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

A 7-dimensional area-minimizing embedded hypersurface M⁷ M 7 will in general have a discrete singular set, and the same is true if M is locally stable provided H⁶(singM) = 0 H 6 ( sing M ) = 0 . We show that if Mᵢ⁷ M i 7 is a sequence of 7D minimal hypersurfaces which are minimizing, stable, or have bounded index, then Mᵢ → M M i → M can limit to a singular M⁷ M 7 with only very controlled geometry, topology, and singular set. We show that one can always “parameterize” a subsequence $$i'$$ i ′ with controlled bi-Lipschitz maps φ i' ϕ i ′ taking φ i'(M1') = Mi' ϕ i ′ ( M 1 ′ ) = M i ′ . As a consequence, we prove the space of smooth, closed, embedded minimal hypersurfaces M in a closed Riemannian 8-manifold (N⁸, g) ( N 8 , g ) with a priori bounds H⁷(M) Λ H 7 ( M ) ≦ Λ and index(M) I index ( M ) ≦ 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.