The Weyl problem in warped product spaces

Li, Chunhe; Wang, Zhizhang

doi:10.4310/jdg/1580526016

Cited by 16 publications

(39 citation statements)

References 31 publications

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“…We emphasize that, when n = 2, given any {Σ t } in (M, g), if Σ 0 admits an isometric embedding into (N,ḡ), there exists a family of {Σ t } in (N,ḡ) satisfying condition (2.2). This is guaranteed by the openness result of solutions to the isometric embedding problem, which is due to Li and Wang [8,9].…”

Section: Evolution Of Quasi-local Massmentioning

confidence: 99%

“…In this section, we apply Formula 2.1, the openness result of the isometric embedding problem [8], and Theorem 1.2 to prove Theorem 1.1.…”

Section: Equality Case Of the Localized Penrose Inequalitymentioning

confidence: 99%

“…We can parametrize {Σ t } so that, as t increases, Σ t evolves in a direction normal to Σ t and has constant unit speed. Applying the openness result of the isometric embedding problem in [8], we obtain a smooth family of 2-surfaces {Σ t } −ǫ<t≤0 in M 3 m so thatΣ 0 = Σ and condition (2.2) is satisfied by {Σ t } and {Σ t }. By (2.3) and the assumption H = H m , we have…”

Section: Equality Case Of the Localized Penrose Inequalitymentioning

confidence: 99%

“…As a key step in their proof, they computed the first variation of the quasi-local energy of Σ ′ in reference to M 3 m . Such a variational consideration is made possible by the openness result of solutions to the isometric embedding problem into warped product space, which is due to Li and Wang [8]. Combining the variation formula with inequality (1.2), Chen-Zhang established the rigidity of Σ in M 3 m .…”

Section: Introductionmentioning

confidence: 99%

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Variation and rigidity of quasi-local mass

Miao

2019

Advances in Theoretical and Mathematical Physics

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Inspired by the work of Chen-Zhang [5], we derive an evolution formula for the Wang-Yau quasi-local energy in reference to a static space, introduced by Chen-Wang-Wang-Yau [4]. If the reference static space represents a mass minimizing, static extension of the initial surface Σ, we observe that the derivative of the Wang-Yau quasi-local energy is equal to the derivative of the Bartnik quasi-local mass at Σ.Combining the evolution formula for the quasi-local energy with a localized Penrose inequality proved in [10], we prove a rigidity theorem for compact 3-manifolds with nonnegative scalar curvature, with boundary. This rigidity theorem in turn gives a characterization of the equality case of the localized Penrose inequality in 3-dimension.

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Section: Evolution Of Quasi-local Massmentioning

confidence: 99%

“…In this section, we apply Formula 2.1, the openness result of the isometric embedding problem [8], and Theorem 1.2 to prove Theorem 1.1.…”

Section: Equality Case Of the Localized Penrose Inequalitymentioning

confidence: 99%

Section: Equality Case Of the Localized Penrose Inequalitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Variation and rigidity of quasi-local mass

Miao

2019

Advances in Theoretical and Mathematical Physics

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“…A natural candidate for the reference space is the Schwarzschild spacetime. In [10], Li and Wang found the following lower bound…”

Section: Introductionmentioning

confidence: 99%

A quasi-local Penrose inequality for the quasi-local energy with static references

Chen

2020

Trans. Amer. Math. Soc.

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The positive mass theorem is one of the fundamental results in general relativity. It states that, assuming the dominant energy condition, the total mass of an asymptotically flat spacetime is non-negative. The Penrose inequality provides a lower bound on mass by the area of the black hole and is closely related to the cosmic censorship conjecture in general relativity. In [14], Lu and Miao proved a quasi-local Penrose inequality for the quasi-local energy with reference in the Schwarzschild manifold. In this article, we prove a quasi-local Penrose inequality for the quasi-local energy with reference in any spherically symmetric static spacetime.

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A free boundary isometric embedding problem in the unit ball

Koerber

2022

Calc. Var.

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We study a free boundary isometric embedding problem for abstract Riemannian two-manifolds with the topology of the disc. Under the assumption of positive Gauss curvature and geodesic curvature of the boundary being equal to one, we show that every such disc may be isometrically embedded into the Euclidean three-space $$\mathbb {R}^3$$ R 3 such that the image of the boundary meets the unit sphere $$\mathbb {S}^2$$ S 2 orthogonally. We also show that the embedding is unique up to rotations and reflections through planes containing the origin. Finally, we define a new Brown-York type quasi-local mass for certain free boundary surfaces and discuss its positivity.

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The Weyl problem in warped product spaces

Cited by 16 publications

References 31 publications

Variation and rigidity of quasi-local mass

Variation and rigidity of quasi-local mass

A quasi-local Penrose inequality for the quasi-local energy with static references

A free boundary isometric embedding problem in the unit ball

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