共形紧Einstein流形上的分析和渐近双曲流形上的预定Q-曲率问题

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].

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。

共形紧Einstein(CCE)流形和渐进双曲(AH)流形上的分析是共形几何中的重要研究课题，与物理中AdS/CFT理论密切相关。本项目主要研究预定共形无穷远CCE度量唯一性和流形分类问题。取得成果：.1) 共形无穷远是球面上齐性空间时：我们证明如果无穷远Yamabe常数适当大，那么内部填充的非正曲率CCE度量全局唯一；如果无穷远Yamabe常数靠近标准球面度量，那么内部填充的CCE空间微分同胚于欧氏球，并且内部填充的CCE度量全局唯一。球面上齐性空间分为三大类，随参数增加难度递增，我们用对称性内延转化为常微分方程组退化两点边值问题依次处理。.2) 对于预定标量曲率渐近双曲度量问题，我们发展了共形曲率流方法。对满足极值原理的方程(Yamabe方程，$\sigma_k$-Ricci方程等)，我们证明始于稳态方程下解的初边值问题随着边值趋于无穷，曲率流内闭一致收敛于Loewner-Nirenberg问题的解。用共形曲率流作为平台实现了始于连续到边的负Ricci曲率度量到止于预定曲率的渐近双曲度量的保持负Ricci曲率度量的连续演化。.研究意义：.1）无穷远为齐性空间情形CCE度量唯一性研究，引入几何分析更广泛的技巧辅助共形几何，拓宽了该问题研究思路。近来张圣容团队对无穷远Yamabe常数足够靠近标准球面度量时CCE度量唯一性给出了共形紧化方法的解决。我们的方法为进一步研究提供特视角，和几何分析研究策略支持。该结论有力佐证了共形无穷远的共形结构对内部CCE流形拓扑和黎曼几何的影响甚至有某种程度的决定这一AdS/CFT理论的几何直观，将继续推动AdS/CFT理论黎曼几何版本的演绎。.2）共形曲率流初边值问题解决广义Loewner-Nirenberg问题的方法，给出了该问题求解的另一种系统的模式化思路，对于满足极值原理的完全非线性方程有效，是一种统一的方法，对研究预定标量曲率型的渐近双曲度量问题提供了有限到无限的连续过度平台，为数值计算提供思路。