Minimal hypersurfaces with bounded index

By Otis Chodosh, Daniel Ketover, D. M'aximo. 2015

52 citations recorded by the source corpus.

We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold $$(M^{n},g)$$(Mn,g), $$3\le n\le 7$$3≤n≤7, can degenerate. Loosely speaking, our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces, up to controlled errors. Several compactness/finiteness theorems follow from our local picture.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1ccf049a06619c289989cd808b4a01f402b27a5a

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Comparing the Morse index and the first Betti number of minimal hypersurfaces (2016, background): “ed orientable stable (= index zero) minimal surface in an orientable three-manifold with positive scalar curvature is necessarily a two-sphere. In these ambient manifolds, Chodosh, Ketover and Máximo [5] have recently proved that the set of closed embedded minimal surfaces with bounded index cannot ”
Morse index of multiplicity one min-max minimal hypersurfaces (2015, methodology): “In this paper, we will use Sharp’s compactness theorem for minimal hypersurfaces with both area and index uniformly bounded from above (see also Carlotto [7], Chodosh-Ketover-Maximo [10] and Li-Zhou [26] for other compactness theorems).”