Bishop-Gromov type inequality on Ricci limit spaces
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.
scientific article; zbMATH DE number 5914719
scientific article