt}}^p (\partial T)$ where at least one such tangent cone is $Q$ copies of a single plane. While an analogous decay statement was proved in [11] 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 technical improvement is in fact needed in [5] 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 $m-2$.

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