Bounds for the first eigenvalue of the horizontal Laplacian in   positively curved sub-Riemannian manifolds
  We establish lower bounds for the first non-zero eigenvalue for the natural geometric sub-elliptic Laplacian operator defined on sub-Riemannian manifolds of step 2 that satisfy a positive curvature condition. The methods are very general and can be applied even when the sub-Riemannian geometry has considerable torsion. 
Robert K. Hladky
math.DG