On the geometry of metric measure spaces with variable curvature bounds

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.