Greiner, Peter C. A Hamiltonian approach to the heat kernel of a subLaplacian on \(S^{2n+1}\). (English) Zbl 1282.53026 Anal. Appl., Singap. 11, No. 6, Article ID 1350035, 62 p. (2013). 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. Reviewer: Nathaniel Eldredge (Greeley) Cited in 5 Documents 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 Keywords:Hamiltonian mechanics; heat kernels; sub-Laplacians; CR manifolds; sub-Riemannian geometry; short time asymptotics; geodesics; Carnot-Carathéodory distance Citations:Zbl 0959.35035 PDFBibTeX XMLCite \textit{P. C. Greiner}, Anal. Appl., Singap. 11, No. 6, Article ID 1350035, 62 p. (2013; Zbl 1282.53026) Full Text: DOI arXiv References: [1] DOI: 10.1007/s00209-008-0436-0 · Zbl 1189.58009 · doi:10.1007/s00209-008-0436-0 [2] DOI: 10.1016/S0021-7824(00)00169-0 · Zbl 0959.35035 · doi:10.1016/S0021-7824(00)00169-0 [3] DOI: 10.1007/BF02786643 · Zbl 1025.58010 · doi:10.1007/BF02786643 [4] DOI: 10.1007/BF02392235 · Zbl 0366.22010 · doi:10.1007/BF02392235 [5] Gaveau B., Bull. Inst. Math. Acad. Sin. (N.S.) 1 pp 79– (2006) [6] DOI: 10.1142/S0219530507001000 · Zbl 1191.53025 · doi:10.1142/S0219530507001000 [7] DOI: 10.1090/fic/052/01 · doi:10.1090/fic/052/01 [8] DOI: 10.1142/S021953050300017X · Zbl 1056.53021 · doi:10.1142/S021953050300017X [9] DOI: 10.1215/S0012-7094-02-11426-4 · Zbl 1072.35130 · doi:10.1215/S0012-7094-02-11426-4 [10] DOI: 10.1063/1.1446664 · Zbl 1059.81089 · doi:10.1063/1.1446664 [11] Ikeda M., Mem. Fac. Eng. Hiroshima Univ. 3 pp 17– (1967) [12] Ikeda M., Mem. Fac. Eng. Hiroshima Univ. 3 pp 31– (1967) [13] Ikeda M., Mem. Fac. Eng. Hiroshima Univ. 3 pp 55– (1967) [14] Ikeda M., Mem. Fac. Eng. Hiroshima Univ. 3 pp 77– (1967) [15] DOI: 10.1007/BF02788796 · Zbl 1063.35062 · doi:10.1007/BF02788796 [16] Strichartz R., J. Diff. Geom. 30 pp 595– (1989) · doi:10.4310/jdg/1214443604 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.