Failure of the measure contraction property via quotients in higher-step sub-Riemannian structures

TL;DR

This paper demonstrates that the measure contraction property (MCP), a synthetic Ricci curvature lower bound, can fail in higher-step sub-Riemannian structures, especially when the distance function lacks Lipschitz regularity, with implications for geometric analysis.

Contribution

It introduces new stability results for MCP under quotients in metric measure spaces and shows MCP failure in specific sub-Riemannian structures, including ideal cases, beyond previously known conditions.

Findings

MCP can fail in higher-step sub-Riemannian geometries.

Failure occurs in examples like Martinet and Engel structures.

Ideal sub-Riemannian structures can also violate MCP.

Abstract

We prove that the synthetic Ricci curvature lower bound known as the measure contraction property (MCP) can fail in sub-Riemannian geometry. This may happen beyond step two, if the distance function is not Lipschitz in charts, and it already occurs in fundamental examples such as the Martinet and Engel structures. Central to our analysis are new results, of independent interest, on the stability of the local MCP under quotients by isometric group actions for general metric measure spaces, developed under a weaker variant of the essential non-branching condition which, in contrast with the classical one, is implied by the minimizing Sard property in sub-Riemannian geometry. As an application, we find sub-Riemannian structures with pre-medium-fat distribution that do not satisfy the MCP, answering a question raised in [L. Rifford, J. \'Ec. polytech. Math. 2023]. Finally, and quite unexpectedly, we show that ideal sub-Riemannian structures can fail the MCP, and this actually happens generically for rank greater than 3 and high dimension.