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

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Nick Edelen University of Notre Dame

A 7-dimensional area-minimizing hypersurface $M$ can have in general a discrete singular set.  The same is true if M is only locally-stable for the area-functional, provided $\haus^6(sing M) = 0$.  In this paper we show that if $M_i$ is a sequence of 7D minimal hypersurfaces with discrete singular set which are minimizing, stable, or have bounded index, and varifold-converge to some $M$, then the geometry, topology, and singular set of the $M_i$ can degenerate in only a very precise manner.  We show that one can always parameterize'' a subsequence $i'$ by ambient, controlled bi-Lipschitz maps taking $\phi_{i'}(M_1) = M_{i'}$.  As a consequence, we prove that the space of closed, $C^2$ embedded minimal hypersurfaces in a closed 8-manifold $(N, g)$ with a priori bounds $\haus^7(M) \leq \Lambda$ and $index(M) \leq I$ 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.