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

Abstract 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 {H}⁶ (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 ₈' ϕ i ′ taking ₈' (M₁') = M₈' 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 {H}⁷ (M) 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.