Abstract:
In this paper, based on an intrinsic definition of asymptotically AdS space-times, we show that the standard anti-de Sitter space-time is the unique strictly stationary asymptotically AdS solution to the vacuum Einstein equations with negative cosmological constant in dimension less than 7. Instead of using the positive energy theorem for asymptotically hyperbolic spaces our approach appeals to the classic positive mass theorem for asymptotically flat spaces.

Abstract:
In this paper we prove that a conformally compact Einstein manifold with the round sphere as its conformal infinity has to be the hyperbolic space. We do not assume the manifolds to be spin, but our approach relies on the positive mass theorem for asymptotic flat manifolds. The proof is based on understanding of positive eigenfunctions and compactifications obtained by positive eigenfunctions.

Abstract:
In this paper we take an approach similar to that in [M] to establish a positive mass theorem for asymptotically hyperbolic spin manifolds admitting corners along a hypersurface. The main analysis uses an integral representation of a solution to a perturbed eigenfunction equation to obtain an asymptotic expansion of the solution in the right order. This allows us to understand the change of the mass aspect of a conformal change of asymptotically hyperbolic metrics.

Abstract:
We solve the classifying problem raised by Fischer and Marsden for Bach flat static spaces. We also prove the conjecture about critical point equations proposed by Besse for Bach flat manifolds. Particularly in dimension 3, we derive an integral identity that allows us to obtain conformal flatness from the vanish of the full divergence of the Cotton tensor for static metrics and metrics satisfying the critical point equation.

Abstract:
In this note we study constant mean curvature surfaces in asymptotically flat 3-manifolds. We prove that, in an asymptotically flat 3-manifold with positive mass, stable spheres of given constant mean curvature outside a fixed compact subset are unique. Therefore we are able to conclude that there is a unique foliation of stable spheres of constant mean curvature in an asymptotically flat 3-manifold with positive mass.

Abstract:
In this paper, we give a sharp spectral characterization of conformally compact Einstein manifolds with conformal infinity of positive Yamabe type in dimension $n+1>3$. More precisely, we prove that the largest real scattering pole of a conformally compact Einstein manifold $(X,g)$ is less than $\ndemi -1$ if and only if the conformal infinity of $(X,g)$ is of positive Yamabe type. If this positivity is satisfied, we also show that the Green function of the fractional conformal Laplacian $P(\alpha)$ on the conformal infinity is non-negative for all $\alpha\in [0, 2]$.

Abstract:
In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators $P_\alpha$ were introduced via the scattering theory for Poincar\'{e} metrics associated with a conformal manifold $(M^n, [g])$. We prove that, on a closed and locally conformally flat manifold with Poincar\'{e} exponent less than $\frac {n-\alpha}2$ for some $\alpha \in [2, n)$, the set of positive smooth solutions to the equation $$ P_\alpha u = u^\frac {n+\alpha}{n-\alpha} $$ is compact in the $C^\infty$ topology. Therefore the existence of positive solutions follows from the existence of Yamabe metrics and a degree theory.

Abstract:
In this note we study the conformal metrics of constant $Q$ curvature on closed locally conformally flat manifolds. We prove that for a closed locally conformally flat manifold of dimension $n\geq 5$ and with Poincar\"{e} exponent less than $\frac {n-4}2$, the set of conformal metrics of positive constant $Q$ and positive scalar curvature is compact in the $C^\infty$ topology.

Abstract:
In this paper we extend the local scalar curvature rigidity result in [6] to a small domain on general vacuum static spaces, which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the paper [10]. We obtain the local scalar curvature rigidity of bounded domains in hyperbolic spaces. We also obtain the global scalar curvature rigidity for conformal deformations of metrics in the domains, where the lapse functions are positive, on vacuum static spaces with positive scalar curvature, and show such domains are maximal, which generalizes the work in [15].

Abstract:
Based on data from the investigation in Jiaozhou bay waters in 1979, the distribution, pollution source and seasonal variation of PHC in Jiaozhou Bay are analyzed. It is showed that in Jiaozhou bay PHC contents arrived the national Category Ⅱof the water quality standard during this year. In summer in the bay the pollution of PHC was heavy, while relatively light in spring. In the coastal waters in the east and the northeast of the bay, the PHC contents in spring surpassed the national Category Ⅱ, and surpassed the national Category Ⅲ. In the coastal waters, in the northeast of the bay, the change of the PHC contents formed the grads: the contents presented the falling trend from the big one to the small, which unveiled that the PHC souce in the bay mainly came from the discharge of the industrial waste water and living sewage.