arXiv subjects

Jie Qing

Publications and source records attributed to Jie Qing.

42 records · Page 3Linked to original sources

arXiv2005

On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds

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.

math.DG↗

arXiv2003

Ricci Curvature Rigidity for Weakly Asymptotically Hyperbolic Manifolds

A rigidity result for weakly asymptotically hyperbolic manifolds with lower bounds on Ricci curvature is proved without assuming that the manifolds are spin. The argument makes use of a quasi-local mass characterization of Euclidean balls from \cite{Miao} \cite{S_T} and eigenfunction compactification ideas from \cite{Qing}.

math.DG↗

arXiv2003

On the uniqueness of the AdS space-time in higher dimensions

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.

math.DG↗

arXiv2003

On the topology of conformally compact Einstein 4-manifolds

In this paper we study the topology of conformally compact Einstein 4-manifolds. When the conformal infinity has positive Yamabe invariant and the renormalized volume is also positive we show that the conformally compact Einstein 4-manifold will have at most finite fundamental group. Under the further assumption that the renormalized volume is relatively large, we conclude that the conformally compact Einstein 4-manifold is diffeomorphic to $B^4$ and its conformal infinity is diffeomorphic to $S^3$.

math.DG↗

arXiv2003

On finiteness of Kleinian groups in general dimension

In this paper we provide a criteria for geometric finiteness of Kleinian groups in general dimension. We formulate the concept of conformal finiteness for Kleinian groups in space of dimension higher than two, which generalizes the notion of analytic finiteness in dimension two. Then we extend the argument in the paper of Bishop and Jones to show that conformal finiteness implies geometric finiteness unless the set of limit points is of Hausdorff dimension n. Furthermore we show that, for a given Kleinian group $Γ$, conformal finiteness is equivalent to the existence of a metric of finite geometry on the Kleinian manifold $Ω(Γ)/Γ$.

math.DG↗

arXiv2003

On the rigidity for conformally compact Einstein manifolds

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.

math.DG↗