Example of a Highly Branching CD Space

Abstract

In Ketterer and Rajala (Potential Anal 42:645–655, 2014) showed an example of metric measure space, satisfying the measure contraction property mathsf {MCP}(0,3), that has different topological dimensions at different regions of the space. In this article I propose a refinement of that example, which satisfies the mathsf {CD}(0,infty ) condition, proving the non-constancy of topological dimension for CD spaces. This example also shows that the weak curvature dimension bound, in the sense of Lott–Sturm–Villani, is not sufficient to deduce any reasonable non-branching condition. Moreover, it allows to answer to some open question proposed by Schultz in (Calc Var Partial Differ Equ 57:1–11, 2018), about strict curvature dimension bounds and their stability with respect to the measured Gromov–Hausdorff convergence.