Unique isoperimetric foliations of asymptotically flat manifolds in all dimensions

Eichmair, Michael; Metzger, Jan

doi:10.1007/s00222-013-0452-5

Cited by 74 publications

(90 citation statements)

References 57 publications

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“…Here, being asymptotically flat means g = e g + O 2 (|x| − 1 2 −ε ) with S = O 0 (|x| −3−ε ), where S the scalar curvature of g . Further properties of this foliation were studied by Huisken-Yau, Corvino-Wu, Eichmair-Metzger, the author, and others [HY96,CW08,EM12,Ner13,Ner15b].…”

Section: Introductionmentioning

confidence: 97%

Geometric characterizations of asymptotic flatness and linear momentum in general relativity

Nerz

2015

Journal of Functional Analysis

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Abstract. In 1996, Huisken-Yau proved that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed surfaces of constant mean curvature (CMC) if it is asymptotically equal to the (spatial) Schwarzschild solution. Later, their decay assumptions were weakened by Metzger, Huang, Eichmair-Metzger, and the author. In this work, we prove the reverse implication, i. e. any three-dimensional Riemannian manifold is asymptotically flat if it possesses a CMC-cover satisfying certain geometric curvature estimates, a uniqueness property, a weak foliation property, and each surface has weakly controlled instability. With the author's previous result that every asymptotically flat manifold possesses a CMC-foliation, we conclude that asymptotic flatness is characterized by existence of such a CMC-cover. Additionally, we use this characterization to give a geometric (i. e. coordinate-free) definition of a (CMC-)linear momentum and prove its compatibility with the linear momentum defined by Arnowitt-Deser-Misner.

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Section: Introductionmentioning

confidence: 97%

Geometric characterizations of asymptotic flatness and linear momentum in general relativity

Nerz

2015

Journal of Functional Analysis

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“…It is well-known that the coordinate CMC-center of mass of a given, suitably asymptotically Euclidean Riemannian 3-manifold (M 3 , g) coincides with its ADM-center of mass. This has been established by Huang [19], Eichmair and Metzger [18], and Nerz [25,Corollary 3.8] under different assumptions on the asymptotic decay. Moreover, it has been demonstrated by Cederbaum that the coordinate CMC-and ADMcenters of mass converge to the Newtonian one in the Newtonian limit c → ∞ [6,Chap.…”

Section: 3mentioning

confidence: 70%

On the Center of Mass of Asymptotically Hyperbolic Initial Data Sets

Cederbaum

Cortier

Sakovich

2015

Ann. Henri Poincaré

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We define the (total) center of mass for suitably asymptotically hyperbolic time-slices of asymptotically anti-de Sitter spacetimes in general relativity. We do so in analogy to the picture that has been consolidated for the (total) center of mass of suitably asymptotically Euclidean time-slices of asymptotically Minkowskian spacetimes (isolated systems). In particular, we unite-an altered version of-the approach based on Hamiltonian charges with an approach based on CMC-foliations near infinity. The newly defined center of mass transforms appropriately under changes of the asymptotic coordinates and evolves in the direction of an appropriately defined linear momentum under the Einstein evolution equations.

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“…Remark 26. The arguments in [12] show that r(Σ) → ∞ as area(Σ) → ∞ when the scalar curvature of (M, g) is positive.…”

Section: Remark 25mentioning

confidence: 99%

“…The seminal results of G. Huisken and S.-T. Yau [22] on the existence of a canonical foliation at infinity of asymptotically flat manifolds and the uniqueness of the leaves of this foliation (which has been refined by J. Qing and G. Tian [27]) has had considerable repercussions on the study of such manifolds, both from a physical and a geometric point of view. We refer the reader to the introductions of [19,12,13,14] for recent accounts on these developments. The goal of this paper is to extend many of these results to asymptotically conical manifolds.…”

Section: Introductionmentioning

confidence: 99%

Isoperimetric structure of asymptotically conical manifolds

Chodosh¹,

Eichmair²,

Volkmann³

2017

J. Differential Geom.

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We study the isoperimetric structure of Riemannian manifolds that are asymptotic to cones with non-negative Ricci curvature. Specifically, we generalize to this setting the seminal results of G. Huisken and S.-T. Yau [22] on the existence of a canonical foliation by volume preserving stable constant mean curvature surfaces at infinity of asymptotically flat manifolds as well as the results of the second-named author with S. Brendle [6] and J. Metzger [13, 14] on the isoperimetric structure of asymptotically flat manifolds. We also include an observation on the isoperimetric cone angle of such manifolds. This result is a natural analogue of the positive mass theorem in this setting.

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Unique isoperimetric foliations of asymptotically flat manifolds in all dimensions

Cited by 74 publications

References 57 publications

Geometric characterizations of asymptotic flatness and linear momentum in general relativity

Geometric characterizations of asymptotic flatness and linear momentum in general relativity

On the Center of Mass of Asymptotically Hyperbolic Initial Data Sets

Isoperimetric structure of asymptotically conical manifolds

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