Example of an Highly Branching CD Space

TL;DR

This paper refines a known metric measure space example to demonstrate that topological dimension can vary in CD spaces and shows limitations of weak curvature bounds in ensuring non-branching.

Contribution

It provides a refined example satisfying CD(0,∞) with variable topological dimension, addressing open questions about curvature bounds and stability under Gromov-Hausdorff convergence.

Findings

Topological dimension varies within the constructed CD space.

Weak curvature dimension bounds do not imply non-branching.

The example answers open questions on curvature bounds and stability.

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 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, about strict curvature dimension bounds and their stability with respect to the measured Gromov Hausdorff convergence.