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

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.

💡 Research Summary

The paper studies the interplay between Gromov‑Hausdorff (GH) convergence and intrinsic flat (IF) convergence for two important classes of metric measure spaces: non‑collapsed RCD(K,N) spaces and non‑collapsed strong Kato limit spaces. Both classes satisfy two structural assumptions introduced by the authors: (ETR) – essential topological regularity, which guarantees the existence of a full‑measure set of regular points each admitting a rectifiable, Euclidean‑like neighbourhood; and (LBD) – local bounded density, which provides a uniform upper bound for the N‑dimensional Hausdorff measure on bounded balls. The authors prove that these assumptions hold for (i) locally non‑collapsed RCD(K,N) spaces without boundary and (ii) non‑collapsed strong Kato limit spaces without boundary.

The central notion of orientability is defined via metric currents in the sense of Ambrosio‑Kirchheim. A space (X,d,ℋⁿ) is called orientable (and without boundary) if there exists a locally integral N‑dimensional current T with set(T)=X, mass measure ‖T‖=ℋⁿ, and zero boundary ∂T=0. This definition parallels the classical picture where an oriented Riemannian manifold gives rise to an integral current whose mass equals the Riemannian volume and whose boundary vanishes by Stokes’ theorem.

The first main technical result (Proposition 1.2) shows a rigidity property for such currents on non‑collapsed RCD spaces: if another locally integral current S has zero boundary in a ball Bₛ(x), then S coincides with an integer multiple of T on that ball. Consequently, Corollary 1.3 yields uniqueness of orientation up to a sign. The proof hinges on a new covering lemma (Proposition 3.4): given a δ‑splitting map u on a regular ball, the ball can be decomposed (up to a null set) into countably many Borel pieces on which u is bi‑Lipschitz onto its image. This allows the authors to write T and S as push‑forwards of Euclidean currents, apply a Euclidean local constancy theorem (Corollary 2.5), and transfer the constancy back to the metric space. The a.e.‑connectedness of the regular set together with (LBD) forces the weight functions of T and S to be locally constant, and the weight of T is forced to be ±1 by examining 0‑dimensional slices.

The second set of results (Theorems 1.5 and 1.6) establishes that, under pointed GH convergence, the GH limit and the IF limit agree for both classes. For a sequence of non‑collapsed RCD(K,N) spaces (or strong Kato limits) with oriented currents T_i, the authors distinguish two cases. In the collapsing case (Hausdorff volume tends to zero) the currents converge to the zero current. In the non‑collapsing case, they prove that the subsequential IF limit is a non‑zero integral current T with ∂T=0, mass ℋⁿ, and set(T)=X, thereby showing that the limit space is orientable and has no boundary in the current sense. The proof combines Matveev‑Portegies’ L¹‑estimates for the multiplicity functions θ_{T_i}, the regularity of δ‑splitting maps, and Bishop–Gromov volume monotonicity to identify the limit mass with the Hausdorff measure. In the non‑compact setting, Lang‑Wenger’s compactness theorem provides a common ambient space where both GH and IF convergence can be realized, reducing the argument to the compact case.

The paper’s contributions are threefold. First, it extends the equivalence GH = IF, previously known for compact non‑collapsed Ricci limit spaces, to the much broader setting of strong Kato limits, where only an integral Kato bound on the negative part of Ricci is required. Second, it provides a streamlined proof of orientation rigidity that relies solely on the push‑forward representation of rectifiable currents, avoiding the need to invoke Sobolev differential forms or topological manifold structures. Third, it demonstrates that the orientation defined via metric currents is stable under pointed GH convergence, a fact that was previously known only for Ricci limits with stronger curvature bounds.

Overall, the work unifies and generalizes several strands of research on metric measure spaces, curvature‑dimension conditions, and geometric measure theory, and it opens the door to further investigations of orientation and flat convergence under even weaker curvature assumptions.