# A.A. Agrachev, U. Boscain, J.-P. Gauthier, F. Rossi, The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups

Title: The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups
Authors: A.A. Agrachev, U. Boscain, J.-P. Gauthier, F. Rossi
Journal title: Func. An.
Year: 2009
Volume: 256
Pages: 2621-2655
Abstract:

We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector, using the Popp’s volume form introduced by Montgomery. This definition generalizes the one of the Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares. We then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly
the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncommutative Fourier transform. We then study some relevant cases: $SU(2)$, $SO(3)$, $SL(2)$ (with the metrics inherited by the Killing form), and the group $SE(2)$ of rototranslations of the plane. Our study is motivated by some recent results about the cut and conjugate loci on these sub-Riemannian manifolds. The perspective is to understand how singularities of the sub-Riemannian distance reflect on the kernel of the corresponding hypoelliptic heat equation.