Bishop-Gromov type inequality on Ricci limit spaces
type: Item
instance of: scholarly article
title: Bishop-Gromov type inequality on Ricci limit spaces
zbMATH Open document ID: 1252.53040
author: Shouhei Honda
published in: Journal of the Mathematical Society of Japan
publication date: 2011-06-29T00:00:00Z
review text: The paper under review deals with theoretical properties of the limit measure on Ricci limit spaces. A Ricci limit space is defined as the Gromov-Hausdorff limit of a sequence of complete, pointed, connected \(n\)-dimensional manifolds with Ricci curvature bounded from below by \(-(n-1)\). The author first analyzes the cut locus of a Ricci limit space. It is proved that the measure of the cut locus of such a space is zero. Moreover, the relationship between the limit space of cut loci and the cut locus of the limit space is established. Several properties of the measure of codimension one \(v_{-1}\) associated to the limit measure \(v\) of a Ricci limit space are proved. A Bishop-Gromov type inequality for \(v_{-1}\) is obtained. Then, finiteness and positivity results are proved for \(v_{-1}\). An analogous result on the co-area formula for Lipschitz maps on Euclidean spaces is obtained.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 5914719
zbMATH Keywords: Ricci curvature
zbMATH Keywords: Gromov-Hausdorff convergence
zbMATH Keywords: geometric measure theory
reviewed by: Cezar Oniciuc
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/06320419
OpenAlex ID: W2034306292
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DOI: 10.2969/JMSJ/06320419
description: scientific article; zbMATH DE number 5914719