Characterization of Low Dimensional $RC{D}^{\ast }(K,N)$ spaces

Abstract

In this paper, we give the characterization of metric measure spaces that satisfy synthetic lower Riemannian Ricci curvature bounds (so called $RC{D}^{\ast }(K,N)$ 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 $RC{D}^{\ast }(K,N)$ 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 Lévy-Gromov's isoperimetric inequality and was previously only known for Ricci limit spaces) which might be also of independent interest.

version 3: 37 pp, to appear in AGMS

References in corpus (2)

• The splitting theorem in non-smooth context
• Measure rigidity of Ricci curvature lower bounds

Cited by in corpus (1)

• On the topology and the boundary of N-dimensional RCD(K,N) spaces