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.

Manuscript

Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces

September 25, 2026 · source · cite as \cite{OAI:Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026}

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.

Result 357 · Differential geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.