2023/07/04 by Magnabosco, Mattia, Rossi, Tommaso · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2307.01820
The Lott-Sturm-Villani curvature-dimension condition 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) 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 C1,1. 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 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 MCP(0,5).