A two-step ideal sub-Riemannian structure with no measure contraction properties

Abstract

We construct a compact, equiregular, ideal sub-Riemannian manifold of step $2$ such that, for every smooth positive measure, the $\mathrm{MCP}(K,N)$ fails for all $K\in \mathbb{R}$ and $N\in (1,\infty )$. This shows that the real-analyticity assumption in the measure contraction theorem of Badreddine and Rifford in arXiv:1712.09900v2 cannot be replaced by smoothness. Moreover, our structure is ideal, i.e. it admits no non-trivial abnormal minimizing geodesics. Although failures of the measure contraction property for ideal structures were recently obtained in the higher-step setting, our construction shows that the phenomenon can already occur on compact, equiregular, ideal structures of step $2$. The proof exploits a differential consequence of the measure contraction property, namely a uniform upper bound for the sub-Laplacian of the squared distance near its base point, and constructs a structure for which this quantity is unbounded.

14 pages, 1 figure