Curvature dimension inequalities and subelliptic heat kernel gradient   bounds on contact manifolds
  We study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain: 1) Geometric conditions ensuring the compactness of the underlying manifold (Bonnet-Myers type results); 2) Volume estimates of metric balls; 3) Gradient bounds and stochastic completeness for the heat semigroup generated by the sub-Laplacian; 4) Spectral gap estimates. 
Fabrice Baudoin, Jing Wang
math.DG math.PR