Example of an Highly Branching CD Space
Abstract
Ketterer and Rajala showed an example of metric measure space, satisfying the measure contraction property $MCP(0,3)$, that has different topological dimensions at different regions of the space. In this article I propose a refinement of that example, which satisfies the $CD(0,\infty)$ condition, proving the nonconstancy of topological dimension for CD spaces. This example also shows that the weak curvature dimension bound, in the sense of LottSturmVillani, is not sufficient to deduce any reasonable nonbranching condition. Moreover, it allows to answer to some open question proposed by Schultz, about strict curvature dimension bounds and their stability with respect to the measured Gromov Hausdorff convergence.
 Publication:

arXiv eprints
 Pub Date:
 January 2021
 arXiv:
 arXiv:2102.00042
 Bibcode:
 2021arXiv210200042M
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Differential Geometry