Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below

2024/05/06 by Elia Brué, Bruè, Elia, Alessandro Pigati +3
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2405.03839

openalex publication_date 2024/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate the topological regularity and stability of noncollapsed Ricci limit spaces (M_iⁿ,g_i,p_i)→ (Xⁿ,d). We confirm a conjecture proposed by Colding and Naber in dimension n=4, showing that the cross-sections of tangent cones at a given point x∈ X⁴ are all homeomorphic to a fixed spherical space form S³/Γ_x, and Γ_x is trivial away from a 0-dimensional set. In dimensions n>4, we show an analogous statement at points where all tangent cones are (n-4)-symmetric. Furthermore, we prove that (n-3)-symmetric noncollapsed Ricci limits are topological manifolds, thus confirming a particular case of a conjecture due to Cheeger, Colding, and Tian. Our analysis relies on two key results, whose importance goes beyond their applications in the study of cross-sections of noncollapsed Ricci limit spaces: (i) A new manifold recognition theorem for noncollapsed \rm RCD(-2,3) spaces. (ii) A cone rigidity result ruling out noncollapsed Ricci limit spaces of the form ℝ^n-3× C(\mathbbRP²).

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