On the fill-in of nonnegative scalar curvature metrics

Shi, Yuguang; Wang, Wenlong; Wei, Guodong; Zhu, Jintian

doi:10.1007/s00208-020-02087-1

Cited by 13 publications

(25 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The regularity assumptions of (g, k) in Theorem 1.1 naturally arise from the fill-in and extension problems (e.g. [8], [44], [54], [58], [57]). For example, let (M 1 , g 1 ), (M 2 , g 2 ) be two Riemannian manifolds with smooth boundary, where ∂M 1 is isometric to ∂M 2 .…”

Section: 1mentioning

confidence: 99%

“…First, in [58], the construction of a scalar flat and asymptotically flat metric and decreasing total mean curvature difference along the radial direction proved in [54] are the main inputs to show that under certain assumptions, if an NNSC fill-in, i.e. (Ω, g) with R g ≥ 0, of Bartnik data (Σ, γ, H) exists, then there is a contradiction to the Riemannian positive mass theorem with corners ( [54], [44]).…”

Section: Non-existence Of Dec Fill-insmentioning

confidence: 99%

“…(cf. [58] Theorem 1.3) Let D SB := (S n−1 , γ, α, H, β) be a spacetime Bartnik data set. If γ is isotopic to γ std in M q psc (S n−1 ) := {η : C q metrics on S n−1 with R η > 0}, where q ≥ 5, then there exists a constant h 0 = h 0 (n, γ) > 0 such that if…”

Section: Non-existence Of Dec Fill-insmentioning

confidence: 99%

“…In [58] and [57], there was a partial affirmative answer given by the parabolic method to construct a metric extension done in [54]. It is shown that if the mean curvature is too large and an NNSC (non-negative scalar curvature) fillin to a Bartnik data set exists, then there is a contradiction to the Riemannian positive mass theorem with corners.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On a spacetime positive mass theorem with corners

Tin-Yau¹

2021

Preprint

View full text Add to dashboard Cite

In this paper we consider the positive mass theorem for general initial data sets satisfying the dominant energy condition which are singular across a piecewise smooth surface. We find jump conditions on the metric and second fundamental form which are sufficient for the positivity of the total spacetime mass. Our method extends that of [30] to the singular case (which we refer to as initial data sets with corners) using some ideas from [31]. As such we give an integral lower bound on the spacetime mass and we characterise the case of zero mass. Our approach also leads to a new notion of quasilocal mass which we show to be positive, extending the work of [54] to the spacetime case. Moreover, we give sufficient conditions under which spacetime Bartnik data sets cannot admit a fill-in satisfying the dominant energy condition. This generalises the work of [58] and [57] to the spacetime setting.

show abstract

Section: 1mentioning

confidence: 99%

Section: Non-existence Of Dec Fill-insmentioning

confidence: 99%

Section: Non-existence Of Dec Fill-insmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On a spacetime positive mass theorem with corners

Tin-Yau¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…For S > 0, the scalar curvature can also be perturbed in a similar way (see more details in the proof of [20,Theorem 1.5]). Now we can assume R g > S and we will construct a metric on a small collar neighborhood of Σ.…”

Section: Corollary 28 (See [12 Corollary 31]) Givenmentioning

confidence: 99%

Nonexistence of the NNSC-cobordism of Bartnik data

Shi

2021

Sci. China Math.

Self Cite

View full text Add to dashboard Cite

In this paper, we consider the problem of the nonnegative scalar curvature (NNSC)-cobordism of, γ 2 , H 2 ). We prove that given two metrics γ 1 and γ 2 on S n−1 (3 n 7) with H 1 fixed, then (S n−1 , γ 1 , H 1 ) and (S n−1 , γ 2 , H 2 ) admit no NNSC-cobordism provided the prescribed mean curvature H 2 is large enough (see Theorem 1.3). Moreover, we show that for n = 3, a much weaker condition that the total mean curvature ∫ S 2 H 2 dµγ 2 is large enough rules out NNSC-cobordisms (see Theorem 1.2); if we require the Gaussian curvature of γ 2 to be positive, we get a criterion for nonexistence of the trivial NNSCcobordism by using the Hawking mass and the Brown-York mass (see Theorem 1.1). For the general topology case, we prove that (Σ n−1 1 , γ 1 , 0) and (Σ n−1 2 , γ 2 , H 2 ) admit no NNSC-cobordism provided the prescribed mean curvature H 2 is large enough (see Theorem 1.5).

show abstract

Isoperimetry for asymptotically flat 3-manifolds with positive ADM mass

Yu¹

2022

Math. Ann.

View full text Add to dashboard Cite

On the fill-in of nonnegative scalar curvature metrics

Cited by 13 publications

References 28 publications

On a spacetime positive mass theorem with corners

On a spacetime positive mass theorem with corners

Nonexistence of the NNSC-cobordism of Bartnik data

Isoperimetry for asymptotically flat 3-manifolds with positive ADM mass

Contact Info

Product

Resources

About