Failure of the Curvature-Dimension Condition in Sub-Finsler Manifolds

The Lott–Sturm–Villani curvature-dimension condition CD(K,N)$$\textsf{CD}(K,N)$$ 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 CD(K,N)$$\textsf{CD}(K,N)$$ 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 an analytic strongly convex norm and with a positive smooth measure. Secondly, we focus on the Sub-Finsler Heisenberg group, proving that curvature-dimension bounds cannot hold also when the reference norm is less regular, in particular when it is of class C1,1$$C^{1,1}$$. Finally, we show the failure of the (weaker) measure contraction property MCP(K,N)$$\textsf{MCP}(K,N)$$ 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 Sub-Riemannian Heisenberg group, which instead satisfies MCP(0,5)$$\textsf{MCP}(0,5)$$.