Abstract
In this thesis, we consider the gradient estimates in pseudohermitian geometry. In chapter 1, we frst give an introduction to pseudohermitian manifolds and derive some Bochner-Type estimates for the later use.

In chapter 2, we introduce some results about the CR sub-Laplacian comparison property.
Secondly, by modifying method of Yau’s gradient estimate and using the CR sub-Laplacian comparison property, we are able to derive the gradient estimate for positive pseudoharmonic functions on a complete noncompact pseudohermitian (2□□+ 1)-manofold which is served as the CR version of Yau’s gradient estimate. As an application of the gradient estimate, we derive the CR analogue of Liouville-type Theorem. In particular, the CR analogue of Liouville-type Theorem holds on the standard Heisenberg (2□□+ 1)-manifolds.

In chapter 3, we introduce a third order operator □□and the CR Paneitz operator □0. Then
we derive another CR Bochner type formula which involves □. We use two kinds of CR Borchner formulae to derive two types of CR Li-Yau gradient estimates on a closed pseudohermitian 3-manifold. As an application, we .rst get a subgradient estimate of logarithm of the positive solution of CR heat equation. Secondly, we have the Harnack inequality and upper bound estimate for the heat kernel. Finally, we obtain Perelman-type entropy formulae for the CR heat equation.

In chapter 4, we introduce the CR Yamabe flow and present the evolution equations under the CR Yamabe flow. Then we prove the CR Li-Yau-Hamilton gradient estimate for CR Yamabe flow. As an application, we are able to get the Harnack inequality for the CR Yamabe flow.

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