Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds (Q464140)

The main result is the following. Theorem. Assume that the three-dimensional contact sub-Riemannian manifold \(M\) is Sasakian. Then the following statements are equivalent: (I) there is a constant \(\mathbb{K}\) such that \(\text{Ric}(\alpha)\geq 2\mathbb{K} H(\alpha)\) for all \(\alpha\) in the cotangent bundle \(T^*M\) of this manifold; (II) the Tanaka-Webster curvature \(k\) is obunded below by \(\mathbb{K}\); (III) the metric measure space \((M,d,\eta)\) satisfies the generalized measure contraction property \(\text{MCP}(\mathbb{K};2,3)\), where \(d\) is the sub-Riemannian distance and \(\eta\) is the Popp' measure. Some consequences of this theorem (for example, local Poincaré inequality and Harnack inequality for sub-Laplacian) are obtained.