2007/01/19 by Kei Funano, Funano, Kei
Mathematics · #31C15 #53C21 #53C23 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Metric Geometry (math.MG) #Point processes and geometric inequalities #math.DG #math.MG #msc:31C15 #msc:53C21 #msc:53C23
paper · pdf · doi:10.48550/arxiv.math/0701535
31 pages,1 figure
openalex publication_date 2007/01/19 · arxiv created 2008/01/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The measure concentration property of an mm-space X is roughly described as that any 1-Lipschitz map on X to a metric space Y is almost close to a constant map. The target space Y is called the screen. The case of Y=ℝ is widely studied in many literature (see \citegromov, \citeledoux, \citemil2, \citemilsch, \citesch, \citetal, \citetal2 and its reference). M. Gromov developed the theory of measure concentration in the case where the screen Y is not necessarily ℝ (cf. \citegromovcat, gromov2, \citegromov). In this paper, we consider the case where the screen Y is a nonpositively curved manifolds. We also show that if the screen Y is so big, then the mm-space X does not concentrate.