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.

arXiv comment37 pages; v2: added section 6.2 on the weighted Heisenberg group, fixed typos. Final version to appear on Mathematische Annalen

References in corpus (11)

• Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds
• Curvature: a variational approach
• A formula for Popp's volume in sub-Riemannian geometry
• Sub-Riemannian curvature in contact geometry
• Sharp measure contraction property for generalized H-type Carnot groups
• Sub-Riemannian interpolation inequalities
• The Sard conjecture on Martinet surfaces
• Comparison theorems for conjugate points in sub-Riemannian geometry
• Measure contraction properties of Carnot groups
• On conjugate times of LQ optimal control problems
• Sub-Riemannian Ricci curvatures and universal diameter bounds for 3-Sasakian manifolds

Cited by in corpus (4)

• Failure of curvature-dimension conditions on sub-Riemannian manifolds via tangent isometries
• Comparison theorems on H-type sub-Riemannian manifolds
• Generalized Bakry-Émery curvature condition and equivalent entropic inequalities in groups
• Unified synthetic Ricci curvature lower bounds for Riemannian and sub-Riemannian structures