Volume and distance comparison theorems for sub-Riemannian manifolds
  In this paper we study global distance estimates and uniform local volume estimates in a large class of sub-Riemannian manifolds. Our main device is the generalized curvature dimension inequality introduced by the first and the third author in \cite{BG1} and its use to obtain sharp inequalities for solutions of the sub-Riemannian heat equation. As a consequence, we obtain a Gromov type precompactness theorem for the class of sub-Riemannian manifolds whose generalized Ricci curvature is bounded from below in the sense of \cite{BG1}. 
Fabrice Baudoin, Michel Bonnefont, Nicola Garofalo, Isidro H. Munive
math.DG math.MG