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

TL;DR

This paper studies the limits of 7-dimensional minimal hypersurfaces with stability or bounded index, showing they converge to controlled singular limits and establishing finiteness of diffeomorphism types in certain classes.

Contribution

It demonstrates that sequences of stable or bounded index minimal hypersurfaces converge to controlled singular limits and proves finiteness of diffeomorphism types under geometric bounds.

Findings

Sequences of such hypersurfaces have controlled geometric and topological limits.

The space of minimal hypersurfaces with bounded volume and index has finitely many diffeomorphism types.

Finiteness persists under metric variations and singular limits.

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 $\phi_{i'}$ taking $\phi_{i'}(M_{1'}) = M_{i'}$. 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 $H^7(M) \leq \Lambda$ 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.