Minimal hypersurfaces with bounded index

Otis Chodosh; Daniel Ketover; Davi Maximo

doi:10.1007/s00222-017-0717-5

Minimal hypersurfaces with bounded index

Otis Chodosh
, Daniel Ketover
, Davi Maximo

Research output: Contribution to journal › Article › peer-review

36 Scopus citations

Abstract

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 (Mⁿ, g) , 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.

Original language	English (US)
Pages (from-to)	617-664
Number of pages	48
Journal	Inventiones Mathematicae
Volume	209
Issue number	3
DOIs	https://doi.org/10.1007/s00222-017-0717-5
State	Published - Sep 1 2017

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1007/s00222-017-0717-5

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abstract = "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 (Mn, g) , 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.",

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AB - 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 (Mn, g) , 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.

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Minimal hypersurfaces with bounded index

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