# A Metric Stability Result for the Very Strict CD Condition

@inproceedings{Magnabosco2021AMS, title={A Metric Stability Result for the Very Strict CD Condition}, author={Mattia Magnabosco}, year={2021} }

In [15] Schultz generalized the work of Rajala and Sturm [13], proving that a weak nonbranching 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 (cf. [11]). 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… Expand

#### One Citation

Optimal maps and local-to-global property in negative dimensional spaces with Ricci curvature bounded from below

- Mathematics
- 2021

In this paper we investigate two important properties of metric measure spaces satisfying the reduced curvature-dimension condition for negative values of the dimension parameter: the existence of a… Expand

#### References

SHOWING 1-10 OF 25 REFERENCES

Example of an Highly Branching CD Space

- Mathematics
- 2021

In [3] 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… Expand

Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows

- Mathematics
- 2015

Aim of this paper is to discuss convergence of pointed metric measure spaces in absence of any compactness condition. We propose various definitions, show that all of them are equivalent and that for… Expand

Equivalent definitions of very strict $CD(K,N)$ -spaces

- Mathematics
- 2019

We show the equivalence of the definitions of very strict $CD(K,N)$ -condition defined, on one hand, using (only) the entropy functionals, and on the other, the full displacement convexity class… Expand

Metric measure spaces with Riemannian Ricci curvature bounded from below

- Mathematics
- 2014

In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X,d,m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler… Expand

NON-BRANCHING GEODESICS AND OPTIMAL MAPS IN STRONG CD(K,∞)-SPACES

- 2013

We prove that in metric measure spaces where the entropy functional is Kconvex along every Wasserstein geodesic any optimal transport between two absolutely continuous measures with finite second… Expand

Riemannian Ricci curvature lower bounds in metric measure spaces with -finite measure

- Mathematics
- 2015

In prior work (4) of the first two authors with Savare, a new Riemannian notion of lower bound for Ricci curvature in the class of metric measure spaces (X,d,m) was introduced, and the corresponding… Expand

Ricci curvature for metric-measure spaces via optimal transport

- Mathematics
- 2004

We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of the… Expand

Failure of the local-to-global property for CD(K,N) spaces

- Mathematics
- 2013

Given any K and N we show that there exists a compact geodesic metric measure space satisfying locally the CD(0,4) condition but failing CD(K,N) globally. The space with this property is a suitable… Expand

On one-dimensionality of metric measure spaces

- Mathematics
- 2019

In this paper, we prove that a metric measure space which has at least one open set isometric to an interval, and for which the (possibly non-unique) optimal transport map exists from any absolutely… Expand

Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below

- Mathematics
- 2014

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces $(X,\mathsf {d},\mathfrak {m})$. Our main results are: A general… Expand