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Math-2271 · mathematics · Analysis

Higher-dimensional quantitative bi-Lipschitz parameterization and a scalar-curvature obstruction

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Find higher-dimensional analogues of the Bonk--Lang theorem on bi-Lipschitz parametrization by Euclidean space, with explicit quantitative control of the bi-Lipschitz constants from the geometric hypotheses. The question asks for a parametrization result, rather than only a continuous or locally topological chart.

Formulation references: [S1], [S2].

Short English statement

Find higher-dimensional analogues of the Bonk--Lang theorem on bi-Lipschitz parametrization by Euclidean space, with explicit quantitative control of the bi-Lipschitz constants from the geometric hypotheses. The question asks for a parametrization result, rather than only a continuous or locally topological chart.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2271-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2271

Bi-Lipschitz charts at regular RCD points: The theorem gives quantitative bi-Lipschitz coordinates in every dimension at regular points of noncollapsed RCD spaces. The source asks a wider higher-dimensional Bonk-Lang analogue, including global parameterization, and does not limit itself to such local regular RCD points.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Higher-dimensional quantitative bi-Lipschitz parameterization and a scalar-curvature obstruction

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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