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

Andrea Mondino, Raquel Perales

TL;DR

The paper extends orientability results for non-smooth spaces to two broad classes: non-collapsed $ ext{RCD}(K,N)$ spaces without boundary and non-collapsed strong Kato limit spaces without boundary. It develops a direct current-based framework, using $ ext{δ}$-splitting maps and (ETR)/(LBD) regularity to prove that orientation is preserved under pointed $ ext{pGH}$ convergence and that pointed $ ext{pGH}$ and local flat limits agree. The approach leverages Ambrosio–Kirchheim and Lang–Wenger currents to pass to limits, showing that limit spaces carry an orienting current with $igl rbracket Tigr rbracket= ext{H}^N$ and $X= ext{set}(T)$, while collapsing cases yield the zero current. The results generalize previous Ricci-limit findings to the Kato-bounded regime, providing robust stability and convergence tools with potential applications to geometric analysis on non-smooth 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.

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

TL;DR

The paper extends orientability results for non-smooth spaces to two broad classes: non-collapsed spaces without boundary and non-collapsed strong Kato limit spaces without boundary. It develops a direct current-based framework, using -splitting maps and (ETR)/(LBD) regularity to prove that orientation is preserved under pointed convergence and that pointed and local flat limits agree. The approach leverages Ambrosio–Kirchheim and Lang–Wenger currents to pass to limits, showing that limit spaces carry an orienting current with and , while collapsing cases yield the zero current. The results generalize previous Ricci-limit findings to the Kato-bounded regime, providing robust stability and convergence tools with potential applications to geometric analysis on non-smooth spaces.

Abstract

We consider metric measure spaces satisfying the properties (ETR), (LBD), and with an almost everywhere connected regular set. In particular, these assumptions are fulfilled by non-collapsed RCD 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.
Paper Structure (20 sections, 25 theorems, 125 equations)

This paper contains 20 sections, 25 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. RCD spaces
  4. Singular strata and boundaries
  5. Ambrosio-Kirchheim's metric currents
  6. Definition and basic properties
  7. Integral currents, integral currents in Euclidean space and the degree of a Lipschitz map with respect to a current
  8. Parametric representation of rectifiable currents
  9. $0$-dimensional slices and weight functions
  10. Intrinsic Flat and Gromov-Hausdorff convergence
  11. $L^1$-Convergence of degrees
  12. Lang--Wenger's currents
  13. Relation between Ambrosio--Kirchheim and Lang--Wenger's currents
  14. Local flat topology and pGH convergence
  15. Bilipschitz charts and regularity properties
  16. ...and 5 more sections

Key Result

Proposition 1

Let $(X,\mathsf{d},\mathscr{H}^N)$ be an $\mathop{\mathrm{RCD}}\nolimits(K,N)$ space without boundary for some $K \in \mathbb{R}$ and $N \in \mathbb N$, that is orientable in the sense of currents by $T \in {\mathbf I}_{\text{\rm loc},\,N}(X), \ \partial T=0$. Then, for any $S \in {\mathbf I}_{\text

Theorems & Definitions (60)

  • Definition 1
  • Proposition 1
  • Corollary 1
  • Definition 2: Carron--Mondello--Tewodrose
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • Remark 2
  • Definition 3
  • Theorem 2.1: $\delta$-splitting vs $\varepsilon$-GH isometry
  • ...and 50 more