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A review of the tangent space in sub-Finsler geometry and applications to the failure of the $\mathsf{CD}$ condition

Mattia Magnabosco, Tommaso Rossi

TL;DR

The paper develops a framework to study curvature-dimension bounds in sub-Finsler spaces via pmGH tangents, showing that the metric-measure tangent at almost every point is a nilpotent approximation (a sub-Finsler Carnot model) equipped with a scaled Lebesgue measure. By proving a stability result for ${\mathsf{CD}}(K,N)$ under pmGH convergence and identifying the tangents as nilpotent models, it deduces the failure of ${\mathsf{CD}}(K,N)$ for 3D-contact sub-Finsler manifolds with bounded measures, via a tangent-space contradiction with the Heisenberg group. The work extends previous results to bounded measures and provides a detailed construction of tangents, privileged coordinates, Ball-Box estimates, and measure-theoretic tangents, with implications for MCP and broader non-Euclidean geometries. These results clarify the limitations of CD-type curvature bounds in non-Riemannian settings and connect infinitesimal sub-Finsler geometry to global synthetic curvature notions.

Abstract

We review the construction of the tangent space to a sub-Finsler manifold in the measured Gromov-Hausdorff sense. Under suitable assumptions on the measure, the metric measure tangent is described by the nilpotent approximation, equipped with a scalar multiple of the Lebesgue measure. We apply this result in the study of the Lott-Sturm-Villani curvature-dimension condition in sub-Finsler geometry. In particular, we show the failure of the $\mathsf{CD}$ condition in 3D-contact sub-Finsler manifolds, equipped with a bounded measure.

A review of the tangent space in sub-Finsler geometry and applications to the failure of the $\mathsf{CD}$ condition

TL;DR

The paper develops a framework to study curvature-dimension bounds in sub-Finsler spaces via pmGH tangents, showing that the metric-measure tangent at almost every point is a nilpotent approximation (a sub-Finsler Carnot model) equipped with a scaled Lebesgue measure. By proving a stability result for under pmGH convergence and identifying the tangents as nilpotent models, it deduces the failure of for 3D-contact sub-Finsler manifolds with bounded measures, via a tangent-space contradiction with the Heisenberg group. The work extends previous results to bounded measures and provides a detailed construction of tangents, privileged coordinates, Ball-Box estimates, and measure-theoretic tangents, with implications for MCP and broader non-Euclidean geometries. These results clarify the limitations of CD-type curvature bounds in non-Riemannian settings and connect infinitesimal sub-Finsler geometry to global synthetic curvature notions.

Abstract

We review the construction of the tangent space to a sub-Finsler manifold in the measured Gromov-Hausdorff sense. Under suitable assumptions on the measure, the metric measure tangent is described by the nilpotent approximation, equipped with a scalar multiple of the Lebesgue measure. We apply this result in the study of the Lott-Sturm-Villani curvature-dimension condition in sub-Finsler geometry. In particular, we show the failure of the condition in 3D-contact sub-Finsler manifolds, equipped with a bounded measure.
Paper Structure (13 sections, 16 theorems, 59 equations)

This paper contains 13 sections, 16 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper.
  3. Preliminaries
  4. The nilpotent approximation of a sub-Finsler manifold
  5. The flag of the distribution.
  6. Privileged coordinates.
  7. Nilpotent approximation.
  8. Ball-Box theorem.
  9. The metric measure tangent to a sub-Finsler manifold
  10. The approximating family of sub-Finsler structures
  11. Bounded measures on a sub-Finsler manifold
  12. Convergence to the metric measure tangent
  13. Applications to the curvature-dimension condition

Key Result

Theorem 1.1

Let $M$ be a complete sub-Finsler manifold, equipped with a bounded measure $\mathfrak m$. Then, for $\mathfrak m$-a.e. $q\in M$, there is a unique metric measure tangent space of $M$ at $q$ and it is its nilpotent approximation equipped with (a multiple of) the Lebesgue measure.

Theorems & Definitions (39)

  • Theorem 1.1
  • Remark 2.1
  • Proposition 2.2
  • Definition 3.1: Privileged coordinates
  • Theorem 3.2
  • Definition 3.3: Nilpotent approximation
  • Remark 3.4
  • Theorem 3.5
  • Theorem 3.6: Ball-Box
  • Remark 3.7
  • ...and 29 more