柯西黎曼 Li-Yau-Hamilton 不等式即其應用

Abstract

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.