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arXiv · 1505.00420

Characterization of Low Dimensional $RCD^*(K,N)$ spaces

Yu Kitabeppu·Sajjad Lakzian
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Abstract

In this paper, we give the characterization of metric measure spaces that satisfy synthetic lower Riemannian Ricci curvature bounds (so called $RCD^*(K,N)$ spaces) with \emph{non-empty} one dimensional regular sets. In particular, we prove that the class of Ricci limit spaces with $Ric \ge K$ and Hausdorff dimension $N$ and the class of $RCD^*(K,N)$ spaces coincide for $N < 2$ (They can be either complete intervals or circles). We will also prove a Bishop-Gromov type inequality ( that is ,roughly speaking, a converse to the L\'{e}vy-Gromov's isoperimetric inequality and was previously only known for Ricci limit spaces) which might be also of independent interest.

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BibTeXRIS

Yu Kitabeppu, Sajjad Lakzian. 2015-05-03. Characterization of Low Dimensional $RCD^*(K,N)$ spaces. https://arxiv.org/abs/1505.00420

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