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

Abstract

AbstractA 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 $${\mathcal {H}}^6(\textrm{sing}M) = 0$$ H 6 ( 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 → 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 ′ with controlled bi-Lipschitz maps $$\phi _{i'}$$ ϕ i ′ taking $$\phi _{i'}(M_{1'}) = M_{i'}$$ 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 $$(N^8, g)$$ ( N 8 , g ) with a priori bounds $${\mathcal {H}}^7(M) \leqq \Lambda $$ H 7 ( M ) ≦ Λ and $$\textrm{index}(M) \leqq I$$ 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.