Comparison theorems on H-type sub-Riemannian manifolds
Fabrice Baudoin, Erlend Grong, Luca Rizzi, Sylvie Vega-Molino
TL;DR
The paper develops uniform sub-Hessian and sub-Laplacian comparison theorems for H-type sub-Riemannian manifolds by using the canonical variation $g_\varepsilon$ and a robust comparison principle for Jacobi fields, then passes to the sub-Riemannian limit to obtain genuine horizontal comparisons. Central to the approach is the structure of H-type foliations with parallel horizontal Clifford structure and the J^2 condition, which yield a natural splitting of horizontal and vertical directions and enable Hessian bounds in Sasakian, Riemannian, and H-type directions. These uniform estimates lead to explicit sub-Riemannian diameter bounds and a sharp sub-Riemannian Bonnet-Myers theorem in the general H-type setting, including non-foliation geometries, by passing to the limit $\varepsilon\to0$ and leveraging $C^\infty$ convergence outside the cut locus. The results extend known cases for contact, quaternionic contact, and 3-Sasakian structures, unify several prior bounds, and provide tools for assessing measure contraction properties and compactness in broader H-type geometries.
Abstract
On H-type sub-Riemannian manifolds we establish sub-Hessian and sub-Laplacian comparison theorems which are uniform for a family of approximating Riemannian metrics converging to the sub-Riemannian one. We also prove a sharp sub-Riemannian Bonnet-Myers theorem that extends to this general setting results previously proved on contact and quaternionic contact manifolds.