Volume and distance comparison theorems for sub-Riemannian manifolds

TL;DR

This paper establishes volume and distance comparison theorems for sub-Riemannian manifolds using generalized curvature inequalities, leading to compactness results based on Ricci curvature bounds.

Contribution

It introduces new global distance and volume estimates for sub-Riemannian manifolds utilizing the generalized curvature dimension inequality.

Findings

Sharp inequalities for sub-Riemannian heat equation solutions

Gromov type precompactness theorem for manifolds with Ricci curvature bounds

Uniform local volume estimates in sub-Riemannian geometry

Abstract

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}.