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# Degeneration of 7-Dimensional Minimal Hypersurfaces with Bounded Index

## [Virtual] Hot Topics: Regularity Theory for Minimal Surfaces and Mean Curvature Flow March 21, 2022 - March 24, 2022

March 21, 2022 (09:00 AM PDT - 09:45 AM PDT)
Speaker(s): Nick Edelen (University of Notre Dame)
Location: MSRI: Online/Virtual
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

#### Degeneration Of 7-Dimensional Minimal Hypersurfaces With Bounded Index

Abstract

A 7D minimal and locally-stable hypersurface will in general have a discrete singular set, provided it has no singularities modeled on a union of half-planes. We show in this talk that the geometry/topology/singular set of these surfaces has uniform control, in the following sense: if $M_i$ is a sequence of 7D minimal hypersurfaces with uniformly bounded index and area, and discrete singular set, then up to a subsequence all the $M_i$ are bi-Lipschitz equivalent, with uniform Lipschitz bounds on the maps. As a consequence, we prove the space of $C^2$ embedded minimal hypersurfaces in a fixed $8$-manifold, having index $\leq I$, area $\leq \Lambda$, and discrete singular set, divides into finitely-many diffeomorphism types.