Sub-Laplacians on sub-Riemannian manifolds

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 ${\mathrm{div}}^{\omega }{\mathrm{grad}}_{\mathcal{H}}$ for some volume form $\omega$ on $M$. We illustrate our results by examples of three Lie groups equipped with a sub-Riemannian structure: $\mathrm{SU}\left(2\right)$, the Heisenberg group and the affine group.

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