非常严格CD条件下的度量稳定性结果
A Metric Stability Result for the Very Strict CD Condition
摘要：
在（Calc.Var.PDE 2018）中，Schultz推广了Rajala和Sturm（Calc.Var.PDE 2014）的工作，证明了弱非分支条件在非常严格的CD空间的更一般的设置中成立。无论如何，类似于强CD条件所发生的情况，非常严格的CD条件对于测量的格罗莫夫-豪斯多夫收敛似乎是不稳定的。在这篇文章中，我证明了非常严格的CD条件下的稳定性结果，假设收敛序列和极限空间上的一些度量要求。证明依赖于\textit{consistent geodesic flow}和\textit{consistent plan selection}的概念，这允许单独对待的静态和动态部分的瓦塞尔斯坦测地线。作为应用，证明了具有结晶范数和勒贝格测度的度量测度空间$\mathbb R^N$满足非常严格的$\ CD(0,\infty)$条件.
英文摘要：
In (Calc.Var.PDE 2018) Schultz generalized the work of Rajala and Sturm (Calc.Var.PDE 2014), proving that a weak non-branching condition holds in the more general setting of very strict CD spaces. Anyway, similar to what happens for the strong CD condition, the very strict CD condition seems not to be stable with respect to the measured Gromov Hausdorff convergence. In this article I prove a stability result for the very strict CD condition, assuming some metric requirements on the converging sequence and on the limit space. The proof relies on the notions of \textit{consistent geodesic flow} and \textit{consistent plan selection}, which allow to treat separately the static and the dynamic part of a Wasserstein geodesic. As an application, I prove that the metric measure space $\mathbb R^N$ equipped with a crystalline norm and with the Lebesgue measure satisfies the very strict $\ CD(0,\infty)$ condition.
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