Consider an area minimizing current modulo $p$ of dimension $m$ in a smooth Riemannian manifold of dimension $m+1$. We prove that its interior singular set is, up to a relatively closed set of dimension at most $m-2$, a $C^{1,α}$ submanifold of dimension $m-1$ at which, locally, $N\leq p$ regular sheets of the current join transversally, each sheet counted with a positive multiplicity $k_i$ so that $\sum_i k_i = p$. This completes the analysis of the structure of the singular set of area minimizing hypersurfaces modulo $p$, initiated by J. Taylor for $m=2$ and $p=3$ and extended by the authors to arbitrary $m$ and all odd $p$ in arXiv:2105.08135. We tackle the remaining case of even $p$ by showing that the set of singular points admitting a flat blow-up is of codimension at least two in the current. First, we prove a structural result for the singularities of minimizers in the linearized problem, by combining an epiperimetric inequality with an analysis of homogeneous minimizers to conclude that the corresponding degrees of homogeneity are always integers; second, we refine Almgren's blow-up procedure to prove that all flat singularities of the current persist as singularities of the $\mathrm{Dir}$-minimizing limit. An important ingredient of our analysis is the uniqueness of flat tangent cones at singular points, recently established by Minter and Wickramasekera in arXiv:2111.11202.
本文证明了在欧氏空间中整$1$ -流模$p$的商群中的每一个等价类都包含一个整流，并给出了它的质量及其边界质量的定量估计.此外，我们表明，这种说法的有效性$m$维积分电流模$p$意味着家庭的$(m-1)$维平面链的形式$pT$，与$T$一个平面链，是封闭的平坦规范。特别地，我们推导出这样的封闭性性质适用于$0$维平坦链，并且，使用Brian White的“The structure of minimizing hypersurfacets mod $4$“中的一个命题，也适用于余维$1$的平坦链。
