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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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