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標題: | 柯西黎曼 Li-Yau-Hamilton 不等式即其應用 CR Li-Yau-Hamilton Inequality and its Applications |
作者: | Yen-Wen Fan 樊彥彣 |
指導教授: | 張樹城(Shu-Cheng Chang) |
關鍵字: | 擬埃爾米特,Li-Yau-Hamilton,Gap 定理,Harnack 不等式, Li-Yau-Hamilton,Gap theorem,CR manifold,Harnack, |
出版年 : | 2014 |
學位: | 博士 |
摘要: | 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. |
URI: | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/5336 |
全文授權: | 同意授權(全球公開) |
顯示於系所單位: | 數學系 |
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