A Hamiltonian approach to the heat kernel of a subLaplacian on \(S^{2n+1}\) (Q2867064)

This paper is concerned with the study of the heat kernel for the Cauchy-Riemann sub-Laplacian \(\Delta_C\) on the sphere \(S^{2n+1} \subset \mathbb C^{n+1}\). The author proceeds along Hamiltonian lines, finding bicharacteristic curves and solving the Hamilton-Jacobi equation to eventually obtain a formula for the heat kernel \(P_C\) for \(\Delta_C\) (Section 4). He is then able to compute short time asymptotics for \(P_C\) via the method of stationary phase (Section 6). Along the way, he also obtains an expression for the Carnot-Carathéodory distance with respect to the corresponding sub-Riemannian structure on \(S^{n+1}\) (Section 5). These computations are preceded by a warm-up Section 3, where the same methods are used to derive the corresponding (known) results for the classical Laplacian \(\Delta_S\) on \(S^{n+1}\) and its Riemannian distance. NEWLINENEWLINENEWLINEThe reader may also be interested in a previous paper by the same author together with \textit{R. W. Beals} and \textit{B. Gaveau} [J. Math. Pures Appl., IX. Sér. 79, No. 7, 633--689 (2000; Zbl 0959.35035)], which uses similar methods to obtain similar results for the sub-Laplacian on Heisenberg groups.