Degeneration of 7-Dimensional Minimal Hypersurfaces Which are Stable or Have a Bounded Index

A 7-dimensional area-minimizing embedded hypersurface \(M^7\) $M^7$ will in general have a discrete singular set, and the same is true if M is locally stable provided \({\mathcal {H}}^6(\textrm{sing}M) = 0\) ${\mathcal{H}}^6(\text{sing}M)=0$ . We show that if \(M_i^7\) $M_i^7$ is a sequence of 7D minimal hypersurfaces which are minimizing, stable, or have bounded index, then \(M_i \rightarrow M\) $M_i\to M$ can limit to a singular \(M^7\) $M^7$ with only very controlled geometry, topology, and singular set. We show that one can always “parameterize” a subsequence \(i'\) $i^{\prime }$ with controlled bi-Lipschitz maps \(\phi _{i'}\) ${\varphi }_{i^{\prime }}$ taking \(\phi _{i'}(M_{1'}) = M_{i'}\) ${\varphi }_{i^{\prime }}(M_{1^{\prime }})=M_{i^{\prime }}$ . As a consequence, we prove the space of smooth, closed, embedded minimal hypersurfaces M in a closed Riemannian 8-manifold \((N^8, g)\) $(N^8,g)$ with a priori bounds \({\mathcal {H}}^7(M) \leqq \Lambda \) ${\mathcal{H}}^7(M)\leqq \mathrm{\Lambda }$ and \(\textrm{index}(M) \leqq I\) $\text{index}(M)\leqq I$ divides into finitely-many diffeomorphism types, and this finiteness continues to hold if one allows the metric g to vary, or M to be singular.