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Math-2669 · mathematics · Geometry

Bi-Lipschitz coordinates at regular RCD points

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each $n\geq 2$, is there L(n) such that any point p of a noncollapsed RCD(K,n) space $(X,d,H^{n}_d)$, whose every pointed tangent is Euclidean $R^{n}$, has an ambient-open neighborhood bi-Lipschitz homeomorphic with constant L(n) to an open Euclidean set? The chart uses the restricted ambient metric, and L(n) is independent of K and the space.

Short English statement

For each $n\geq 2$, is there L(n) such that any point p of a noncollapsed RCD(K,n) space $(X,d,H^{n}_d)$, whose every pointed tangent is Euclidean $R^{n}$, has an ambient-open neighborhood bi-Lipschitz homeomorphic with constant L(n) to an open Euclidean set? The chart uses the restricted ambient metric, and L(n) is independent of K and the space.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2669-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2669

Bi-Lipschitz charts at regular RCD points: For noncollapsed $\mathrm{RCD}(K,n)$ spaces with $n\ge2$, every regular point is claimed to have an open Euclidean bi-Lipschitz neighborhood. The distortion bound depends only on dimension, and the neighborhood carries the restricted ambient distance. The parameters, regularity assumptions and normalization in this question are retained in applying that assertion.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Bi-Lipschitz coordinates at regular RCD points

OpenAI's claim

Bi-Lipschitz charts at regular RCD points: For noncollapsed $\mathrm{RCD}(K,n)$ spaces with $n\ge2$, every regular point is claimed to have an open Euclidean bi-Lipschitz neighborhood. The distortion bound depends only on dimension, and the neighborhood carries the restricted ambient distance.

Mathematical statement of the claimed result

Mathematical claim 1.
For each integer $n\ge2$ there is $L_n\ge1$ with the following claimed property. In any noncollapsed $\operatorname{RCD}(K,n)$ space $(X,d,\mathcal H_d^n)$, $K\in\mathbb R$, if every pointed Gromov--Hausdorff tangent at $p$ is Euclidean $(\mathbb R^n,|\cdot|,0)$, then open neighborhoods $p\in U\subset X$, $V\subset\mathbb R^n$ admit a homeomorphism $F:U\to V$ with \[L_n^{-1}d(x,z)\le|F(x)-F(z)|\le L_nd(x,z)\quad(x,z\in U).\] The constant depends only on $n$; the distance is the ambient distance restricted to $U$. Here $\operatorname{RCD}(K,n)$ uses the unreduced $\operatorname{CD}(K,n)$ condition together with quadratic Cheeger energy, and the measure remains exactly $\mathcal H_d^n$.

Outline of the argument

Bi-Lipschitz charts at regular RCD points: Poisson replacements and two-center estimates produce summable directional defects for local metric matrices. An iteration on a common open coordinate image corrects the pullback metric using inverse-transpose Jacobians and smooth covector fields. Diagonal second-derivative recurrences and higher-derivative bounds keep the corrections summable even when their square roots are not, yielding invertible bi-Lipschitz coordinates.

Formal verification

Bi-Lipschitz charts at regular RCD points: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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