球面熱核跡數在圓作用下之漸進展開

Abstract

在這篇論文中，我們啟發自Jochen Brüning 及Ernst Heintze 的論文[2]，延伸他們對於式子1.1 的討論，研究在特定例子下該漸進展開的係數。我們透過拉普拉思算子、熱核與群作用的基本性質，將原式轉變成以幾何性質定義的量，並且不斷透過泰勒展開式將積分化簡以求得目標係數。最後我們會發現積分內的函數如何影響係數。

In this thesis, we are inspired by Jochen Brüning and Ernst Heintze’s work [2] and extend their result to achieve the coefficients of the asymptotic expansion of equation 1.1 in [2] in a specific condition. Our result will be based on their work. We will reduce the ordinary formula into an integration defined by some geometric objects via Laplacian, heat kernel and group action. Therefore, we use Taylor expansion to deal with the integration. We will find the relation between the functions in integration and the coefficients as our result.