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

TL;DR

This paper investigates the convergence properties of certain metric measure spaces, specifically RCD(K,N) and Kato spaces, establishing stability of orientation and equivalence of different notions of limits.

Contribution

It introduces new stability results for orientation in metric currents and proves the equivalence of Gromov-Hausdorff and flat limits for these spaces.

Findings

Orientation is stable under pointed Gromov-Hausdorff convergence.

Gromov-Hausdorff limit coincides with the local flat limit.

Applicable to non-collapsed RCD(K,N) and strong Kato limit spaces.

Abstract

We consider metric measure spaces $(X,\mathsf{d},\mathscr{H}^N)$ 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.