A Metric Stability Result for the Very Strict CD Condition

TL;DR

This paper proves a stability result for the very strict CD condition under certain metric assumptions and demonstrates that Euclidean spaces with crystalline norms satisfy this condition.

Contribution

It establishes the first stability result for the very strict CD condition and introduces methods to handle static and dynamic parts of Wasserstein geodesics.

Findings

Stability of the very strict CD condition under specific metric convergence.

Euclidean spaces with crystalline norms satisfy the very strict CD(0,∞) condition.

Introduction of consistent geodesic flow and plan selection techniques.

Abstract

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.