柯西黎曼山邊問題：奇異柯西黎曼山邊方程的正則性、超曲面的柯西黎曼不變量與仿埃米爾特質量

Abstract

本論文包含一個主要研究主題與兩個次要研究主題。主要研究主題是探討柯西黎曼山邊方程的正則性理論在強擬凸仿埃米爾特流型上並且具有光滑且非奇異的邊界。第一個次要主題為在五維柯西黎曼流型中構造超曲面的柯西黎曼面積不變量；第二個次要主題則探討仿埃米爾特質量的性質。

關於主要研究主題，奇異柯西黎曼山邊度量是研究超曲面之柯西離黎曼不變量的重要工具。我們證明，在具有光滑且非奇異邊界的強擬凸仿埃米爾特流型上，奇異雅馬貝方程存在唯一解，並分析給出其邊界漸近行為，同時將 柯西黎曼面積不變量E2推廣至高維情況。

在五維的柯西黎曼面積不變量方面，我運用柯西黎曼流型的卡當幾何結構，為嵌入於五維 CR 流形中的超曲面建構新的不變量。

最後，在仿埃米爾特質量的研究中，我在柯西黎曼史瓦西上構造出具線性奇異性的切向柯恩拉普拉斯算子的顯式解。

This dissertation consists of one primary topic and two subsidiary topics. The main focus is on the regularity theory for the CR singular Yamabe equation on strongly pseudoconvex (spc) pseudohermitian manifolds with smooth, non-singular boundaries. One subsidiary topic involves the construction of CR area invariants for hypersurfaces in 5-dimensional CR manifolds. The other addresses certain properties of the p-mass.

For the primary topic, the singular Yamabe metric serves as a fundamental tool for studying CR invariants of hypersurfaces. We establish the existence, uniqueness, and boundary asymptotic behavior of singular Yamabe solutions on spc CR manifolds with smooth, non-singular boundaries and provide a generalization of the CR invariant energy E2 to higher-dimensional spc CR manifolds.

Regarding the CR area invariant in five dimensions, I exploit the Cartan geometric structure of CR manifolds to construct new invariants associated with embedded hypersurfaces in 5-dimensional CR manifolds.

Finally, in the context of the p-mass, I present explicit solutions with line singularities for the tangential Kohn Laplacian on the CR Schwarzschild model.