Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations, Part II

TL;DR

This paper explores the implications of curvature-dimension inequalities on sub-Riemannian manifolds derived from Riemannian foliations, focusing on heat semigroup properties and geometric-analytic bounds.

Contribution

It extends the curvature-dimension inequality framework to sub-Riemannian settings and establishes new bounds and inequalities related to heat semigroups and geometry.

Findings

Bounds for the gradient of heat semigroup

Entropy and Poincaré inequalities derived

Li-Yau type inequality established

Abstract

Using the curvature-dimension inequality proved in Part~I, we look at consequences of this inequality in terms of the interaction between the sub-Riemannian geometry and the heat semigroup $P_t$ corresponding to the sub-Laplacian. We give bounds for the gradient, entropy, a Poincar\'e inequality and a Li-Yau type inequality. These results require that the gradient of $P_t f$ remains uniformly bounded whenever the gradient of $f$ is bounded and we give several sufficient conditions for this to hold.