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Non-compactness of the space of minimal hypersurfaces. (English) Zbl 1386.49059

Author’s abstract: We show that the space of min-max minimal hypersurfaces is non-compact when the manifold has an analytic metric of positive Ricci curvature and dimension \(3\leq n+1\leq 7\). Furthermore, we show that bumpy metrics with positive Ricci curvature admit minimal hypersurfaces with unbounded index+area. When combined with the recent work of F. C. Marques and A. Neves, we then deduce some new properties regarding the infinitely many minimal hypersurfaces they found.

MSC:

49Q05 Minimal surfaces and optimization
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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