Needle decompositions in Riemannian geometry

TL;DR

This paper extends the localization technique from convex geometry to Riemannian manifolds with Ricci curvature bounds, utilizing log-concave measures and Monge mass transfer to analyze geometric properties.

Contribution

It generalizes the localization method to Riemannian manifolds with Ricci curvature bounds, connecting log-concavity and geodesic foliations.

Findings

Log-concave measures arise from conditioning the volume measure on manifolds with non-negative Ricci curvature.

The Monge mass transfer problem is integral to the analysis of these manifolds.

The method provides new tools for geometric and curvature-related problems in Riemannian geometry.

Abstract

The localization technique from convex geometry is generalized to the setting of Riemannian manifolds whose Ricci curvature is bounded from below. In a nutshell, our method is based on the following observation: When the Ricci curvature is non-negative, log-concave measures are obtained when conditioning the Riemannian volume measure with respect to an integrable geodesic foliation. The Monge mass transfer problem plays an important role in our analysis.