Bi-Lipschitz coordinates at every regular RCD point

1 · The essentials

The manuscript studies spaces that may have an uneven shape but satisfy precise restrictions on curvature and dimension. It focuses on points where every possible limiting view, obtained by zooming in, is ordinary flat space. The main claim is that each such point has a whole open neighborhood that can be described using ordinary coordinates. These coordinates control distances in both directions: they neither stretch nor shrink them by more than a finite factor. The distances being compared are those of the original space, not distances recalculated by staying inside the neighborhood. The claim applies under a specific assumption linking the space’s measure to its dimension, and only in dimensions at least two. The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, think of making a street map of a small district with a guarantee that every distance changes by no more than a fixed factor in either direction. The claim concerns having such a map around every qualifying point, rather than merely finding that the district looks flat when viewed at smaller and smaller scales.

Why it matters

The manuscript connects a condition about all limiting views at a point with a distance-controlled coordinate description on an actual open neighborhood. It describes this as resolving the regular-point bi-Lipschitz conjecture in the noncollapsed setting. The distinction matters: the conclusion concerns a neighborhood with the original space’s distances, not just its limiting shape.

2 · Going further

The precise setting is a noncollapsed $RCD (K, n)$ space with $K \in R$ and an integer $n \geq 2$ . The reference measure is exactly $H^{n}$ . A point is regular here when all its pointed tangents are Euclidean. The claimed conclusion is that every such point belongs to an open neighborhood $U$ admitting a bi-Lipschitz homeomorphism $f : U \to V$ , where $V$ is an open subset of $R^{n}$ .

Writing $d$ for the original space’s distance, the distance comparison means that a finite constant $L \geq 1$ satisfies $L^{- 1} d (x, y) \leq ∣ f (x) - f (y) ∣ \leq L d (x, y)$ for all $x, y \in U$ . Crucially, $U$ carries the restriction of $d$ : the statement is not about replacing it with a distance obtained from routes constrained to remain in $U$ .

The abstract describes the result as resolving the regular-point bi-Lipschitz conjecture in the noncollapsed setting, but it does not describe the argument or its main construction. There is also a difference in the stated strength of the distance bound: the repository summary gives a point-dependent finite constant, whereas the abstract says that the constant depends only on $n$ . The supplied text does not explain this difference.

Only the main manuscript is supplied. No companion manuscripts or relationships between companion results are described, so no further account of the argument or of supporting results can be given from this source.

Useful notions

Distances and open neighborhoods
Euclidean coordinates
Limits obtained by zooming in
Bi-Lipschitz distance comparisons
Hausdorff measure

What was known before / what is new

Before

The abstract identifies the regular-point bi-Lipschitz statement as a conjecture in the noncollapsed setting. The supplied text gives no earlier partial results, historical attribution, or account of previously available coordinate descriptions.

New

The manuscript claims that every regular point in the stated class has an open neighborhood bi-Lipschitz homeomorphic to an open subset of Euclidean space. The neighborhood uses the restricted ambient distance. The abstract additionally claims a bi-Lipschitz constant depending only on the dimension, while the repository summary states only a point-dependent finite constant.

Limits and assumptions

The claim requires a noncollapsed $RCD (K, n)$ space, reference measure exactly $H^{n}$ , real $K$ , and integer $n \geq 2$ . Regularity requires every pointed tangent to be Euclidean, not merely one tangent. The supplied statement does not address nonregular points, spaces outside the noncollapsed setting, or dimension one, and it identifies no further open problems. The discrepancy between the two descriptions of the bi-Lipschitz constant is not explained in the supplied text. The abstract supplies no argument to assess, and the main result is not formalized in Lean and has not been formally verified.

Mini glossary

Regular point: A point for which all pointed tangents are Euclidean.
Pointed tangent: A limiting view of a space obtained by zooming in around a chosen point.
Bi-Lipschitz homeomorphism: A one-to-one coordinate correspondence onto its target that bounds distance changes in both directions by a finite multiplicative factor.
Restricted ambient distance: The original space’s distance applied to pairs of points in the neighborhood.
Noncollapsed setting: The setting in this statement where the reference measure is exactly the dimension-matched Hausdorff measure $H^{n}$ .

Keywords : RCD spaces · Regular points · Bi-Lipschitz coordinates

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves that every regular point of a noncollapsed $RCD (K, n)$ 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 $H^{n}$ , and the chart compares ambient distances with a point-dependent finite constant.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces

25/09/2026

To be verified

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 $RCD (K, n)$ 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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