Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds

created by han1 on 23 Jan 2019
modified on 05 Feb 2019

[BibTeX]

Preprint

Inserted: 23 jan 2019
Last Updated: 5 feb 2019

Year: 2019

ArXiv: 1902.00942 PDF
Notes:

This is a further work after http:/cvgmt.sns.itpaper3594

In this paper we investigate Lott-Sturm-Villani's synthetic lower Ricci curvature bound on Riemannian manifolds with boundary. We prove several new measure rigidity results, which completely characterize ${\rm CD}(K, \infty)$ condition and non-collapsed ${\rm CD}(K, N)$ condition on Riemannian manifolds with boundary. In particular, using recent results on $L^1$-optimal transportation theory, we prove that ${\rm CD}(K, \infty)$ condition implies geodesical convexity of the support of the reference measure.