The Penrose Inequality and Positive Mass Theorem with Charge for Manifolds with Asymptotically Cylindrical Ends

Jaracz, Jaroslaw S.

doi:10.1007/s00023-020-00927-z

Cited by 2 publications

(2 citation statements)

References 17 publications

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“…Novel features of Theorem 3.4 include the lower bound (3.11) for the difference m − |Q|, which does not rely on an energy condition, as well as a new approach to the rigidity statement which does not appear to be fully resolved in all cases [28]. Other related results may be found in [1, Theorem 2.1], [46,Theorem 2], and [47, Theorem 2].…”

Section: Statement Of Resultsmentioning

confidence: 99%

Spacetime Harmonic Functions and Applications to Mass

Bray¹,

Hirsch²,

Kazaras³

et al. 2021

Preprint

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In the pioneering work of Stern [73], level sets of harmonic functions have been shown to be an effective tool in the study of scalar curvature in dimension 3. Generalizations of this idea, utilizing level sets of so called spacetime harmonic functions as well as other elliptic equations, are similarly effective in treating geometric inequalities involving the ADM mass. In this paper, we survey recent results in this context, focusing on applications of spacetime harmonic functions to the asymptotically flat and asymptotically hyperbolic versions of the spacetime positive mass theorem, and additionally introduce a new concept of total mass valid in both settings which is encoded in interpolation regions between generic initial data and model geometries. Furthermore, a novel and elementary proof of the positive mass theorem with charge is presented, and the level set approach to the Penrose inequality given by Huisken and Ilmanen is related to the current developments. Lastly, we discuss several open problems.

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Section: Statement Of Resultsmentioning

confidence: 99%

Spacetime Harmonic Functions and Applications to Mass

Bray¹,

Hirsch²,

Kazaras³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The inequality has been extended to include charge and angular momentum and proven under certain conditions, but all of these require some kind of non-negativity for the scalar curvature. See for example [8,14,16,17].…”

Section: The Penrose Conjecturementioning

confidence: 99%

Nonexistence of solutions to the coupled generalized Jang equation/zero divergence system

Jaracz

2023

Class. Quantum Grav.

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In [5], Bray and Khuri proposed coupling the generalized Jang equation to several different auxiliary equations. The solutions to these coupled systems would then imply the Penrose inequality. One of these involves coupling the generalized Jang equation to $\overline{div}(\phi q)=0$, as this would guarantee the non-negativity of the scalar curvature in the Jang surface. This coupled system of equations has not received much attention, and we investigate its solvability. We prove that there exists a spherically symmetric initial data set for the Einstein equations for which there do not exist smooth radial solutions to the system having the appropriate asymptotics for application to the Penrose inequality.

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The Penrose Inequality and Positive Mass Theorem with Charge for Manifolds with Asymptotically Cylindrical Ends

Cited by 2 publications

References 17 publications

Spacetime Harmonic Functions and Applications to Mass

Spacetime Harmonic Functions and Applications to Mass

Nonexistence of solutions to the coupled generalized Jang equation/zero divergence system

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