Generic Regularity of Minimal Hypersurfaces in Dimension 8

2020/12/10 by Yangyang Li, Li, Yangyang, Zhihan Wang +1
Mathematics · #49Q05 #53A10 #Algebraic Geometry and Number Theory #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2012.05401

openalex publication_date 2020/12/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we show that every 8-dimensional closed Riemmanian manifold with C^∞-generic metrics admits a smooth minimal hypersurface. This generalized previous results by N. Smale and Chodosh-Liokumovich-Spolaor. Different from their local perturbation techniques, our construction is based on a global perturbation argument in and a novel geometric invariant which counts singular points with suitable weights.

Citations

• Normal and Integral Currents
• Curvature estimates for minimal hypersurfaces
• Minimal Varieties in Riemannian Manifolds
• Existence of hypersurfaces with prescribed mean curvature I - Generic min-max
• Generic Transversality of Minimal Submanifolds and Generic Regularity of Two-Dimensional Area-Minimizing Integral Currents
• Singular behavior and generic regularity of min-max minimal hypersurfaces

Related

• Minimal hypersurfaces for generic metrics in dimension 8
• Existence of Infinitely Many Minimal Hypersurfaces in Higher-dimensional Closed Manifolds with Generic Metrics
• Regularity of sets of least perimeter in Riemannian manifolds
• Generic regularity for minimizing hypersurfaces in dimension 11
• On isolated singularities and generic regularity of min-max CMC hypersurfaces