研究期間：10208~10307;The CR Paneitz operator plays a key role in CR geometry. By means of properties of this operator,e.g.its nonnegativity, we can simplify some estimates and thus get some important results.In this project, we would like to know how many CR manifolds are associated with a nonnegative Paneitz operator.In addition, we would like to convert this problem into a CR embedding problem and see how much they relate to each other. Our recent work reveals such a phenomenon. We will start with studing some examples and some special case. To review a lot of articles about the subject of embedding problem and try to find out how the Paneitz operator come in. We also want to find a natural method to prove the nonnegativity of the Paneitz operator when the pseudohermitian torsion is free. The “natural” here means that it can be extended to show this in a more general case. Finally, we would like to build a general theory by means of this studing of examples and special cases..