A counterexample to gluing theorems for MCP metric measure spaces

Abstract : Perelman's doubling theorem asserts that the metric space obtained by gluing along their boundaries two copies of an Alexandrov space with curvature $\geq \kappa$ is an Alexandrov space with the same dimension and satisfying the same curvature lower bound. We show that this result cannot be extended to metric measure spaces satisfying synthetic Ricci curvature bounds in the $\mathrm{MCP}$ sense. The counterexample is given by the Grushin half-plane, which satisfies the $\mathrm{MCP}(0,N)$ if and only if $N\geq 4$, while its double satisfies the $\mathrm{MCP}(0,N)$ if and only if $N\geq 5$.
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https://hal.archives-ouvertes.fr/hal-01635434
Contributor : Luca Rizzi <>
Submitted on : Wednesday, November 15, 2017 - 10:58:37 AM
Last modification on : Thursday, February 7, 2019 - 5:15:19 PM

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Luca Rizzi. A counterexample to gluing theorems for MCP metric measure spaces. Bulletin of the London Mathematical Society, London Mathematical Society, In press, 50 (5), pp.781-790. ⟨10.1112/blms.12186⟩. ⟨hal-01635434⟩

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