Generic regularity of homologically area minimizing hypersurfaces in eight dimensional manifolds (Q1905458)

Consider a smooth, compact manifold \(N\) of dimension \(n+1\) with nontrivial homology class \(H_n (N, \mathbb{Z})\). For \(k\geq 3\) let \(M^k\) denote the class of \(C^k\) metrics on \(N\), endowed with the \(C^k\) topology, and let \(|\cdot |_k\) be a norm defining the topology. Given \(\alpha\in H_n (N, \mathbb{Z})\), \(\alpha\neq 0\), the subclass \({\mathcal F}^k_\alpha\) of \(M^k\) consists of all metrics \(g\) such that there exists a smooth area minimizing (relative to \(g\)) \(n\)-dimensional, integer multiplicity current \(T\) homologous to \(\alpha\). The main result of this paper states that for \(n=7\), \({\mathcal F}^k_\alpha\) contains an open subset being dense in \(M^k\).