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A Hamiltonian approach to the heat kernel of a subLaplacian on \(S^{2n+1}\). (English) Zbl 1282.53026

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.
The reader may also be interested in a previous paper by the same author together with R. W. Beals and 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.

MSC:

53C17 Sub-Riemannian geometry
35H20 Subelliptic equations
35C15 Integral representations of solutions to PDEs
35K08 Heat kernel
32V20 Analysis on CR manifolds
70H20 Hamilton-Jacobi equations in mechanics
34K10 Boundary value problems for functional-differential equations

Citations:

Zbl 0959.35035
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References:

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