Minimal hypersurfaces with bounded index

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.

Final version

References in corpus (6)

• Effective versions of the positive mass theorem
• On the topology and index of minimal surfaces
• Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area
• Compactness of minimal hypersurfaces with bounded index
• Effective Finiteness of irreducible Heegaard splittings of non Haken 3-manifolds
• Index and topology of minimal hypersurfaces in R^n

Cited by in corpus (16)

• Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area
• Compactness and generic finiteness for free boundary minimal hypersurfaces (I)
• Geometric convergence results for closed minimal surfaces via bubbling analysis
• Morse index, Betti numbers and singular set of bounded area minimal hypersurfaces
• Stable minimal hypersurfaces in ${\mathbf{R}}^5$
• Index and topology of minimal hypersurfaces in R^n
• Generic scarring for minimal hypersurfaces along stable hypersurfaces
• Deformations of Singular Minimal Hypersurfaces I, Isolated Singularities
• Minimal hypersurfaces with arbitrarily large area
• Erratum to "Minimal surfaces in finite volume noncompact hyperbolic $3$-manifolds"
• A note on minimal surfaces with bounded index
• The p-widths of a surface
• Comparing the Morse index and the first Betti number of minimal hypersurfaces
• The systole of large genus minimal surfaces in positive Ricci curvature
• Appearance of stable minimal spheres along the Ricci flow in positive scalar curvature
• Ricci curvature and minimal hypersurfaces with large Betti numbers