指導教授：張樹城臺灣大學：數學研究所樊彥彣Fan, Yen-WenYen-WenFan2014-11-302018-06-282014-11-302018-06-282014http://ntur.lib.ntu.edu.tw//handle/246246/264018這篇文章包含三大部分，第一部分證明矩陣形式的 Li-Yau-Hamilton Harnack 不等式。第二部份延續第一部分的工作，推廣至(1,1)-form 形式的 Li-Yau-Hamilton Harnack 不等式。第三部份將應用這不等式証明柯西黎曼上的 Gap 定理。In the first part of thesis, we first derive the CR analogue of matrix Li-Yau-Hamilton inequality for a positive solution to the CR heat equation in a closed pseudohermitian (2n+1)- manifold with nonnegative bisectional curvature and bitorsional tensor. We then obtain the CR Li-Yau gradient estimate in a standard Heisenberg group. Finally, we extend the CR matrix Li-Yau-Hamilton inequality to the case of Heisenberg groups. As a consequence, we derive the Hessian comparison property in the standard Heisenberg group. In the second part, we study the CR Lichnerowicz-Laplacian heat equation deformation of (1; 1)-tensors on a complete strictly pseudoconvex CR (2n+1)-manifold and derive the linear trace version of Li-Yau-Hamilton inequality for positive solutions of the CR Lichnerowicz- Laplacian heat equation. We also obtain a nonlinear version of Li-Yau-Hamilton inequality for the CR Lichnerowicz-Laplacian heat equation coupled with the CR Yamabe flow and trace Harnack inequality for the CR Yamabe flow. In the last part, by applying a linear trace Li-Yau-Hamilton inequality for a positive (1; 1)-form solution of the CR Hodge-Laplace heat equation and monotonicity of the heat equation deformation, we obtain an optimal gap theorem for a complete strictly pseudocovex CR (2n+1)-manifold with nonnegative pseudohermitian bisectional curvature and vanishing torsion. We prove that if the average of the Tanaka-Webster scalar curvature over a ball of radius r centered at some point o decays as o(r^-2 ), then the manifold is flat.1. Abstract v 2. Introduction 1 2.1. CR Li-Yau Gradient Estimate and Harnack Inequality 2 2.2. CR Matrix Li-Yau-Hamilton Inequality 4 2.3. CR Linear Trace Li-Yau-Hamilton Inequality and Gap Theorem 6 2.4. The Coupled CR Yamabe Flow 7 3. Preliminary 10 4. CR Matrix Li-Yau-Hamilton Harnack Inequality 12 4.1. CR Matrix Li-Yau-Hamilton Inequality 15 4.2. The CR Gradient Estimate and Harnack inequality in Heisenberg Groups 20 4.3. Complete noncompact case 25 5. Linear Trace Li-Yau-Hamilton inequality 31 5.1. The CR Bochner-Weitzenbock Formula 35 5.2. Linear Trace Li-Yau-Hamilton Inequality 39 5.3. Nonlinear Version for Li-Yau-Hamilton Inequality 50 6. CR Gap Theorem 58 6.1. CR Moment-Type Estimates 59 6.2. CR Lichnerowicz-Laplacian heat equation 63 6.3. Proof of CR Optimal Gap Theorem 67 Appendix A. 71 References 74756411 bytesapplication/pdf論文公開時間：2014/07/29論文使用權限：同意無償授權擬埃爾米特Li-Yau-HamiltonGap 定理Harnack 不等式thesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/264018/1/ntu-103-D98221003-1.pdf