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 74

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