Perelman's doubling theorem asserts that the metric space obtained by gluing along their boundaries two copies of an Alexandrov space with curvature ≥κ 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 MCP sense. The counterexample is given by the Grushin half-plane, which satisfies the MCP(0,N) if and only if N≥4, while its double satisfies the MCP(0,N) if and only if N≥5.
A counterexample to gluing theorems for MCP metric measure spaces / Rizzi, L.. - In: BULLETIN OF THE LONDON MATHEMATICAL SOCIETY. - ISSN 0024-6093. - 50:5(2018), pp. 781-790. [10.1112/blms.12186]
A counterexample to gluing theorems for MCP metric measure spaces
Rizzi L.
2018
Abstract
Perelman's doubling theorem asserts that the metric space obtained by gluing along their boundaries two copies of an Alexandrov space with curvature ≥κ 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 MCP sense. The counterexample is given by the Grushin half-plane, which satisfies the MCP(0,N) if and only if N≥4, while its double satisfies the MCP(0,N) if and only if N≥5.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
