# Bakry-\'Emery curvature and model spaces in sub-Riemannian geometry

## TL;DR

This paper develops comparison theorems for sub-Riemannian curvature using a new Bakry-Émery curvature notion, leading to sharp measure contraction results for specific manifolds, advancing understanding of geometric analysis in sub-Riemannian spaces.

## Contribution

It introduces a sub-Riemannian Bakry-Émery curvature concept and applies it to establish comparison theorems and measure contraction properties in sub-Riemannian geometry.

## Key findings

- Comparison theorems for sub-Riemannian distortion coefficients
- Sub-Laplacian comparison theorem for sub-Riemannian distance
- Sharp measure contraction property for 3-Sasakian manifolds

## 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-\'Emery 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.