Minimal hypersurfaces with bounded index

2015/09/30 by Otis Chodosh, Daniel Ketover, Davi Máximo +1
Mathematics · #Bounded function #Compact space #Compactness theorem #Computer science #Degenerate energy levels #Dimension (graph theory) #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Index (typography) #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Minimal surface #Physics #Pure mathematics #Riemannian manifold #Uniform boundedness #math.DG #math.GT

paper · pdf · doi:10.1007/s00222-017-0717-5

published as Invent. Math., Vol. 209, No. 3, pp. 617--664 (2017) · Final version

openalex publication_date 2017/02/14 · arxiv created 2017/08/21 · arxiv updated 2019/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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 of dimension at most seven, 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 follows our local picture.

Citations

• The existence of a black hole due to condensation of matter
• Curvature estimates for minimal hypersurfaces
• On the First Variation of a Varifold
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• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature
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