Sub-Laplacians on sub-Riemannian manifolds

TL;DR

This paper compares various natural sub-Laplacians on sub-Riemannian manifolds, establishing conditions for their equivalence and illustrating with examples from Lie groups.

Contribution

It introduces a canonical sub-Laplacian related to horizontal Brownian motion and compares it with divergence-based operators, clarifying their relationships.

Findings

The canonical sub-Laplacian can be defined without a measure.

Conditions are provided for different sub-Laplacians to coincide.

Examples include SU(2), Heisenberg, and affine groups.

Abstract

We consider different sub-Laplacians on a sub-Riemannian manifold $M$. Namely, we compare different natural choices for such operators, and give conditions under which they coincide. One of these operators is a sub-Laplacian we constructed previously in \cite{GordinaLaetsch2014a}. This operator is canonical with respect to the horizontal Brownian motion, we are able to define the sub-Laplacian without some a priori choice of measure. The other operator is $\operatorname{div}^{\omega} \operatorname{grad}_{\mathcal{H}}$ for some volume form $\omega$ on $M$. We illustrate our results by examples of three Lie groups equipped with a sub-Riemannian structure: $\operatorname{SU}\left( 2 \right)$, the Heisenberg group and the affine group.