Failure of topological rigidity results for the measure contraction\n property

2014/03/12 by C. Ketterer, Ketterer, Christian, Tapio Rajala +1
Computer Science · Mathematics · #49Q20 (Secondary) #53C23 (Primary) 28A33 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1403.3105

openalex publication_date 2014/03/12 · openalex created_date 2025/10/27 · openalex updated_date 2026/07/28

Abstract

We give two examples of metric measure spaces satisfying the measure\ncontraction property MCP(K,N) but having different topological dimensions at\ndifferent regions of the space. The first one satisfies MCP(0,3) and contains a\nsubset isometric to \ℝ, but does not topologically split. The second\nspace satisfies MCP(2,3) and has diameter \π, which is the maximal possible\ndiameter for a space satisfying MCP(N-1,N), but is not a topological spherical\nsuspension. The latter example gives an answer to a question by Ohta.\n

Citations

• Ricci curvature for metric-measure spaces via optimal transport
• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Local Poincaré inequalities from stable curvature conditions on metric spaces
• Ricci curvature of Markov chains on metric spaces
• The splitting theorem in non-smooth context
• Non-branching geodesics and optimal maps in strong CD(K,∞)-spaces
• Decomposition of geodesics in the Wasserstein space and the\n globalization property

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• Failure of the measure contraction property via quotients in higher-step sub-Riemannian structures
• Sharp Poincaré inequalities under Measure Contraction Property