A note on asymptotically flat metrics on ℝ³ which are scalar-flat and admit minimal spheres

Corvino, Justin

doi:10.1090/s0002-9939-05-07926-8

Cited by 26 publications

(24 citation statements)

References 17 publications

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“…This can be proved by various means (cf. [7]), and one such way is to use the following proposition from [9], which relies on the Riemannian Penrose Inequality [2,12], cf. [3].…”

Section: Apparent Horizonsmentioning

confidence: 99%

Construction of 𝑁-body time-symmetric initial data sets in general relativity

Chruściel¹,

Corvino²,

Isenberg³

2011

Complex Analysis and Dynamical Systems IV

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Given a collection of N asymptotically Euclidean ends with zero scalar curvature, we construct a Riemannian manifold with zero scalar curvature and one asymptotically Euclidean end, whose boundary has a neighborhood isometric to the disjoint union of a specified collection of sub-regions of the given ends. An application is the construction of time-symmetric solutions of the constraint equations which model N -body initial data.

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“…This can be proved by various means (cf. [7]), and one such way is to use the following proposition from [9], which relies on the Riemannian Penrose Inequality [2,12], cf. [3].…”

Section: Apparent Horizonsmentioning

confidence: 99%

Construction of 𝑁-body time-symmetric initial data sets in general relativity

Chruściel¹,

Corvino²,

Isenberg³

2011

Complex Analysis and Dynamical Systems IV

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“…The problem of stability for the Positive Mass Theorem has been studied by the first author in [19], by Finster with Bray and Kath in [5], [9] [8] and by Corvino in [6]. The work of Finster and his collaborators mainly focuses on using the ADM mass to obtain L 2 bounds on curvature.…”

Section: Introductionmentioning

confidence: 99%

Stability of the positive mass theorem for rotationally symmetric Riemannian manifolds

Lee

Sormani

2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

179

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We study the stability of the Positive Mass Theorem using the Intrinsic Flat Distance. In particular we consider the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces whose boundaries are either outermost minimal hypersurfaces or are empty. We prove that a sequence of these manifolds whose ADM masses converge to zero must converge to Euclidean space in the pointed Intrinsic Flat sense. In fact we provide explicit bounds on the Intrinsic Flat Distance between annular regions in the manifold and annular regions in Euclidean space by constructing an explicit filling manifold and estimating its volume. In addition, we include a variety of propositions that can be used to estimate the Intrinsic Flat distance between Riemannian manifolds without rotationally symmetry. Conjectures regarding the Intrinsic Flat stability of the Positive Mass Theorem in the general case are proposed in the final section.

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“…Finster [Fin09] removed the dependence on the isoperimetric constant and obtained the L 2 bound of the curvature tensor with the exception of a set of small surface area. J. Corvino [Cor05] proved that a particular bound on the mass and sectional curvature of a threedimensional asymptotically flat manifold of nonnegative scalar curvature implies the manifold is diffeomorphic to R 3 . Under the assumption of conformal flatness and zero scalar curvature outside a compact set, the second author [Lee09] proved that if a sequence of smooth asymptotically flat metrics of nonnegative scalar curvature has mass approaching zero, then the sequence converges in smooth topology to the Euclidean metric outside a compact set.…”

Section: Introductionmentioning

confidence: 99%

Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space

Huang

Lee

Sormani

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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The rigidity of the Positive Mass Theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We study the stability of this statement for spaces that can be realized as graphical hypersurfaces in E n+1 . We prove (under certain technical hypotheses) that if a sequence of complete asymptotically flat graphs of nonnegative scalar curvature has mass approaching zero, then the sequence must converge to Euclidean space in the pointed intrinsic flat sense. The appendix includes a new Gromov-Hausdorff and intrinsic flat compactness theorem for sequences of metric spaces with uniform Lipschitz bounds on their metrics.

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A note on asymptotically flat metrics on ℝ³ which are scalar-flat and admit minimal spheres

Cited by 26 publications

References 17 publications

Construction of 𝑁-body time-symmetric initial data sets in general relativity

Construction of 𝑁-body time-symmetric initial data sets in general relativity

Stability of the positive mass theorem for rotationally symmetric Riemannian manifolds

Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space

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