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HEAT KERNELS OF THE SUB-LAPLACIAN AND THE LAPLACIAN ON NILPOTENT LIE GROUPS

    Work partially supported by the Grant-in-Aid for Scientific Research (No. 17540202 C), The Ministry of Education, Culture, Sports, Science and Technology, Japan.

    https://doi.org/10.1142/9789812773609_0008Cited by:4 (Source: Crossref)
    Abstract:

    Since the heat kernel of the sub-Laplacian on Heisenberg group was constructed in an explicit integral form by A. Hulanicki, we have several ways to construct the heat kernel for the sub-Laplacian and the Laplacian on 2-step nilpotent Lie groups. In this note we explain a method effectively employed by Beals-Gaveau-Greiner, the so called complex Hamilton-Jacobi theory, and illustrate the construction of the heat kernel for general 2-step cases. We discuss the solution of the generalized Hamilton-Jacobi equation and a quantity similar to the van Vleck determinant and their roles in the integral expression of the heat kernel. We expect this method will work also for 3-step cases to construct the heat kernel together with the theory of elliptic functions. So as an example, we consider the solution of the generalized Hamilton-Jacobi equation for the lowest dimensional 3-step nilpotent Lie group (Engel group). Then we discuss a hierarchy of heat kernels for the three dimensional Heisenberg group and Heisenberg manifolds as a simple example.

    Dedication: Dedicated to Krzysztof P. Wojciechowski on his 50 th birthday
    Keywords:
    AMSC: Primary 35K05, Secondary 22E25