The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups

By Andrei A. Agrachev, Ugo V. Boscain, Jean-Paul André Gauthier, Francesco Rossi. 2008

187 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b6934b6a5f4bb4c41cb49569ce4cf40f1e45a569

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Citations in the corpus

The subelliptic heat kernel on the CR sphere (2011, background): “To put things in perspective let us observe that the study of explicit expressions for subelliptic heat kernels has generated a great amount of work (see [1], [2] [4], [3], [5] [7], [11] and the references therein).”
Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations (2014, background): “rs [25] and [55]. Besides the Hopf ﬁbrations, there are of course many other situations where sub-Riemannian heat kernels may computed more or less explicitely. We mention in particular the reference [1] which deals with the case of unimodular Lie groups. 3.1 Heisenberg group One of the simplest non”
Small time heat kernel asymptotics at the sub-Riemannian cut locus (2012, background): “The problem of relating the sub-Riemannian distance to the heat-kernel is an old problem (see for instance [6, 13, 18, 21, 22, 23, 30, 36, 38, 39, 44, 50, 52]).”
On the Hausdorff volume in sub-Riemannian geometry (2010, methodology): “Recall that there exists a canonical isomorphism of 1-dimensional vector spaces (see [2] for details) n ∧ (T ∗ qM) ' n ∧ (grq(∆) ∗).”
Trace heat kernel asymptotics in 3D contact sub-Riemannian geometry (2011, methodology): “From this we can explicitly compute the sub-Laplacian in the 3D contact case (see also [4], formula (8)).”
Sub-Riemannian structures on 3D lie groups (2010, methodology): “Moreover in the non-unimodular case the generalized Fourier transform method, used in [3], cannot apply .”
A Formula for Popp’s Volume in Sub-Riemannian Geometry (2012, background): “More in general, in [3], the authors prove that for left-invariant structures on unimodular Lie groups the sub-Laplacian is a sum of squares.”