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

我们阐明了一般绝热膨胀理论在研究与拉普拉斯有关的一些问题中的有用性和强大性。特别地，在此理论的基础上，我们得到了与接触黎曼流形上的科恩-罗西拉普拉斯算子相关的热核的每一个导数的渐近展开的系数公式。为了将系数明确地描述为由曲率、扭转（厄米特坦诺连接）和尼jenhuis张量构成的任意高阶的通用多项式，我们只需要将微积分的基本知识添加到公式中。（传统的方法需要各种渊博的知识，对系数的研究可能很少。）我们还发现，该理论对调查所谓的CR-Yamabe问题很有用。

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.