Generalized Ricci Curvature Bounds for Three Dimensional Contact Subriemannian manifolds

Measure contraction property is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. In this paper, we discover sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property.

💡 Research Summary

The paper addresses the problem of extending Ricci curvature lower bounds, a cornerstone of Riemannian geometry, to the much less regular setting of sub‑Riemannian manifolds. Specifically, the authors focus on three‑dimensional contact sub‑Riemannian manifolds, which are equipped with a two‑dimensional bracket‑generating distribution Δ and a compatible metric g on Δ, together with a distinguished Reeb vector field X₀. Because the metric is only defined on Δ, classical notions of Ricci curvature do not apply directly, and one must look for alternative synthetic curvature conditions. The authors adopt the Measure Contraction Property (MCP) as a synthetic analogue of a Ricci lower bound. MCP(K,N) asserts that for any Borel probability measure μ and any t∈