這篇博士論文主要包含三部分. 第一部分討論在擬埃爾米特流形上Li-Yau的梯度估計. 後面兩部分主要是討論Yau均值化定理於CR流形上的推廣. 其內容可以概述如下.
首先,我們將於第一章介紹擬埃爾米特流形中的基本概念與符號.
在接續下來的第二章節中,從CR Bôchner 公式,可以推得廣義曲率維度不等式,進而可以用Cao-Yau的方法,推出在擬埃爾米特流形上Li-Yau的梯度估計. 其方法不僅提供了更精確的Li-Yau梯度估計且允許擬埃爾米特的扭度非零. 作為Li-Yau的梯度估計的應用,我們有CR Harnack不等式以及CR 熱核上界估計. 此外,在廣義曲率維度不等式的假定下,我們把Cao-Yau梯度估計推廣到高階的情況.
第三章中,我們首先給出CR熱方程解的存在性與唯一性的一個充分條件. 經由選定適宜的初始條件,考慮CR熱方程的解及CR單調公式,可使我們得到CR粗略維度估計. 若把證明過程中的估計更加細緻化, 則我們可以得到CR最佳維度估計.
於第四章初始,我們先給出CR subhessian比較定理; 作為其第一個應用,可以得到CR subLaplacian比較定理. 由CR subhessian比較定理, 我們亦可以得出這章節的主要定理-CR三圓定理. 作為CR三圓定理的推論, 我們得到兩個CR 最佳單調公式與CR最佳維度估計及其剛性定理.

The thesis consists of three topics mainly. One is about the CR Li-Yau gradient estimate. The latter ones are concerned with the CR analogues of Yau's uniformization problems. They could be briefly described as follows.
In Chapter 1, we introduce the basic notions in pseudohermitian geometry and the necessary notations adopted in the later chapters.
In Chapter 2, we first derive the generalized curvature-dimension inequality with the help of the CR Bochner formula. Then it enables us to derive the CR Li-Yau gradient estimate for positive solutions to CR heat equation by modifying the Cao-Yau's method. Actually Cao-Yau's method not only provides a new way to obtain more precise gradient estimate than before but allows the pseudohermitian torsion to be nonvanishing. As applications, we have the CR version of Li-Yau Harnack inequality and upper bound estimate for the CR heat kernel. And, under the assumption of the generalized curvature-dimension inequality, we are able to confirm the Li-Yau gradient estimate for the sum of squares of vector fields up to higher step.
In Chapter 3, we give a sufficient condition of the existence of the solution to CR heat equation to ensure its solution could be expressed in the convolution of the initial data with the CR heat kernel. Subsequently, by taking the appropriate initial condition of CR heat equation, we can get the rough dimension estimate. If modifying the estimate about the vanishing order of CR-holomorphic functions utilized in the rough dimension estimate, we are able to obtain the sharp dimension estimate.
In Chapter 4, we first deduce the CR subhessian comparison; as a byproduct, it enables us to obtain the CR subLaplacian comparison. And then we derive the CR three-circle theorem. As applications, it enables us to deduce two CR sharp monotonicity formulas and the CR sharp dimension estimate with its rigidity.

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