| Summary: | 博士 === 國立臺灣大學 === 數學研究所 === 102 === 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.
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