[Paper Review] Failure of the curvature-dimension condition in sub-Finsler manifolds

Mattia Magnabosco, Tommaso Rossi|arXiv (Cornell University)|Jul 4, 2023

Advanced Differential Geometry ResearchPhysics and Astronomy3 citations

TL;DR

This paper demonstrates that the curvature-dimension condition $\mathsf{CD}(K,N)$ fails in sub-Finsler manifolds, even under smooth, strongly convex norms and in the $C^{1,1}$-regular Heisenberg group. By adapting Juillet's method and introducing novel geometric tools, the authors prove that volume contraction along geodesics violates the required convexity, and further show that the weaker $\mathsf{MCP}(K,N)$ condition also fails under singular norms—highlighting fundamental incompatibility of these synthetic curvature bounds with sub-Finsler geometry.

ABSTRACT

The Lott-Sturm-Villani curvature-dimension condition $\mathsf{CD}(K,N)$ provides a synthetic notion for a metric measure space to have curvature bounded from below by $K$ and dimension bounded from above by $N$. It has been recently proved that this condition does not hold in sub-Riemannian geometry for every choice of the parameters $K$ and $N$. In this paper, we extend this result to the context sub-Finsler geometry, showing that the $\mathsf{CD}(K,N)$ condition is not well-suited to characterize curvature in this setting. Firstly, we show that this condition fails in (strict) sub-Finsler manifolds equipped with a smooth strongly convex norm and with a positive smooth measure. Secondly, we focus on the sub-Finsler Heisenberg group, proving that curvature-dimension bounds can not hold also when the reference norm is less regular, in particular when it is of class $C^{1,1}$. The strategy for proving these results is a non-trivial adaptation of the work of Juillet [Rev. Mat. Iberoam., 37(1):177-188, 2021], and it requires the introduction of new tools and ideas of independent interest. Finally, we demonstrate the failure of the (weaker) measure contraction property $\mathsf{MCP}(K,N)$ in the sub-Finsler Heisenberg group, equipped with a singular strictly convex norm and with a positive smooth measure. This result contrasts with what happens in the \sr Heisenberg group, which instead satisfies $\mathsf{MCP}(0,5)$.

Motivation & Objective

• To investigate whether the synthetic curvature-dimension condition $\mathsf{CD}(K,N)$ holds in sub-Finsler geometry, extending known failures in sub-Riemannian settings.
• To establish that $\mathsf{CD}(K,N)$ fails even in smooth, strictly convex sub-Finsler manifolds with smooth measures.
• To analyze the failure of $\mathsf{CD}(K,N)$ in the sub-Finsler Heisenberg group with $C^{1,1}$-regular norms.
• To prove that the weaker $\mathsf{MCP}(K,N)$ condition also fails under singular, strictly convex norms in the Heisenberg group.
• To demonstrate that branching geodesics can occur uniquely even with unique minimizers, challenging classical intuition.

Proposed method

• Adaptation of Juillet's approach to construct a family of geodesics with abnormal sub-segments to violate $\mathsf{CD}(K,N)$.
• Construction of a set $\mathscr{A}$ of initial conditions generating unique geodesics that remain in the $z=0$ plane for small $t$, violating volume contraction estimates.
• Use of convex trigonometry and the dual norm $\Omega^\circ$ to analyze geodesic behavior in the Heisenberg group.
• Analysis of the exponential map and end-point map to characterize extremal curves and their regularity.
• Introduction of a new geometric inequality involving concave $C^1$ functions to prove strict monotonicity of area-to-length-squared ratios, crucial for contradiction.
• Application of the Brunn–Minkowski inequality $\mathsf{BM}(K,N)$ to derive a contradiction via the measure of $t$-midpoints $M_t(A,B)$.

Experimental results

Research questions

• RQ1 Does the $\mathsf{CD}(K,N)$ condition hold in smooth sub-Finsler manifolds equipped with a smooth, strongly convex norm and a positive smooth measure?
• RQ2 Can $\mathsf{CD}(K,N)$ be satisfied in the sub-Finsler Heisenberg group when the norm is only $C^{1,1}$-regular?
• RQ3 Does the weaker $\mathsf{MCP}(K,N)$ condition hold in the sub-Finsler Heisenberg group under a singular, strictly convex norm?
• RQ4 What role does the geometry of the dual norm $\Omega^\circ$ play in the failure of curvature bounds?
• RQ5 Can unique geodesics branch in sub-Finsler geometry, and what does this imply for synthetic curvature conditions?

Key findings

• The $\mathsf{CD}(K,N)$ condition fails in all smooth, strictly convex sub-Finsler manifolds with a positive smooth measure, regardless of $K$ and $N$.
• In the sub-Finsler Heisenberg group with a $C^{1,1}$-regular norm, the $\mathsf{CD}(K,N)$ condition fails due to violation of volume contraction estimates along geodesics.
• The $\mathsf{MCP}(K,N)$ condition fails in the sub-Finsler Heisenberg group when the norm is singular and strictly convex, contrasting with the sub-Riemannian case where $\mathsf{MCP}(0,5)$ holds.
• A family of geodesics exists that remain in the plane $\{y=0,z=0\}$ for small $t$, violating the expected volume growth and contradicting the $\mathsf{CD}(K,N)$ inequality.
• Branching geodesics occur uniquely in the singular sub-Finsler Heisenberg group, even though each geodesic is individually unique, challenging classical expectations.
• A new geometric inequality is proven: for a $C^1$ concave function $f$, the ratio $a(s)/d(s)^2$ is strictly decreasing, which is essential for deriving the contradiction in the proof.