本项目将探讨拟Hermitian流形上的若干几何分析问题，涉及拟调和映射的存在性和拟Hermitian流形的紧性问题等。拟调和映照是调和映照在拟Hermitian流形上的自然推广，它不仅提供了一种重要的次椭圆偏微分方程，而且是研究拟Hermitian几何结构的重要工具。我们将探讨紧致带边和完备非紧拟Hermitian流形上拟调和映照的存在性及其几何应用。各种特殊几何结构的紧性是几何分析的重要问题，项目将探讨一般拟Hermitian流形及特殊拟Hermitian结构的紧性，如：拟Einstein结构的紧性，以期得到若干分类和CR版本的单值化型定理。为了实现以上项目目标，我们将研究CR Sobolev不等式、Carnot-Caratheodory距离的次拉普拉斯比较定理和热核估计等。

This project will study some geometric problems on pseudo-Hermitian manifolds, such as existence of pseudo-harmonic maps and compactness of pseudo-Hermitian manifolds. One natural generalization of harmonic maps on pseudo-Hermitian manifolds is the pseudo-harmonic map, which is an important subelliptic differential equations and provides some important tools to explore pseudo-Hermitian geometry. We will investigate the existences of pseudo-harmonic maps from both complete noncompact pseudo-Hermitian manifolds and compact pseudo-Hermitian manifolds with boundary. Some geometric applications will be studied. The compactness of various geometric structures is one of important problems in geometric analysis. In order to establish some classification and CR uniformization theorems, this project will explore the compactness of general pseudo-Hermitian manifolds and some special ones, such as the compactness of pseudo-Einstein manifolds. For these purposes, we will study CR Sobolev inequality, sub-Laplacian comparison theorem of Carnot-Caratheodory distance and heat kernel estimates.