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Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces
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Abstract
We resolve the regular-point bi-Lipschitz conjecture in the noncollapsed setting. For every integer n ≥ 2 and every real K, every regular point of a noncollapsed $\mathrm{RCD}(K,n)$ space has an open neighborhood bi-Lipschitz homeomorphic to an open subset of ℝn. The bi-Lipschitz constant depends only on n, and the neighborhood uses the restricted ambient distance.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Possibly relevant literature (found by the agent, not checked)
- G. De Philippis and J. Núñez-Zimbrón, The behavior of harmonic functions at singular points of RCD spaces (arXiv:1909.05220, 2019). Honda and Zhang (arXiv:2608.16021, Section 3) cite this work for the observation that bi-Lipschitz harmonic charts need not exist on noncollapsed RCD spaces, which bears on the paper's remark that its limiting coordinates are not claimed to have bounded Laplacian.
Digests and expositions
- Latent Seminar, Bi-Lipschitz coordinates at regular RCD points: controlling distortion across scales (2026-10-07). This exposition describes the manuscript's harmonic coordinates, directional Bochner estimates, and iterative coordinate corrections, explicitly distinguishing its selected proof outlines from independent verification of the complete proof.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 357
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026,
author = {{OpenAI}},
title = {{Bi-Lipschitz Coordinates at Regular Points of Noncollapsed $\mathrm{RCD}$ Spaces}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026/paper.pdf}{OAI:Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026}},
year = {2026}
}Family 357: Bi-Lipschitz coordinates at every regular RCD point
Proves that every regular point of a noncollapsed $\mathop{\mathrm{RCD}}\nolimits (K,n)$ space, for K ∈ ℝ and integer n ≥ 2, has an open neighborhood bi-Lipschitz to an open subset of ℝn. Regularity requires all pointed tangents to be Euclidean, the reference measure is exactly $\mathcal H^n$, and the chart compares ambient distances with a point-dependent finite constant.
This is the only paper in this family.