高度分支的CD空间的示例

摘要：

Ketterer和Rajala给出了一个度量测度空间的例子，满足测度收缩性质$MCP(0,3)$，在空间的不同区域具有不同的拓扑维数。在这篇文章中，我提出了一个改进的例子，它满足$CD(0,\infty)$条件，证明了非恒定的CD空间的拓扑维数。这个例子还表明，弱曲率维数界，在Lott-Sturm-Villani的意义下，是不足以推出任何合理的非分支条件。此外，它允许回答一些公开的问题，Schultz提出的，严格的曲率维数界和它们的稳定性方面的测量格罗莫夫-豪斯多夫收敛。

英文摘要：

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.

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