Generic scarring for minimal hypersurfaces along stable hypersurfaces

2020/06/04 by Antoine Song, Xin Zhou, Song, Antoine +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.AP #math.DG

paper · pdf · doi:10.48550/arxiv.2006.03038

v2: final version, to appear in GAFA

openalex publication_date 2020/06/04 · arxiv created 2021/09/09 · arxiv updated 2021/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Mⁿ⁺¹ be a closed manifold of dimension 3≤ n+1≤ 7. We show that for a C^∞-generic metric g on M, to any connected, closed, embedded, 2-sided, stable, minimal hypersurface S⊂ (M,g) corresponds a sequence of closed, embedded, minimal hypersurfaces \Σ_k\ scarring along S, in the sense that the area and Morse index of Σ_k both diverge to infinity and, when properly renormalized, Σ_k converges to S as varifolds. We also show that scarring of immersed minimal surfaces along stable surfaces occurs in most closed Riemannian 3-manifods.

Citations

• Ricci flow with surgery on three-manifolds
• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature
• A dichotomy for minimal hypersurfaces in manifolds thick at infinity

Cited by

• Deformations of Totally Geodesic Foliations and Minimal Surfaces in Negatively Curved 3-Manifolds

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