arXiv subjects

Allen Juntao Fang

Publications and source records attributed to Allen Juntao Fang.

9 recordsLinked to original sources

arXiv2026

Teukolsky on slowly-rotating Kerr-de Sitter in the vanishing $Λ$ limit

As a first step towards resolving a vanishing cosmological constant black hole stability conjecture, we prove energy, Morawetz and rp-weighted estimates for solutions to the Teukolsky equations on a slowly-rotating Kerr-de Sitter background, which we derive using an extension of the non-integrable formalism of [GKS24]. The main feature of our estimates is their uniformity with respect to the cosmological constant $Λ>0$ (thus allowed to tend to 0), while they hold on the whole domain of outer communications, extending up to $Λ^{-\frac{1}{2}}$. As an application of our result, we recover well-known corresponding estimates for solutions to Teukolsky on a slowly-rotating Kerr background in the limit $Λ\to 0$.

math.AP↗

arXiv2026

Wave decay and horizon instability on strongly charged extremal Kerr-Newman black holes

We prove the first boundedness and pointwise decay result for the scalar wave equation on rotating extremal black holes without any symmetry assumptions. The result applies to slowly rotating (equivalently, strongly charged) extremal Kerr-Newman spacetimes. We establish uniform energy boundedness, integrated local energy decay, and a hierarchy of boundary-weighted estimates at the extremal horizon and at null infinity, from which inverse-polynomial pointwise decay follows in the entire exterior region. As a consequence, we also prove the expected Aretakis instability: for generic initial data, suitable transversal derivatives fail to decay along the event horizon, and higher transversal derivatives blow up asymptotically. The proof uses the $b$-structure of the wave operator near the two boundary hypersurfaces, together with a treatment of normally hyperbolic trapping on extremal Kerr--Newman.

math.AP↗

arXiv2025

Mass-Centered GCM Framework in Perturbations of Kerr(-Newman)

The nonlinear stability problem for black hole solutions of the Einstein equations critically depends on choosing an appropriate geometric gauge. In the vacuum setting, the use of Generally Covariant Modulated (GCM) spheres and hypersurfaces has played a central role in the proof of stability for slowly rotating Kerr spacetime. In this work, we develop an alternative GCM framework, that we call mass-centered, designed to overcome the breakdown of the standard GCM construction in the charged case, where electromagnetic-gravitational coupling destroys the exceptional behavior of the $\ell=1$ mode of the center-of-mass quantity used in the vacuum analysis. This construction is aimed at the nonlinear stability of Reissner-Nordström and Kerr-Newman spacetimes. Our approach replaces transport-based control of the center-of-mass quantity with a sphere-wise vanishing condition on a renormalized $\ell=1$ mode, yielding mass-centered GCM hypersurfaces with modified gauge constraints. The resulting elliptic-transport system remains determined once an $\ell=1$ basis is fixed via effective uniformization and provides an alternative construction in vacuum in the uncharged limit.

gr-qc↗

arXiv2025

Einstein-Maxwell Equations on Mass-Centered GCM Hypersurfaces

The resolution of the nonlinear stability of black holes as solutions to the Einstein equations relies crucially on imposing the right geometric gauge conditions. In the vacuum case, the use of Generally Covariant Modulated (GCM) spheres and hypersurfaces has been successful in the proof of stability for slowly rotating Kerr spacetime. For the charged setting, our companion paper introduced an alternative mass-centered GCM framework, adapted to the additional difficulties of the Einstein-Maxwell system. In this work, we solve the Einstein-Maxwell equations on such a mass-centered spacelike GCM hypersurface, which is equivalent to solving the constraint equations there. We control all geometric quantities of the solution in terms of some seed data, corresponding to the gauge-invariant fields describing coupled gravitational-electromagnetic radiation in perturbations of Reissner-Nordström or Kerr-Newman, first identified by the second author and expected to be governed by favorable hyperbolic equations. This provides the first step toward controlling gauge-dependent quantities in the nonlinear stability analysis of the Reissner-Nordström and Kerr-Newman families.

gr-qc↗

arXiv2025

On the uniqueness of Kerr-de Sitter spacetimes

In this paper, we prove a series of results concerning the uniqueness of Kerr-de Sitter as a family of smooth stationary black hole solutions to the nonlinear Einstein vacuum equations with positive cosmological constant $Λ$. The results only assume smoothness rather than analyticity of the solution in question. The results use a two-sided approach to rigidity, requiring assumptions on both the event horizon and the cosmological horizon (or a neighborhood thereof) to formulate an appropriate unique continuation argument to prove the rigidity of Kerr-de Sitter.

gr-qc↗

arXiv2025

Spacelike initial data for black hole stability

We construct initial data suitable for the Kerr stability conjecture, that is, solutions to the constraint equations on a spacelike hypersurface with boundary entering the black hole horizon that are arbitrarily decaying perturbations of a Kerr initial data set. This results from a more general perturbative construction on any asymptotically flat initial data set with the topology of $\mathbb{R}^3\setminus\{r<1\}$ enjoying some analyticity near and at the boundary. In particular, we design a suitable mixed boundary condition for the elliptic operator of the conformal method in order to exclude the Killing initial data sets (KIDS).

math.AP↗

arXiv2025

Initial data for Minkowski stability with arbitrary decay

We construct and parametrize solutions to the constraint equations of general relativity in a neighborhood of Minkowski spacetime with arbitrary prescribed decay properties at infinity. We thus provide a large class of initial data for the results on stability of Minkowski which include a mass term in the asymptotics. Due to the symmetries of Minkowski, a naive linear perturbation fails. Our construction is based on a simplified conformal method, a reduction to transverse traceless perturbations and a nonlinear fixed point argument where we face linear obstructions coming from the cokernels of both the linearized constraint operator and the Laplace operator. To tackle these obstructions, we introduce a well-chosen truncated black hole around which to perturb. The control of the parameters of the truncated black hole is the most technical part of the proof, since its center of mass and angular momentum could be arbitrarily large.

math.AP↗

arXiv2022

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

In this paper, we provide a new proof of nonlinear stability of the slowly-rotating Kerr-de Sitter family of black holes as a family of solutions to the Einstein vacuum equations with cosmological constant $Λ>0$, originally established by Hintz and Vasy in their seminal work [arXiv:1606.04014]. Using the linear theory developed in an upcoming companion paper, we prove the nonlinear stability of slowly-rotating Kerr-de Sitter using a bootstrap argument, avoiding the need for a Nash-Moser argument, and requiring initial data small only in the $H^6$ norm.

math.AP↗

arXiv2022

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

In this paper, we prove that the slowly-rotating Kerr-de Sitter family of black holes are linearly stable as a family of solutions to the Einstein vacuum equations with $Λ>0$ in harmonic (wave) gauge. This article is part of a series that provides a novel proof of the full nonlinear stability of the slowly-rotating Kerr-de Sitter family. This paper and its follow-up offer a self-contained alternative approach to nonlinear stability of the Kerr-de Sitter family from the original work of Hintz and Vasy by interpreting quasinormal modes as $H^k$ eigenvalues of an operator on a Hilbert space, and using integrated local energy decay estimates to prove the existence of a spectral gap. In particular, we avoid the construction of a meromorphic continuation of the resolvent. We also do not compactify the spacetime, thus avoiding the use of $b$-calculus and instead only use standard pseudo-differential arguments in a neighborhood of the trapped set; and avoid constraint damping altogether. The methods in the current paper offer an explicit example of how to use the vectorfield method to achieve resolvent estimates on a trapping background.

gr-qc↗