Gromov-Hausdorff and intrinsic flat convergence of RCD(K,N) and Kato spaces

Abstract

We consider metric measure spaces $(X,\mathsf{d},{\mathcal{H}}^N)$ satisfying the properties (ETR), (LBD), and with an almost everywhere connected regular set. In particular, these assumptions are fulfilled by non-collapsed RCD$(K,N)$ spaces without boundary, as well as by non-collapsed strong Kato limit spaces without boundary. For both classes, we study orientability in the sense of metric currents, establish stability of orientation under pointed Gromov--Hausdorff convergence, and show that the pointed Gromov--Hausdorff limit coincides with the local flat limit.