A counterexample to gluing theorems for MCP metric measure spaces

TL;DR

This paper demonstrates that the gluing property for Alexandrov spaces with curvature bounds does not extend to MCP metric measure spaces, using the Grushin half-plane as a counterexample.

Contribution

It provides a counterexample showing the failure of gluing theorems for MCP spaces, highlighting limitations in synthetic Ricci curvature theory.

Findings

The Grushin half-plane satisfies MCP(0,N) iff N ≥ 4.

Its double satisfies MCP(0,N) iff N ≥ 5.

Gluing does not preserve MCP curvature bounds in general.

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 $N\geq 5$.