Degeneration of 7-dimensional minimal hypersurfaces which are stable or have bounded index

Abstract

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