[Paper Review] Equivalence of two different notions of tangent bundle on rectifiable metric measure spaces

Nicola Gigli, Enrico Pasqualetto|arXiv (Cornell University)|Nov 29, 2016

Advanced Topology and Set Theory4 citations

TL;DR

This paper establishes the isometric equivalence between two notions of tangent bundle on rectifiable metric measure spaces: the abstract tangent module from Sobolev calculus and the space of $L^2$-sections of the Gromov-Hausdorff tangent bundle. The authors prove that for strongly $\mathfrak{m}$-rectifiable spaces—such as ${\sf RCD}^*(K,N)$ spaces—pointed-Gromov-Hausdorff blow-ups yield Euclidean tangent spaces $\mathfrak{m}$-a.e., and the resulting measurable sections form a space isometric to the tangent module, thereby unifying geometric and analytic approaches to differentiability.

ABSTRACT

We prove that for a suitable class of metric measure spaces, the abstract notion of tangent module as defined by the first author can be isometrically identified with the space of $L^2$-sections of the `Gromov-Hausdorff tangent bundle'. The class of spaces $({ m X},{\sf d},{\mathfrak m})$ we consider are PI spaces that for every $\varepsilon>0$ admit a countable collection of Borel sets $(U_i)$ covering ${\mathfrak m}$-a.e.\ ${ m X}$ and corresponding $(1+\varepsilon)$-biLipschitz maps $φ_i:U_i o{\mathbb R}^{k_i}$ such that $(φ_i)_*{\mathfrak m}\lower3pt\hbox{$|_{U_i}$}\ll\mathcal L^{k_i}$. This class is known to contain ${\sf RCD}^*(K,N)$ spaces. Part of the work we carry out is that to give a meaning to notion of $L^2$-sections of the Gromov-Hausdorff tangent bundle, in particular explaining what it means to have a measurable map assigning to ${\mathfrak m}$-a.e.\ $x\in { m X}$ an element of the pointed-Gromov-Hausdorff limit of the blow-up of ${ m X}$ at $x$.

Motivation & Objective

• To establish a rigorous connection between two distinct notions of tangent bundle: the abstract tangent module from Sobolev calculus and the geometric Gromov-Hausdorff tangent bundle.
• To define a canonical measurable structure on the Gromov-Hausdorff tangent bundle using charts in strongly $\mathfrak{m}$-rectifiable spaces.
• To show that for $\mathfrak{m}$-a.e. $x \in \rm{X}$, the pointed-Gromov-Hausdorff limit of rescaled spaces at $x$ is a Euclidean space, enabling a geometric interpretation of the tangent bundle.
• To prove that the space of $L^2$-sections of the Gromov-Hausdorff tangent bundle is isometrically isomorphic to the tangent module $L^2(T\rm{X})$.
• To resolve the ambiguity in identifying tangent spaces via arbitrary isometries by constructing a canonical measurable structure using biLipschitz charts.

Proposed method

• Introduce the concept of a measurable Banach bundle structure on the Gromov-Hausdorff tangent bundle using a countable family of $(1+\varepsilon)$-biLipschitz charts $\varphi_i: U_i \to \mathbb{R}^{k_i}$.
• Define $L^2$-sections of the Gromov-Hausdorff tangent bundle as measurable maps assigning to $\mathfrak{m}$-a.e. $x \in \rm{X}$ an element of the pointed-Gromov-Hausdorff limit of the rescaled space at $x$.
• Use the existence of $\varepsilon_n$-atlases with compact domains and density points to construct a sequence $r_n \downarrow 0$ such that rescaled charts $\Phi_n(x, \cdot)$ approximate the identity in the Gromov-Hausdorff sense.
• Apply Lemma 1.2 to control the distance between points and their projections in the charts, ensuring that the rescaled charts $\Phi_n$ converge uniformly to an isometry on small balls.
• Prove that for $\mathfrak{m}$-a.e. $x$, the rescaled spaces $B_{r_n R}(x)/r_n$ converge in the pointed-Gromov-Hausdorff sense to $\mathbb{R}^k$ with the standard Euclidean metric.
• Establish an isometric isomorphism between $L^2(T\rm{X})$ and $L^2(T_{\rm{GH}}\rm{X})$ by showing that the differential of Sobolev functions corresponds precisely to measurable sections of the Gromov-Hausdorff tangent bundle.

Experimental results

Research questions

• RQ1 Can the abstract tangent module defined via $L^\infty$-modules and Sobolev calculus be geometrically realized as sections of the Gromov-Hausdorff tangent bundle?
• RQ2 Under what conditions does the pointed-Gromov-Hausdorff limit of rescaled spaces at $\mathfrak{m}$-a.e. point $x$ yield a Euclidean space?
• RQ3 How can one canonically define a measurable structure on the bundle of tangent spaces obtained via Gromov-Hausdorff limits, avoiding arbitrary isometric identifications?
• RQ4 Is there a natural isometric identification between the space of $L^2$-sections of the Gromov-Hausdorff tangent bundle and the tangent module $L^2(T\rm{X})$?
• RQ5 To what extent do the notions of differentiability via weak derivatives (analytic) and via blow-ups (geometric) coincide in singular but rectifiable metric measure spaces?

Key findings

• For $\mathfrak{m}$-a.e. $x \in \rm{X}$, the pointed-Gromov-Hausdorff limit of the rescaled space $B_{r_n R}(x)/r_n$ as $r_n \downarrow 0$ is isometric to $\mathbb{R}^k$ for some $k$, with the limit depending only on the local dimension at $x$.
• The space of $L^2$-sections of the Gromov-Hausdorff tangent bundle, denoted $L^2(T_{\rm{GH}}\rm{X})$, is well-defined and admits a canonical measurable structure via the biLipschitz charts.
• There exists a sequence $r_n \downarrow 0$ such that the rescaled charts $\Phi_n(x, \cdot)$ satisfy $\big{|}\big{|}\Phi_n(x,y_0) - \Phi_n(x,y_1)\big{|}_{\mathbb{R}^k} - \frac{{\sf d}(y_0,y_1)}{r_n}\big{|} \leq \varepsilon$ uniformly on $B_{r_n R}(x)$ for large $n$.
• The image of the rescaled charts $\Phi_n(x, \cdot)$ $\varepsilon$-dense in the ball $B_{R-\varepsilon}(0_{\mathbb{R}^k})$, ensuring that the limit captures the full tangent space.
• The tangent module $L^2(T\rm{X})$ and the space of $L^2$-sections $L^2(T_{\rm{GH}}\rm{X})$ are isometrically isomorphic, with the isomorphism given by the differential of Sobolev functions.
• The result generalizes Rademacher’s theorem in the sense that both analytic and geometric notions of differentiability coincide $\mathfrak{m}$-a.e. under strong rectifiability assumptions.