Study on the general adiabatic expansion theory for Laplacian and its some applications

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We clarified the usefulness and powerfulness of the general adiabatic expansion theory in studying some subjects related to Laplacian. In particular, based on the theory we obtained a formula for the coefficients of the asymptotic expansion of every derivative of the heat kernel associated to the Kohn-Rossi Laplacian on contact Riemannian manifold. In order to describe the coefficients explicitly up to an arbitrarily high order as universal polynomials built from the curvature, the torsion (of hermitian Tanno connection) and the Nijenhuis tensor, we need only a basic knowledge of calculus added to the formula. (The conventional method requires various kinds of profound knowledge and there may be few investigations on the coefficients.) We found also that the theory is useful for investigating the so-called CR-Yamabe problem.