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

TL;DR

This paper characterizes low-dimensional $RCD^*(K,N)$ spaces with non-empty one-dimensional regular sets, showing their equivalence to Ricci limit spaces for $N<2$, and establishes a new Bishop-Gromov inequality.

Contribution

It provides a characterization of low-dimensional $RCD^*(K,N)$ spaces with one-dimensional regular sets and proves a new Bishop-Gromov inequality for these spaces.

Findings

Ricci limit spaces with $Ric \,\geq K$ and Hausdorff dimension $N<2$ coincide with $RCD^*(K,N)$ spaces.

Classified $RCD^*(K,N)$ spaces as either complete intervals or circles in low dimensions.

Established a Bishop-Gromov type inequality for $RCD^*(K,N)$ spaces, extending known results.

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.