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Sub-Riemannian spectral distance

arXiv:2606.12804v1 Announce Type: cross Abstract: We study eigenvalues and eigenfunctions of the ``div-grad type" sub-Laplacian with respect to Popp's volume on a compact equiregular sub-Riemannian manifold $M$. Since Popp's volume is canonically determined by the sub-Riemannian structure of $M$, the spetra of the sub-Laplacian carry geometric meanings. In this paper, we first embed $M$ into the Hilbert space of square-summable sequences using eigenfunctions and then define a spectral distance between two compact equiregular sub-Riemannian manifolds. Our result is a sub-Riemannian analogue of Berard-Besson-Gallot's classical work in the Riemannian case.