On the geometry of metric measure spaces with variable curvature bounds

TL;DR

This paper introduces a new curvature-dimension condition for metric measure spaces with variable curvature bounds, extending classical results and establishing stability and geometric properties.

Contribution

It generalizes Sturm's curvature condition to variable bounds, proving key geometric inequalities and stability under convergence and tensorization.

Findings

Proves sharp Bishop-Gromov volume comparison

Establishes a generalized Bonnet-Myers theorem

Demonstrates stability under measured Gromov-Hausdorff convergence

Abstract

Motivated by a classical comparison result of J. C. F. Sturm we introduce a curvature-dimension condition CD(k,N) for general metric measure spaces and variable lower curvature bound k. In the case of non-zero constant lower curvature our approach coincides with the celebrated condition that was proposed by K.-T. Sturm. We prove several geometric properties as sharp Bishop-Gromov volume growth comparison or a sharp generalized Bonnet-Myers theorem (Schneider's Theorem). Additionally, our curvature-dimension condition is stable with respect to measured Gromov-Hausdorff convergence, and it is stable with respect to tensorization of finitely many metric measure spaces provided a non-branching condition is assumed.