Bakry-Émery curvature and model spaces in sub-Riemannian geometry

Abstract

We prove comparison theorems for the sub-Riemannian distortion coefficients appearing in interpolation inequalities. These results, which are equivalent to a sub-Laplacian comparison theorem for the sub-Riemannian distance, are obtained by introducing a suitable notion of sub-Riemannian Bakry-Émery curvature. The model spaces for comparison are variational problems coming from optimal control theory. As an application we establish the sharp measure contraction property for 3-Sasakian manifolds satisfying a suitable curvature bound.

37 pages; v2: added section 6.2 on the weighted Heisenberg group, fixed typos. Final version to appear on Mathematische Annalen