CATALOGUE 357
Bi-Lipschitz coordinates at every regular RCD point
Official catalogue 357 concerns open bi-Lipschitz coordinates at every prescribed regular point of a noncollapsed $\operatorname{RCD}(K,n)$ space. It contains one manuscript. The proposed proof compensates for the scale-dependent degeneration of harmonic coordinates in their value space, obtaining all-pairs metric bounds and an open image.
The article explains how the two main inputs combine, summable directional pinching, iteration in value space, and all pairs, the limit and openness. The distinction from known a.e. rectifiability on measurable pieces is the ambient-open domain and same-dimensional open image at every prescribed regular point.
Paper in this catalogue
These summaries give the principal scope; consult the cited original results for the complete hypotheses and statements.
Page counts include references. On narrow screens, scroll the table sideways.
| # | Manuscript and overview | Principal claim | Role and available overview |
|---|---|---|---|
| 01 | Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces →Short name: Bi-Lipschitz Coordinates at Regular RCD PointsSeptember 25, 2026 · 43 pagesOriginal paper (GitHub) ↗ | For integer n≥2, noncollapsed full RCD(K,n) with reference measure Hⁿ, every point whose pointed GH tangents are all Rⁿ has an ambient-open neighborhood $L_n$-bi-Lipschitz to an open subset of Rⁿ. $L_n$ depends only on n; charts use restricted ambient distance. A full-measure open region containing all regular points has a countable atlas. Theorem 1.1; Corollary 1.2 · pp. 1–2 ↗ | Open harmonic coordinates feed cumulative matrix control of Bochner mass and summable directional defects into inverse-transpose updates in value space. Finite-scale derivative estimates then yield both distance bounds for every pair and openness of the limit. Read the proof overview → |
Sources and checking scope
The official subject is Differential geometry, with catalogue range 333–361; catalogue 357 appears on overview page 38. The reference snapshot is official repository commit 1581d4cbdd69a0faf34b2089614fe2a814316a6a. The manuscript is the 43-page version by OpenAI dated September 25, 2026.
OpenAI Research Catalog · Differential geometry, Catalogue 357 · p. 38
Theorem 1.1 in the manuscript states that $L_n$ depends only on dimension, while the neighborhood radius may depend on the space and point. This is distinguished from the official overview's abbreviated description of a point-dependent constant; the article and table follow the manuscript's quantifiers. Corollary 1.2 gives a full-measure open region containing all regular points and a countable atlas, without asserting that the exact regular locus is open.
The principal statements and proof connections were compared with the manuscript, together with statements and uses of the main external inputs in harmonic approximation, heat-kernel estimates, noncollapsed volume theory and invariance of domain. Exhaustive external comparison for local Poisson regularity, Hessian calculus, the Bochner inequality and distance Laplacian comparison, as well as independent verification of all proofs, remains incomplete. See the article's checking scope. Relations with other catalogues are uninvestigated; having one manuscript does not imply absence of external dependencies.