Positive mass theorem on manifolds admitting corners along a hypersurface

Miao, Pengzi

doi:10.4310/atmp.2002.v6.n6.a4

Cited by 194 publications

(305 citation statements)

References 7 publications

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“…The key assumption of positive semidefiniteness of the sum of the second fundamental forms of the boundaries in the gluing procedure of Kosovskiȋ [2002] and Perelman [1997] has also been used by Miao [2002]. He obtained a metric of bounded scalar curvature on the union of two manifolds with boundary, assuming that each manifold had bounded scalar curvature and that the boundaries satisfied the sum condition for mean curvature.…”

Section: Remarks On Gluingmentioning

confidence: 99%

An extension procedure for manifolds with boundary

Wong¹

2008

Pacific J. Math.

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This paper introduces an isometric extension procedure for Riemannian manifolds with boundary, which preserves some lower curvature bound and produces a totally geodesic boundary. As immediate applications of this construction, one obtains in particular upper volume bounds, an upper intrinsic diameter bound for the boundary, precompactness, and a homeomorphism finiteness theorem for certain classes of manifolds with boundary, as well as a characterization up to homotopy of Gromov-Hausdorff limits of such a class.

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Section: Remarks On Gluingmentioning

confidence: 99%

An extension procedure for manifolds with boundary

Wong¹

2008

Pacific J. Math.

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“…Nevertheless, through a delicate estimate, they were able to prove the existence of harmonic spinors to furnish the proof of the positivity of the total mass. Positive mass theorems on manifolds with discontinuities have been proved by Shi-Tam [18] and Miao [13].…”

Section: Introductionmentioning

confidence: 99%

A generalization of Liu-Yau's quasi-local mass

Wang¹,

Yau²

2007

Communications in Analysis and Geometry

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In [11,12], Liu and the second author propose a definition of the quasi-local mass and prove its positivity. This is demonstrated through an inequality which in turn can be interpreted as a total mean curvature comparison theorem for isometric embeddings of a surface of positive Gaussian curvature. The Riemannian version corresponds to an earlier theorem of Shi and Tam [18]. In this article, we generalize such an inequality to the case when the Gaussian curvature of the surface is allowed to be negative. This is done by an isometric embedding into the hyperboloid in the Minkowski space and a future-directed time-like quasi-local energy-momentum is obtained.

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“…It was known to many experts that the positive mass theorem holds in this general setting and in fact this fact was used by Bunting and Masood-ul-alam [2,9,8] to prove uniqueness for various blackhole solutions. But there had been no detailed proofs in the literature until two recent papers by Miao [7] who uses a mollification argument to reduce it to the regular case and by Shi-Tam [11] generalizing Witten's spinor argument.…”

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confidence: 99%

On the Uniqueness of the ADS Spacetime

Wang

2005

Acta Math Sinica

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The uniqueness of the ADS spacetime among all static vacuum spacetimes with the same conformal infinity is proved for dimension n ≤ 7. For dimension n > 7, the same result is established under the spin assumption. In Einstein's theory of general relativity with a negative cosmological constant Λ, a vacuum spacetime is a solution to the equation R ab = Λg ab and the lowest-energy solution is the anti-de Sitter spacetime. Moreover it was proved by Bocher-GibbonsHorowitz [1] in 3 + 1 dimensions that the anti-de Sitter spacetime is the unique static, asymptotically anti-de Sitter vacuum.To give the precise statement of their theorem, let us first recall that an (n + 1) dimensional static spacetime (N, g) has the formwhere (M, g) is a Riemannian manifold and V is a positive function on M . The vacuum Einstein equation (without loss of generality we always take the negative cosmological constant Λ to be −n)Ric (g) = −ng

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Positive mass theorem on manifolds admitting corners along a hypersurface

Cited by 194 publications

References 7 publications

An extension procedure for manifolds with boundary

An extension procedure for manifolds with boundary

A generalization of Liu-Yau's quasi-local mass

On the Uniqueness of the ADS Spacetime

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