# Seminars & Colloquia Calendar

Joint Princeton-Rutgers Seminar on Geometric PDE's

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

#### Nick Edelen, University of Notre Dame

Location:  zoom
Date & time: Monday, 24 May 2021 at 1:00PM - 2:00PM

Abstract: 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.

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