Mathematics

On a Riemannian manifold, a constant lower bound on the Ricci curvature tensor can be formulated in (at least) three other equivalent ways: the validity of the Bakry-Émery gradient estimate for the heat flow, the Wasserstein contractivity property of the dual heat flow on probability measures, and the convexity property of the Boltzmann entropy along Wasserstein geodesics. The equivalence between these three properties can be established in a much more general framework, the realm of the now-called CD spaces. Although very broad, the CD condition is not met by the natural metric-measure structure of a sub-Riemannian manifold. Nevertheless, a rich family of sub-Riemannian manifolds naturally endowed with a group structure, Carnot groups and the SU(2) group, do satisfy a weaker version of the Bakry-Émery gradient estimate for the heat flow. In this talk, I will discuss the relationship between this weak version of the Bakry-Émery gradient estimate on metric-measure groups of sub-Riemannian type with both the weak contractivity property of the dual heat flow and the weak convexity property of the entropy.