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

Qing, Jie; Тиан, Ганг

doi:10.1090/s0894-0347-07-00560-7

Cited by 52 publications

(85 citation statements)

References 9 publications

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“…A more general uniqueness result was proven by Qing and Tian [13]. Metzger [12] generalized the previous results to manifolds whose metrics are small perturbations of strongly asymptotically flat metrics.…”

Section: Introductionmentioning

confidence: 74%

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Foliations by Stable Spheres with Constant Mean Curvature for Isolated Systems with General Asymptotics

Huang

2010

Commun. Math. Phys.

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Abstract. We prove the existence and uniqueness of constant mean curvature foliations for initial data sets which are asymptotically flat satisfying the Regge-Teitelboim condition near infinity. It is known that the (Hamiltonian) center of mass is well-defined for manifolds satisfying this condition. We also show that the foliation is asymptotically concentric, and its geometric center is the center of mass. The construction of the foliation generalizes the results of Huisken-Yau, Ye, and Metzger, where strongly asymptotically flat manifolds and their small perturbations were studied.

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Section: Introductionmentioning

confidence: 74%

“…Let k be a positive constant andû = u + k. Then multiplyingû p−1 on the both sides of (4.11), 13) wheref = f + k −1 h. Integrating (4.13) and using…”

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confidence: 99%

Foliations by Stable Spheres with Constant Mean Curvature for Isolated Systems with General Asymptotics

Huang

2010

Commun. Math. Phys.

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“…The optimal result that one could hope for is a = 0; namely, the foliation is unique outside a fixed compact set. We remark that it is proven true for strongly asymptotically flat manifolds by Qing and Tian [18], and it remains open for general asymptotics. …”

Section: Sketch Of Proofmentioning

confidence: 88%

“…Those authors also proved that the foliation is unique under some conditions. A more general uniqueness result was obtained by Qing and Tian [18]. For strongly asymptotically flat metric which is conformally flat near infinity, Corvino and Wu proved the geometric center of Huisken-Yau's foliation is equal to center of mass [9], and we later removed the condition of being conformally flat [11].…”

Section: Introductionmentioning

confidence: 84%

On the center of mass in general relativity

Ji¹,

Poon²,

Yang³

et al. 2012

Fifth International Congress of Chinese Mathematicians

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Abstract. The classical notion of center of mass for an isolated system in general relativity is derived from the Hamiltonian formulation and represented by a flux integral at infinity. In contrast to mass and linear momentum which are well-defined for asymptotically flat manifolds, center of mass and angular momentum seem less well-understood, mainly because they appear as the lower order terms in the expansion of the data than those which determine mass and linear momentum. This article summarizes some of the recent developments concerning center of mass and its geometric interpretation using the constant mean curvature foliation near infinity. Several equivalent notions of center of mass are also discussed.

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“…The definition pioneered a very active area of research with an extensive volume of literatures by now [51,42,40,32,35,26,24,6,7,34,8].…”

Section: Introductionmentioning

confidence: 99%

Energy, momentum, and center of mass in general relativity

Wang

2016

Surveys in Differential Geometry

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Abstract. These notions in the title are of fundamental importance in any branch of physics. However, there have been great difficulties in finding physically acceptable definitions of them in general relativity since Einstein's time. I shall explain these difficulties and progresses that have been made. In particular, I shall introduce new definitions of center of mass and angular momentum at both the quasi-local and total levels, which are derived from first principles in general relativity and by the method of geometric analysis. With these new definitions, the classical formula p = mv is shown to be consistent with Einstein's field equation for the first time. This paper is based on joint work [14,15] with Po-Ning Chen and Shing-Tung Yau.

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

Cited by 52 publications

References 9 publications

Foliations by Stable Spheres with Constant Mean Curvature for Isolated Systems with General Asymptotics

Foliations by Stable Spheres with Constant Mean Curvature for Isolated Systems with General Asymptotics

On the center of mass in general relativity

Energy, momentum, and center of mass in general relativity

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