Bi-Lipschitz coordinates at every regular RCD point

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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.

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What the papers state

Theorem 1.1

In Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces

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For every integer n\ge2 there is a constant L_n\ge1 with the following property. Let K\in\R, and let (X,d,\mathcal H_d^n) be a noncollapsed \RCD(K,n) space. Let p\in X. If every pointed Gromov–Hausdorff tangent of X at p is pointed-isometric to (\R^n,|\cdot|,0), then there exist an open neighborhood U of p, an open set V\subset\R^n, and a homeomorphism F:U\to V such that

L_n^{-1}d(x,z)\le |F(x)-F(z)|\le L_nd(x,z) \qquad (x,z\in U).

The distance on the left and right is the original distance d restricted to U.

Corollary 1.2

In Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces

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Under the hypotheses of Theorem 1.1, there is an open set \mathcal U\subset X containing every n-regular point, with \mathcal H_d^n(X\setminus\mathcal U)=0, and a countable atlas of L_n-bi-Lipschitz charts from ambient-open subsets of \mathcal U onto open subsets of \R^n. Each chart uses the restricted ambient distance, and its transition maps are L_n^2-bi-Lipschitz.

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Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces

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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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29 references in Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces

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