Calculus of Variations and Geometric Measure Theory

B. Han

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

All comments are welcome !


Abstract:

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.


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