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

Ma, Shiguang

doi:10.2140/pjm.2011.252.145

Cited by 13 publications

(14 citation statements)

References 13 publications

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“…For CMC-surfaces in asymptotically Euclidean Riemannian manifolds, uniqueness has been proven under much weaker assumptions on the CMC-surfaces σ with H ( σ ) ≡ 2 /σ than σ ∈ A (a, b, η) (but under stronger asymptotic decay assumptions than C 2 1 /2+εasymptotic flatness), most notably by Ye [48], Qing and Tian [43], Ma [33], Carlotto, Chodosh, and Eichmair [9], and Chodosh and Eichmair [15]. It would certainly be interesting to understand uniqueness of STCMC-surfaces under weaker assumptions on the surfaces; note however that STCMC-surfaces do not naturally arise from a variational principle and in particular that the STCMC-stability operator L H derived in Lemma 1 is non-selfadjoint.…”

Section: Remark 11mentioning

confidence: 99%

On center of mass and foliations by constant spacetime mean curvature surfaces for isolated systems in General Relativity

Cederbaum

Sakovich

2021

Calc. Var.

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We propose a new foliation of asymptotically Euclidean initial data sets by 2-spheres of constant spacetime mean curvature (STCMC). The leaves of the foliation have the STCMC-property regardless of the initial data set in which the foliation is constructed which asserts that there is a plethora of STCMC 2-spheres in a neighborhood of spatial infinity of any asymptotically flat spacetime. The STCMC-foliation can be understood as a equivariant relativistic generalization of the CMC-foliation suggested by Huisken and Yau (Invent Math 124:281–311, 1996). We show that a unique STCMC-foliation exists near infinity of any asymptotically Euclidean initial data set with non-vanishing energy which allows for the definition of a new notion of total center of mass for isolated systems. This STCMC-center of mass transforms equivariantly under the asymptotic Poincaré group of the ambient spacetime and in particular evolves under the Einstein evolution equations like a point particle in Special Relativity. The new definition also remedies subtle deficiencies in the CMC-approach to defining the total center of mass suggested by Huisken and Yau (Invent Math 124:281–311, 1996) which were described by Cederbaum and Nerz (Ann Henri Poincaré 16:1609–1631, 2015).

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Section: Remark 11mentioning

confidence: 99%

On center of mass and foliations by constant spacetime mean curvature surfaces for isolated systems in General Relativity

Cederbaum

Sakovich

2021

Calc. Var.

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“…We remark that much progress has been made recently in developing analogues of the results of [38,57] in general asymptotically flat Riemannian 3-manifolds, see e.g. [36,43,53]. D. Christodoulou and S.-T. Yau [20] have noted that the Hawking mass of volume-preserving stable CMC spheres in asymptotically flat Riemannian 3-manifolds with non-negative scalar curvature is non-negative.…”

Section: Introductionmentioning

confidence: 99%

Effective versions of the positive mass theorem

2016

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The study of stable minimal surfaces in Riemannian 3-manifolds (M, g) with non-negative scalar curvature has a rich history. In this paper, we prove rigidity of such surfaces when (M, g) is asymptotically flat and has horizon boundary. As a consequence, we obtain an effective version of the positive mass theorem in terms of isoperimetric or, more generally, closed volume-preserving stable CMC surfaces that is appealing from both a physical and a purely geometric point of view. We also include a proof of the following conjecture of Schoen: An asymptotically flat Riemannian 3-manifold with non-negative scalar curvature that contains an unbounded area-minimizing surface is isometric to flat R 3 .

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“…The class of surfaces to which the uniqueness result applies was subsequently extended to all volume-preserving stable CMC surfaces lying outside of a sufficiently large set by J. Qing and G. Tian [QT07]. See also [Hua09,Hua10,Ma11,Ma12] for results along these lines for metrics with more general asymptotics.…”

Section: Introductionmentioning

confidence: 99%

Large Isoperimetric Regions in Asymptotically Hyperbolic Manifolds

Chodosh

2015

Commun. Math. Phys.

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We show the existence of isoperimetric regions of sufficiently large volumes in general asymptotically hyperbolic three manifolds. Furthermore, we show that large coordinate spheres in compact perturbations of Schwarzschild-anti-deSitter are uniquely isoperimetric. This is relevant in the context of the asymptotically hyperbolic Penrose inequality.Our results require that the scalar curvature of the metric satisfies Rg ≥ −6, and we construct an example of a compact perturbation of Schwarzschild-anti-deSitter without Rg ≥ −6 so that large centered coordinate spheres are not isoperimetric. The necessity of scalar curvature bounds is in contrast with the analogous uniqueness result proven by Bray for compact perturbations of Schwarzschild, where no such scalar curvature assumption is required.This demonstrates that from the point of view of the isoperimetric problem, mass behaves quite differently in the asymptotically hyperbolic setting compared to the asymptotically flat setting. In particular, in the asymptotically hyperbolic setting, there is an additional quantity, the "renormalized volume," which has a strong effect on the large-scale geometry of volume.I am very grateful to my advisor, Simon Brendle, for suggesting this problem, as well as for his invaluable assistance, support, and many suggestions on a preliminary version of this paper. Additionally, I would like to thank Michael Eichmair and Brian White for patiently answering numerous questions, as well as for their continued encouragement. Finally, I would like to acknowledge

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

Cited by 13 publications

References 13 publications

On center of mass and foliations by constant spacetime mean curvature surfaces for isolated systems in General Relativity

On center of mass and foliations by constant spacetime mean curvature surfaces for isolated systems in General Relativity

Effective versions of the positive mass theorem

Large Isoperimetric Regions in Asymptotically Hyperbolic Manifolds

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