共形紧Einstein流形(CCE流形)是共形几何中重要的研究课题，与物理中AdS/CFT理论关系密切。本项目主要研究预定共形无穷远的CCE度量的唯一性和CCE流形的分类问题。.(1) 研究预定共形无穷远边值（包括齐性边值和球面扰动边值）对应的内部非负曲率CCE度量的唯一性和一般的内部填充CCE度量的全局唯一性。思路是用CCE流形的几何质心和调和坐标结合来fix gauge，将Einstein方程组化为内部非退化的椭圆型方程组。其中齐性边值情形问题用对称性内延化为常微分方程组两点边值问题来处理。已有部分成果[29][31]。.(2) 研究重整化体积对四维CCE流形的拓扑和微分结构的影响。.作为CCE流形的推广，渐近双曲流形上预定常值Q-曲率问题的研究有利于CCE度量的研究。我们考虑.(3) 研究渐近双曲流形上给定共形类中预定常值Q-曲率的渐近双曲度量的存在性。

The study on conformally compact Einstein(CCE) manifolds has been important in conformal geometry, and it is closely related to the AdS/CFT theory in physics. This program focuses on uniqueness of CCE metrics with prescribed conformal infinity and the classification problem of the CCE manifolds. Namely,.(1) Uniqueness of (non-positively curved) CCE metrics with prescribed conformal infinity(which is a homogeneous metric on the sphere or a perturbation metric of the round metric). The approach is to fix gauge of the Einstein equations by the geometric center of gravity of the CCE manifold and the harmonic coordinates, so that the Einstein system becomes a system of elliptic differential equations. In particular, for homogeneous conformal infinity, we extend the symmetry into the CCE manifold and deform the problem to a two-point boundary value problem of a system of ODEs. For partial results see [29],[31]..(2) Consider how the renormalized volume influences the topology and differential structures of the CCE manifolds of dimension four...Consider the asymptotically hyperbolic(AH) manifolds as a generalization of the CCE manifolds. The study of the prescribed constant Q-curvature problem on AH manifolds is helpful to the existence and uniqueness problem of CCE metrics. And as a generalization of Yamabe problem on AH manifolds, it has its own interests. We study.(3) The existence of constant Q-curvature metric which is asymptotically hyperbolic in the conformal class of a given AH manifold. For known perturbation results, see [28].