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 Gordina and Laetsch (Trans. Amer. Math. Soc., 2015). This operator is canonical with respect to the horizontal Brownian motion; we are able to define this sub-Laplacian without some a priori choice of measure. The other operator is \(\operatorname {div}^{\omega } \operatorname {grad}_{\mathcal {H}}\) for some volume form ω on M. We illustrate our results by examples of three Lie groups equipped with a sub-Riemannian structure: SU(2), the Heisenberg group and the affine group.
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This research was supported in part by NSF Grant DMS-1007496.
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Gordina, M., Laetsch, T. Sub-Laplacians on Sub-Riemannian Manifolds. Potential Anal 44, 811–837 (2016). https://doi.org/10.1007/s11118-016-9532-7
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DOI: https://doi.org/10.1007/s11118-016-9532-7