Abstract
The thesis consists of three topics mainly. One is about the CR Li-Yau gradient estimate. The latter ones are concerned with the CR analogues of Yau's uniformization problems. They could be briefly described as follows. In Chapter 1, we introduce the basic notions in pseudohermitian geometry and the necessary notations adopted in the later chapters. In Chapter 2, we first derive the generalized curvature-dimension inequality with the help of the CR Bochner formula. Then it enables us to derive the CR Li-Yau gradient estimate for positive solutions to CR heat equation by modifying the Cao-Yau's method. Actually Cao-Yau's method not only provides a new way to obtain more precise gradient estimate than before but allows the pseudohermitian torsion to be nonvanishing. As applications, we have the CR version of Li-Yau Harnack inequality and upper bound estimate for the CR heat kernel. And, under the assumption of the generalized curvature-dimension inequality, we are able to confirm the Li-Yau gradient estimate for the sum of squares of vector fields up to higher step. In Chapter 3, we give a sufficient condition of the existence of the solution to CR heat equation to ensure its solution could be expressed in the convolution of the initial data with the CR heat kernel. Subsequently, by taking the appropriate initial condition of CR heat equation, we can get the rough dimension estimate. If modifying the estimate about the vanishing order of CR-holomorphic functions utilized in the rough dimension estimate, we are able to obtain the sharp dimension estimate. In Chapter 4, we first deduce the CR subhessian comparison; as a byproduct, it enables us to obtain the CR subLaplacian comparison. And then we derive the CR three-circle theorem. As applications, it enables us to deduce two CR sharp monotonicity formulas and the CR sharp dimension estimate with its rigidity.