Observable concentration of mm-spaces into nonpositively curved manifolds

TL;DR

This paper investigates how measure concentration in metric measure spaces behaves when the target space is a nonpositively curved manifold, revealing that larger screens prevent concentration.

Contribution

It extends measure concentration theory to nonpositively curved manifolds as screens and shows that large screens inhibit concentration phenomena.

Findings

Measure concentration fails with large nonpositively curved screens.

The theory generalizes previous results for real-valued screens.

Nonpositive curvature influences concentration properties.

Abstract

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 $Y=\mathbb{R}$ is widely studied in many literature (see \cite{gromov}, \cite{ledoux}, \cite{mil2}, \cite{milsch}, \cite{sch}, \cite{tal}, \cite{tal2} and its reference). M. Gromov developed the theory of measure concentration in the case where the screen $Y$ is not necessarily $\mathbb{R}$ (cf. \cite{gromovcat}, {gromov2}, \cite{gromov}). 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.