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Measure contraction property and curvature-dimension condition on sub-Finsler Heisenberg groups

Samuël Borza, Mattia Magnabosco, Tommaso Rossi, Kenshiro Tashiro

Abstract

In this paper, we investigate the validity of synthetic curvature-dimension bounds in the sub-Finsler Heisenberg group, equipped with a positive smooth measure. Firstly, we study the measure contraction property, in short $\mathsf{MCP}$, proving that its validity depends on the norm generating the sub-Finsler structure. Indeed, we show that, if it is neither $C^1$ nor strongly convex, the associated Heisenberg group does not satisfy $\mathsf{MCP}(K,N)$ for any pair of parameters $K \in \mathbb{R}$ and $N \in (1,\infty)$. On the contrary, we prove that the sub-Finsler Heisenberg group, equipped with a $C^{1,1}$ and strongly convex norm, and with the Lebesgue measure, satisfies $\mathsf{MCP}(0,N)$ for some $N \in (1,\infty)$. Additionally, we provide a lower bound on the optimal dimensional parameter, and we also study the case of $C^1$ and strongly convex norms. Secondly, we address the validity of the curvature-dimension condition pioneered by Sturm and Lott-Villani, in short $\mathsf{CD}(K,N)$. We show that the sub-Finsler Heisenberg group, equipped with a $C^1$ and strongly convex norm, and with a positive smooth measure, does not satisfy the $\mathsf{MCP}(K,N)$ condition for any pair of parameters $K \in \mathbb{R}$ and $N \in (1,\infty)$. Combining this result with our findings regarding the measure contraction property, we conclude the failure of the $\mathsf{CD}$ condition in the Heisenberg group for every sub-Finsler structure.

Measure contraction property and curvature-dimension condition on sub-Finsler Heisenberg groups

Abstract

In this paper, we investigate the validity of synthetic curvature-dimension bounds in the sub-Finsler Heisenberg group, equipped with a positive smooth measure. Firstly, we study the measure contraction property, in short , proving that its validity depends on the norm generating the sub-Finsler structure. Indeed, we show that, if it is neither nor strongly convex, the associated Heisenberg group does not satisfy for any pair of parameters and . On the contrary, we prove that the sub-Finsler Heisenberg group, equipped with a and strongly convex norm, and with the Lebesgue measure, satisfies for some . Additionally, we provide a lower bound on the optimal dimensional parameter, and we also study the case of and strongly convex norms. Secondly, we address the validity of the curvature-dimension condition pioneered by Sturm and Lott-Villani, in short . We show that the sub-Finsler Heisenberg group, equipped with a and strongly convex norm, and with a positive smooth measure, does not satisfy the condition for any pair of parameters and . Combining this result with our findings regarding the measure contraction property, we conclude the failure of the condition in the Heisenberg group for every sub-Finsler structure.
Paper Structure (22 sections, 39 theorems, 248 equations, 3 figures)

This paper contains 22 sections, 39 theorems, 248 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The CD(K,N) and the MCP(K,N) conditions
  4. The CD(K,N) condition.
  5. The Brunn--Minkowski inequality.
  6. The MCP(K,N) condition.
  7. Scaling and stability properties.
  8. Convex trigonometry
  9. Convex trigonometry associated with a norm
  10. The sub-Fin-sler geometry of the Heisenberg group
  11. Geodesics in the Heisenberg group
  12. The Jacobian of the exponential map
  13. The MCP condition in the Heisenberg group
  14. Failure of the measure contraction property in the sub-Finsler Heisenberg group
  15. Failure for non-C1 norms
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\mathbb{H}$ be the sub-Fin-sler Heisenberg group, equipped with a norm ${\|\!\cdot\!\|}$ which is not $C^1$ or not strongly convex, and let $\mathfrak m$ be a positive smooth measure on $\mathbb{H}$. Then, the metric measure space $(\mathbb{H}, \mathsf d, \mathfrak m)$ does not satisfy the meas

Figures (3)

  • Figure 1: Values of the generalized trigonometric functions $\cos_\Omega$ and $\sin_\Omega$.
  • Figure 2: Representation of the correspondence $\theta \ext@arrow 9999{ \arrowfill@\leftarrow\relbar\rightarrow}{}{\Omega} \psi$.
  • Figure 3: The flat part of $\partial\Omega^\circ$.

Theorems & Definitions (99)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Theorem 1.6
  • Conjecture 1.7
  • Definition 2.1: $\mathsf{CD}(K,N)$ condition
  • Definition 2.2: Midpoints
  • Definition 2.3: Brunn--Minkowski inequality
  • ...and 89 more