A positive mass theorem for manifolds with boundary

Hirsch, Sven; Miao, Pengzi

doi:10.2140/pjm.2020.306.185

Cited by 15 publications

(14 citation statements)

References 26 publications

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“…Decay (75) coupled with (16) (18) and (19) and one can show, with similar estimates as before, that the first two terms of this sum tend to 0 for r → +∞. It is also easy to see, using ( 13) and (19), that…”

Section: Appendixsupporting

confidence: 73%

“…We remark that our results are not based on the Positive Mass Theorem. By contrast, we observe that using this celebrated result, more precisely a consequence of it contained in [16,Theorem 1.5], one can prove the following uniqueness statement. We refer the reader to Definition 1.1 for the notation and terminology.…”

Section: Introductionmentioning

confidence: 91%

“…Proof of Theorem 1.2. By condition (8) and by the fact that D 2 g0 u ≡ 0 on ∂M , which in turn implies H g0 ∂M ≡ 0, we have that the hypothesis of [16,Theorem 1.5] are fulfilled, so that m ADM ≥ C.…”

Section: A Black-hole Uniqueness Theorem For Sub-static Manifoldsmentioning

confidence: 99%

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A geometric capacitary inequality for sub-static manifolds with harmonic potentials

Agostiniani¹,

Mazzieri²,

Oronzio³

2020

Preprint

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In this paper, we prove that associated with a sub-static asymptotically flat manifold endowed with a harmonic potential there is a one-parameter family {F β } of functions which are monotone along the level-set flow of the potential. Such monotonicity holds up to the optimal threshold β = n−2 n−1 and allows us to prove a geometric capacitary inequality where the capacity of the horizon plays the same role as the ADM mass in the celebrated Riemannian Penrose Inequality.

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Section: Appendixsupporting

confidence: 73%

Section: Introductionmentioning

confidence: 91%

See 1 more Smart Citation

A geometric capacitary inequality for sub-static manifolds with harmonic potentials

Agostiniani¹,

Mazzieri²,

Oronzio³

2020

Preprint

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“…Remark 1.3. If R ≥ 0 and H ≤ −2 ∂G ∂ν , (1.6) implies m(g) ≥ 0, which is a special case of results in [14].…”

Section: Introductionmentioning

confidence: 90%

Mass of asymptotically flat $3$-manifolds with boundary

Hirsch,

Miao,

Tsang

2020

Preprint

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We study the mass of asymptotically flat 3-manifolds with boundary using the method of Bray-Kazaras-Khuri-Stern [5]. More precisely, we derive a mass formula on the union of an asymptotically flat manifold and fill-ins of its boundary, and give new sufficient conditions guaranteeing the positivity of the mass. Motivation to such consideration comes from studying the quasi-local mass of the boundary surface. If the boundary isometrically embeds in the Euclidean space, we apply the formula to obtain convergence of the Brown-York mass along large surfaces tending to ∞ which include the scaling of any fixed coordinate-convex surface.Proposition 1.1. Let (M, g) be an asymptotically flat 3-manifold with boundary Σ. Let u be a harmonic function on M which is asymptotic to one of the asymptotically flat coordinate functions at infinity. Let Σ t = u −1 (t) be the level set of u and ν be the infinity pointing unit P. Miao acknowledges the support of NSF Grant DMS-1906423.

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“…This is proved in Section 2. The basic idea is to use a certain Green's function for the Laplacian based at the orbifold points, which we use to reduce to the positive mass theorem for manifolds with concave boundary due to Hirsch-Miao [HM20] using the fundamental work of Schoen-Yau [SY79, SY81, SY17].…”

Section: Introductionmentioning

confidence: 99%

Conformally prescribed scalar curvature on orbifolds

Ju¹,

Viaclovsky²

2021

Preprint

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We study the prescribed scalar curvature problem in a conformal class on orbifolds with isolated singularities. We prove a compactness theorem in dimension 4, and an existence theorem which holds in dimensions n ≥ 4. This problem is more subtle than the manifold case since the positive mass theorem does not hold for ALE metrics in general. We also determine the U(2)-invariant Leray-Schauder degree for a family of negative-mass orbifolds found by LeBrun.

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A positive mass theorem for manifolds with boundary

Cited by 15 publications

References 26 publications

A geometric capacitary inequality for sub-static manifolds with harmonic potentials

A geometric capacitary inequality for sub-static manifolds with harmonic potentials

Mass of asymptotically flat $3$-manifolds with boundary

Conformally prescribed scalar curvature on orbifolds

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