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On the rectifiability of $\mathsf{CD}(K,N)$ and $\mathsf{MCP}(K,N)$ spaces with unique tangents

Mattia Magnabosco, Andrea Mondino, Tommaso Rossi

TL;DR

The paper investigates rectifiability for CD$(K,N)$ and MCP$(K,N)$ spaces under pointwise Ahlfors regularity and a.e. unique metric tangent assumption. It proves that CD$(K,N)$ spaces with Hausdorff dimension $n<5$ are $( rak m,n)$-rectifiable, while non-collapsed MCP$(K,N)$ spaces with Hausdorff dimension $N$ are $( rak m,N)$-rectifiable; the core argument reduces to analyzing tangent spaces, showing that low-dimensional sub-Finsler Carnot tangents must be finite-dimensional Banach spaces, and applying Bate’s criterion. A key ingredient is a no-$ ext{MCP}$ result for noncommutative sub-Finsler Carnot groups, which forces tangents in the MCP setting to be commutative, hence Banach spaces, enabling rectifiability. The work relies on Le Donne’s tangent framework and a Carnot-group conjecture linking $ ext{CD}$ on groups to finite-dimensional Banach-structure, and it highlights pathways to stratified rectifiability in the CD context and the role of non-collapsed hypotheses in MCP. Together, these results advance understanding of geometric regularity in synthetic curvature spaces and identify sharp obstructions via Carnot-group geometry.

Abstract

We prove rectifiability results for $\mathsf{CD}(K,N)$ and $\mathsf{MCP}(K,N)$ metric measure spaces $(\mathsf{X},\mathsf{d},\mathfrak{m})$ with pointwise Ahlfors regular reference measure $\mathfrak{m}$ and with $\mathfrak{m}$-almost everywhere unique metric tangents. In particular, we show rectifiability if (i) $(\mathsf{X},\mathsf{d},\mathfrak{m})$ is $\mathsf{CD}(K,N)$ for an arbitrary $N$ and has Hausdorff dimension $n<5$, or (ii) $(\mathsf{X},\mathsf{d},\mathfrak{m})$ is $\mathsf{MCP}(K,N)$ and non-collapsed, namely it has Hausdorff dimension $N$. Our strategy is based on the failure of the $\mathsf{CD}$ condition in sub-Finsler Carnot groups, on a new result on the failure of the non-collapsed $\mathsf{MCP}$ on sub-Finsler Carnot groups, and on the recent breakthrough by Bate [Invent. Math., 230(3):995-1070, 2022].

On the rectifiability of $\mathsf{CD}(K,N)$ and $\mathsf{MCP}(K,N)$ spaces with unique tangents

TL;DR

The paper investigates rectifiability for CD and MCP spaces under pointwise Ahlfors regularity and a.e. unique metric tangent assumption. It proves that CD spaces with Hausdorff dimension are -rectifiable, while non-collapsed MCP spaces with Hausdorff dimension are -rectifiable; the core argument reduces to analyzing tangent spaces, showing that low-dimensional sub-Finsler Carnot tangents must be finite-dimensional Banach spaces, and applying Bate’s criterion. A key ingredient is a no- result for noncommutative sub-Finsler Carnot groups, which forces tangents in the MCP setting to be commutative, hence Banach spaces, enabling rectifiability. The work relies on Le Donne’s tangent framework and a Carnot-group conjecture linking on groups to finite-dimensional Banach-structure, and it highlights pathways to stratified rectifiability in the CD context and the role of non-collapsed hypotheses in MCP. Together, these results advance understanding of geometric regularity in synthetic curvature spaces and identify sharp obstructions via Carnot-group geometry.

Abstract

We prove rectifiability results for and metric measure spaces with pointwise Ahlfors regular reference measure and with -almost everywhere unique metric tangents. In particular, we show rectifiability if (i) is for an arbitrary and has Hausdorff dimension , or (ii) is and non-collapsed, namely it has Hausdorff dimension . Our strategy is based on the failure of the condition in sub-Finsler Carnot groups, on a new result on the failure of the non-collapsed on sub-Finsler Carnot groups, and on the recent breakthrough by Bate [Invent. Math., 230(3):995-1070, 2022].
Paper Structure (14 sections, 19 theorems, 116 equations)

This paper contains 14 sections, 19 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Strategy of the proofs.
  3. Structure of the paper.
  4. Preliminaries
  5. Synthetic Curvature-Dimension bounds
  6. Sub-Finsler Carnot groups
  7. Dilations on Carnot groups.
  8. The Heisenberg group H.
  9. Classification of low-dimensional Carnot groups.
  10. Rectifiability of CD(K,N) spaces with unique tangents
  11. On the notion of convergence of metric measure spaces
  12. Metric measure tangents
  13. Rectifiability of CD(K,N) spaces with unique tangents
  14. Rectifiability of non-collapsed MCP(K,N) spaces with unique tangents

Key Result

Theorem 1.1

Given $K\in\mathbb{R}$ and $N\in (1,\infty)$, let $(\mathsf{X},\mathsf d,\mathfrak m)$ be a $\mathsf{CD} (K,N)$ space. Assume that: Then:

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: magnabosco2023failureborza2024measure
  • Conjecture : Conjecture \ref{['conj:carnot_nocd']}
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • ...and 38 more