Nonlinear stability of the slowly-rotating Kerr-de Sitter family

Fang, Allen Juntao

doi:10.48550/arxiv.2112.07183

Cited by 6 publications

(7 citation statements)

References 11 publications

(24 reference statements)

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“…Unlike their Λ > 0 and Λ = 0 counterparts, whose exterior stability properties have been intensely studied over the past two decades [Vas13,Dya16,HV18,Sch16,Mav21a,Mav21b,Fan21,Fan22] and [DHR19b, ABBM19, HHV21, DHRT21, KS20, KS21], the stability properties of Kerr-adS spacetimes have remained more elusive. A first distinct feature of the Λ < 0 case is that stability is to be understood in the context of an initial boundary value problem for the Einstein equations, for which boundary conditions have to be imposed at the conformal infinity of the spacetime (see [Fri95,EK19]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In this paper, we prove mode stability for the initial boundary value problem of Theorem 1.4. Mode stability -in conjunction with the theory of scattering resonances 7 -has been a fundamental ingredient to obtain the decay of solutions to wave-type equations: see for example [HHV21] for the proof of the linear stability of Kerr with slow rotation |a| M , or [HV18,Fan21,Fan22] for the proof of the linear and non-linear stability of the slowly rotating Kerr-de Sitter solutions, which all rely on resonance expansions and the identification of obstructions to mode stability. In another framework, quantitative versions of mode stability (see [Shl15,Tei20]) were used in the proof of the decay of solutions to the wave and Teukolsky equations on Kerr in the full subextremal range |a| < M in [DRS16] and [ST20].…”

Section: Mode Stability On the Real Axis And The Main Theoremmentioning

confidence: 99%

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Mode stability for the Teukolsky equations on Kerr-anti-de Sitter spacetimes

Graf¹,

Holzegel²

2022

Preprint

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We prove that there are no non-stationary (with respect to the Hawking vectorfield) real mode solutions to the Teukolsky equations on all (3+1)-dimensional subextremal Kerr-anti-de Sitter spacetimes. We further prove that stationary solutions do not exist if the black hole parameters satisfy the Hawking-Reall bound and a √ −Λ < √ 3 20 . We conclude with the statement of mode stability which preludes boundedness and decay estimates for general solutions which will be proven in a separate paper. Our boundary conditions are the standard ones which follow from fixing the conformal class of the metric at infinity and lead to a coupling of the two Teukolsky equations. The proof relies on combining the Teukolsky-Starobinsky identities with the coupled boundary conditions. In the stationary case the proof exploits elliptic estimates which fail if the Hawking-Reall bound is violated. This is consistent with the superradiant instabilities expected in that regime.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Mode Stability On the Real Axis And The Main Theoremmentioning

confidence: 99%

Mode stability for the Teukolsky equations on Kerr-anti-de Sitter spacetimes

Graf¹,

Holzegel²

2022

Preprint

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“…We recall in this context that the proofs of the nonlinear stability of slowly rotating Kerrde Sitter and Kerr-Newman-de Sitter black holes by the author and Vasy [HV18,Hin18] (see also [Fan21]) are of this latter kind: rough exponential bounds for waves on asymptotically Kerr-Newman-de Sitter spacetimes are obtained using a simple energy estimate, and high regularity in such exponentially weighted spaces is proved by microlocal means (i.e. propagation of regularity in phase space).…”

Section: 2mentioning

confidence: 99%

Linear waves on asymptotically flat spacetimes. I

Hintz¹

2023

Preprint

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We introduce a novel framework for the analysis of linear wave equations on non-stationary asymptotically flat spacetimes, under the assumptions of mode stability and absence of zero energy resonances for a stationary model operator. Our methods apply in all spacetime dimensions and to tensorial equations, and they do not require any symmetry or almost-symmetry assumptions on the spacetime metrics or on the wave type operators. Moreover, we allow for the presence of terms which are asymptotically scaling critical at infinity, such as inverse square potentials. For simplicity of presentation, we do not allow for normally hyperbolic trapping or horizons.In the first part of the paper, we study stationary wave type equations, i.e. equations with time-translation symmetry, and prove pointwise upper bounds for their solutions. We establish a relationship between pointwise decay rates and weights related to the mapping properties of the zero energy operator. Under a nondegeneracy assumption, we prove that this relationship is sharp by extracting leading order asymptotic profiles at late times. The main tool is the analysis of the resolvent at low energies.In the second part, we consider a class of wave operators without time-translation symmetry which settle down to stationary operators at a rate t −δ * as an appropriate hyperboloidal time function t * tends to infinity. The main result is a sharp solvability theory for forward problems on a scale of polynomially weighted spacetime L 2 -Sobolev spaces. The proof combines a regularity theory for the non-stationary operator with the invertibility of the stationary model established in the first part. The regularity theory is fully microlocal and utilizes edge-b-analysis near null infinity, as developed in joint work with Vasy, and 3b-analysis in the forward cone.

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“…We also mention Mavrogiannis's works [88][89][90] on quasilinear wave equations on cosmological black hole background. The global nonlinear stability of slowly rotating Kerr-de Sitter spacetimes was proved by Hintz-Vasy [64], followed by the work of Fang [41,42]. 1.3.…”

Section: Introductionmentioning

confidence: 99%

The linear stability of weakly charged and slowly rotating Kerr-Newman family of charged black holes

McClements¹

2023

Preprint

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In this paper, we prove the linear stability of weakly charged and slowly rotating Kerr-Newman black holes under coupled gravitational and electromagnetic perturbations. We show that the solutions to the linearized Einstein-Maxwell equations decay at an inverse polynomial rate to a linearized Kerr-Newman solution plus a pure gauge term. This work builds on the framework developed in [57] for the study of the Einstein vacuum equations. We work in the generalized wave map and Lorenz gauge. The proof involves the analysis of the resolvent of the Fourier transformed linearized Einstein-Maxwell operator on asymptotically flat spaces, which relies on recent advances in microlocal analysis and non-elliptic Fredholm theory developed in [109]. The most delicate part of the proof is the description of the resolvent at low frequencies.

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Nonlinear stability of the slowly-rotating Kerr-de Sitter family

Cited by 6 publications

References 11 publications

Mode stability for the Teukolsky equations on Kerr-anti-de Sitter spacetimes

Mode stability for the Teukolsky equations on Kerr-anti-de Sitter spacetimes

Linear waves on asymptotically flat spacetimes. I

The linear stability of weakly charged and slowly rotating Kerr-Newman family of charged black holes

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