Weakly non-collapsed RCD spaces are strongly non-collapsed

2021/10/05 by Camillo Brena, Brena, Camillo, Nicola Gigli +5
Mathematics · #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2110.02420

openalex publication_date 2021/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that any weakly non-collapsed RCD space is actually non-collapsed, up to a renormalization of the measure. This confirms a conjecture raised by De Philippis and the second named author in full generality. One of the auxiliary results of independent interest that we obtain is about the link between the properties - tr(Hessf)=Δf on U\subsetX for every f sufficiently regular, - \mathfrakm=c\mathscrHⁿ on U\subsetX for some c>0, where U⊂ X is open and X is a - possibly collapsed - RCD space of essential dimension n.

Citations

• Ricci curvature for metric-measure spaces via optimal transport
• On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces
• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Local Poincaré inequalities from stable curvature conditions on metric spaces
• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products
• The splitting theorem in non-smooth context
• A Bishop-type inequality on metric measure spaces with Ricci curvature bounded below
• Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows
• Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces
• Sobolev mappings between RCD spaces and applications to harmonic maps: a heat kernel approach
• Parallel transport on non-collapsed RCD(K,N) spaces

Related

• Non-collapsed spaces with Ricci curvature bounded from below
• A characterization of non-collapsed RCD(K, N) spaces via Einstein tensors
• Weakly noncollapsed RCD spaces with upper curvature bounds
• From almost smooth spaces to RCD spaces
• Volume bounds for the quantitative singular strata of non collapsed RCD metric measure spaces