Key ingredients of the proof

b0= (M,0,0)∈B, M,0>0, (4.1) be parameters for a Schwarzschild–de Sitter spacetime. We describe the three main ingre-dients using which we will prove the non-linear stability of slowly rotating Kerr–de Sit-ter black holes: first, UEMS—the mode stability for the ungauged Einstein equation, D_gb

0(Ric+Λ)(r)=0—see §4.1; second, SCP—the existence of a hyperbolic formulation of the linearized Einstein equation for which the constraint propagation equation ex-hibits mode stability, in a form which is stable under perturbations—see§4.2; and third, ESG—quantitative high-energy bounds for the linearized Einstein equations which are stable under perturbations—see§4.3. We recall that the key for the non-linear problem is to understand the linear stability problem in a robust perturbative framework, and the ingredients SCP and ESG will allow us to set up such a framework, which then also makes UEMS a stable property.

Motivated by the discussion in§1.1of our approach to the non-linear stability prob-lem, the linearized gauged Einstein equation which we will study to establish the linear stability of slowly rotating Kerr–de Sitter spacetimes is

Lbr= (Dg_b(Ric+Λ)−δ˜^∗Dg_bΥ)r= 0, (4.2) where Υ(g) is the gauge 1-form defined in (3.35), and where ˜δ^∗ is a modification ofδ^∗_gb, which we define in (4.5) below. For proving the linear stability of the metricgb, using the linearization of the gauge condition Υ(g)−Υ(gb)=0 aroundg=gb leads to the condition Dg_bΥ(r)=0, and hence to the form ofLb given here.

4.1. Mode stability for the ungauged Einstein equation

Theorem4.1. (UEMS—Ungauged Einstein equation: mode stability) Let b0be the parameters of a Schwarzschild–de Sitter black hole as in (4.1).

(1) Let σ∈C, Imσ>0,σ6=0, and suppose that h(t∗, x) =e^−iσt∗h0(x),

with h₀∈C^∞(Y, S²T_Y^∗Ω),is a mode solution of the linearized Einstein equation D_gb

0(Ric+Λ)(h) = 0. (4.3)

Then there exists a1-form ω(t_∗, x)=e^−iσt∗ω0(x),with ω0∈C^∞(Y, T_Y^∗Ω),such that h=δ^∗_gb

0ω.

(2) For all k∈N0, and all generalized mode solutions h(t_∗, x) =

Thus, (1) asserts that any mode solution h of the linearized Einstein equations which is exponentially growing, or non-decaying and oscillating, is a pure gauge solution h=δ_g^∗

b0ω=¹₂L_ω]g_b0coming from an infinitesimal diffeomorphism (Lie derivative), i.e. does not constitute a physical degree of freedom for the linearized Einstein equation. On the other hand, (2) asserts that mode solutions with frequencyσ=0, and possibly containing polynomially growing terms, are linearized Kerr–de Sitter metrics, up to a Lie derivative.

Thus, the only non-decaying mode solutions of the linearized Einstein equation (4.3), up to Lie derivatives, are the linearized Kerr–de Sitter metricsg_b⁰

0(b⁰), linearized around the Schwarzschild–de Sitter metricgb₀.

Observe here that a small displacement of the center of a black hole with parameters b=(M,a), preserving its mass and its angular momentum vector, corresponds to pulling back the metric gb on M by a translation; the restriction of the pull-back metric to Ω is well defined. Therefore, on the linearized level, the change in the metric due to an infinitesimal displacement of the black hole is given by the Lie derivative ofgb along a translation vector field. This justifies our restriction of the space B of black hole parameters to only include mass and angular momentum, but not the center of mass.

The particular form of the Kerr–de Sitter metricsgb and its linearizations used here is rather arbitrary: ifφb:M!M is a smooth family of diffeomorphisms that commute with translations int∗, then one obtains another smooth family of Kerr–de Sitter metrics on the extended spacetimeMby setting ˜gb:=φ^∗_bgb. Given a solutionhof (4.3), we have

In order to see the invariance of (2) underφb, we in addition compute

˜

hence the linearized metrics indeed agree under the diffeomorphism φb up to a Lie de-rivative, as desired. In particular, UEMS is independent of the specific choice of the functionscb,±and ˜cb,± in (3.15).

We stress that Theorem4.1only concerns the mode stability of a (single) Schwarz-schild–de Sitter black hole; the assumption doesnotconcern the mode stability of nearby slowly rotating Kerr–de Sitter spacetimes. We give the proof of the theorem in§7.

4.2. Mode stability for the constraint propagation equation

Recall from (2.12), or rather its modification, taking the modification ofδ^∗_g into account, that for a solutions r of the linearized gauged Einstein equation (4.2) (with arbitrary Cauchy data), the linearized gauge 1-form%=Dg_bΥ(r) solves the equation

e^CPg_b %≡2δ_gbG_gb˜δ^∗%= 0. (4.4) The asymptotic behavior of general solutions%∈C^∞(M, T^∗M) of this equation depends on the specific choice of ˜δ^∗. We will show the following result.

Theorem 4.2. (SCP—Stable constraint propagation) One can choose δ˜^∗, equal to δ_g^∗_b

0 up to order-zero terms,such that there exists α>0with the property that smooth so-lutions%∈C^∞(M, T^∗M)of the equation e^CPg_b0%=0decay exponentially in t_∗with rate α, that is, %=O(e^−αt∗).

In fact, we will give a very concrete definition of ˜δ^∗. Namely, we will show that δ˜^∗u:=δ^∗_g

b0u+γ dt_∗·u−¹₂eγhu, dt∗iGb0g_b0 (4.5) works, for e<1 close to 1 and for sufficiently large γ >0, for allb near b0. We give the proof of Theorem4.2in §8.

We recall from§1.1that the usefulness of SCP stems from the observation that, for a non-decaying smooth mode solutionrof the equationLb0r=0, the constraint propagation equatione^CPg_b0Dg_b0Υ(r)=0 impliesDg_b0Υ(r)=0, and hencerin fact solves the linearized ungauged Einstein equationD_gb

0(Ric+Λ)r=0. We can then appeal to UEMS to obtain very precise information about r. Thus, all non-decaying resonant states of L_b0 are pure gauge solutions, plus linearized Kerr–de Sitter metricsg_b⁰

0(b⁰); and furthermore all non-decaying resonant statesrsatisfy the linearized gauge conditionD_gb

0Υ(r)=0.

In fact, we will show in§10, using the additional ingredient ESG below, that SCP is sufficient to deduce the mode stability, and in fact linear stability, of slowly rotating Kerr–de Sitter black holes as well.

Note here that, as in§4.1, the above theoremonly concerns a (fixed)Schwarzschild–

de Sitter metric.

4.3. High-energy estimates for the linearized gauged Einstein equation The final key ingredient ensures that solutions of the linear equations which appear in the non-linear iteration scheme have asymptotic expansions up to exponentially decaying remainder terms. Specifically, we need this to be satisfied for solutions of the linearized gauged Einstein equation (4.2) for b=b0, and for perturbations of this equation. Since the linear equations we will need to solve are always invariant under translations int_∗ up to operators with exponentially decaying coefficients, the following theorem suffices for this purpose.

Theorem4.3. (ESG—Essential spectral gap for the linearized gauged Einstein equa-tion; informal version) For the choice of δ˜^∗in SCP,the linearized Einstein operator L_b0 defined in (4.2) has a positive essential spectral gap; that is, there exists α>0 and in-tegers N_L>0, d_j>1 for 16j6N_L, and n<sub>j`</sub>>0 for 16j6N<sub>L</sub> and 16`6d_j, as well as smooth sectionsa_j`k∈C^∞(Y, S²T_Y^∗Ω),such that solutions rof the equation L_b0r=0with smooth initial data on Σ₀ have an asymptotic expansion

r=

N_L

X

j=1 dj

X

`=1

rj`

^nj`

X

k=0

e^−iσjt∗t^k_∗aj`k(x)

+r⁰ (4.6)

in Ω, with rj`∈Cand r⁰=O(e^−αt∗).

The same holds true, with possibly different NL, dj, nj` and aj`k, but the same constant α>0,for the operator Lb,with b∈UB.

In order to give a precise and quantitative statement, which in addition will be straightforward to check, we recall from [140] and, directly related to the present context, from [72] that the only difficulty in obtaining an asymptotic expansion (4.6) is the precise understanding of the operator Lb₀ at the trapped set Γ⊂T^∗M\o defined in (3.30);

we discuss this in detail in §5. Concretely, one is set if, for a fixed t_∗-independent positive definite inner product on the bundleS²T^∗M, one can find a stationary, elliptic pseudodifferential operator (ps.d.o.) Q∈Ψ⁰(M; End(S²T^∗M)), defined microlocally near Γ, with microlocal inverseQ⁻, such that in the coordinates (3.29), we have

±|σ|⁻¹σ1

1

2i QLb₀Q⁻−(QLb₀Q⁻)^∗

< αΓId (4.7) in Γ^±=Γ∩{±σ <0}(recall (3.31)), whereα_Γis a positive constant satisfyingα_Γ<¹₂ν_min, with ν_min the minimal expansion rate of the H_Gb

0-flow in the normal directions at Γ, defined in [55, equation (5.1)] and explicitly computed for Schwarzschild–de Sitter space-times in [54,§3] and [72,§2]. If one arranges the estimate (4.7), it also implies the same estimate at the trapped set for perturbations ofL_b0 within any fixed finite-dimensional

family of stationary, second-order, principally scalar differential operators; in particular, it holds forLb withb∈UB, where we possibly need to shrink the neighborhoodUB ofb0. (In the latter case, one in fact does not need to use the structural stability of the trapped set, since one can check it directly for Kerr–de Sitter spacetimes, see [54, §3], building on [147,§2] and [140,§6]. See the discussion at the end of§9.1for further details.)

Observe that condition (4.7) only concerns principal symbols; thus, checking it amounts to an algebraic computation, which is most easily done using the framework of pseudodifferential inner products; see [72] and Remark5.2.

Theorem4.4. (ESG—Essential spectral gap for the linearized gauged Einstein equa-tion; precise version) Fix αΓ>0. Then, there exist a neighborhood UB of b0 and a sta-tionary ps.d.o. Qsuch that(4.7)holds for Lb with b∈UB. Moreover,for any fixed s>¹₂, there exist constants α>0,CL<∞and C >0 such that, for all b∈UB, the estimate

kukH^s

hσi−16Chσi²kLˆb(σ)uk_Hs−1

hσi−1 (4.8)

holds for all u for which the norms on both sides are finite, provided Imσ >−α and

|Reσ|>CL, as well as for Imσ=−α. Here, H_~^s≡H_~^s(Y, S²T_Y^∗Ω) is the semiclassical Sobolev space of distributions which are extendible at ∂Y (see Appendix A.3), defined using the volume density induced by the metric gb0,and using a fixed stationary positive definite inner product on S²T^∗M.

Moreover, by reducing α>0 if necessary, one can arrange that all resonances σ of Lb₀, i.e. poles of the meromorphic family Lˆb₀(σ)⁻¹, which satisfy Imσ >−α, in fact satisfy Imσ>0.

Remark 4.5. The dependence of the decay rate αon the regularitys is very mild:

the inequality that needs to be satisfied is s>¹₂+βα, withβ being the larger value of βb₀,± defined in (3.27). In particular, if this holds for some sand αas in Theorem4.4, then it continuous to hold for all larger values ofsas well. The size ofα>0 is thus really only restricted by the location of resonances in the lower half-plane.

It is crucial here that there exists a choice of ˜δ^∗which makes SCP hold and for which at the same time ESG is valid as well; this turns out to be very easy to arrange. We will prove this theorem in§9.

That Theorem 4.3 is a consequence of Theorem 4.4 follows from a representation of solutions of the linear equation L_b0r=0 in the t_∗-Fourier domain, and shifting the contour in the inverse Fourier transform from a line Imσ1 to Imσ=−α, which is justified by the estimate (4.8). The asymptotic expansion is then a consequence of the residue theorem: the exponents σ_j∈C are the poles of ˆL_b0(σ)⁻¹, n_j` is one more than

the order of the pole, anddj is the rank of the resonance; see§5.1.1for definitions, and the beginning of§5.2.2for a sketch of the contour-shifting argument.

Under the assumptions of this theorem, the operators L_b have only finitely many resonances σ, in the half-space Imσ>−α, and no resonances σ have Imσ=−α. By general perturbation arguments discussed in§5, the total rank of the resonances ofL_b in Imσ >−αthus remains constant; see Proposition5.11. The essential spectral gap of Lb is, by definition, the supremum over all α>0 such thate Lb has only finitely many resonances in the half-space Imσ >−α.e

The regularity assumption in ESG, s>¹₂, which we will justify in §9.2, is due to radial point estimates at the event and cosmological horizons intersected with future infinityX (microlocally: near∂Rb), as we explain in§5.1. The powerhσi² on the other hand is due to the loss of one power of the semiclassical parameter in the estimate at the semiclassical trapped set, which forL_b0 can be identified with Γ∩{σ=∓1}.