# Azimuthal instabilities on extremal Kerr

Dejan Gajic  Institut für Theoretische Physik, Universität Leipzig,
Brüderstrasse 16, 04103 Leipzig, Deutschland<EMAIL_ADDRESS>
###### Abstract.

We prove the existence of instabilities for the geometric linear wave equation
on extremal Kerr spacetime backgrounds, which describe stationary black holes
rotating at their maximally allowed angular velocity. These instabilities can
be associated to non-axisymmetric azimuthal modes and are stronger than the
axisymmetric instabilities discovered by Aretakis in [Are15]. The existence of
non-axisymmetric _instabilities_ follows from a derivation of very precise
_stability_ properties of solutions: we determine therefore the precise,
global, leading-order, late-time behaviour of solutions supported on a bounded
set of azimuthal modes via energy estimates in both physical and frequency
space. In particular, we obtain sharp, uniform decay-in-time estimates and we
determine the coefficients and rates of inverse-polynomial late-time tails
everywhere in the exterior of extremal Kerr black holes. We also demonstrate
how non-axisymmetric instabilities leave an imprint in the radiation on future
null infinity via the coefficients appearing in front of slowly decaying and
oscillating late-time tails.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries: geometry and foliations
  3. 3 Preliminaries: wave equation
  4. 4 Precise statements of the main theorems
  5. 5 A sketch of the proofs and an overview of the main new ideas
  6. 6 Frequency-space analysis
  7. 7 Integrated energy estimates
  8. 8 $r$-weighted and $(r-M)^{-1}$-weighted energy estimates
  9. 9 Energy decay for $K$-derivatives
  10. 10 Elliptic estimates and $K$-inversion theory
  11. 11 Decay estimates and late-time asymptotics
  12. 12 Azimuthal instabilities
  13. A Estimates for $\square_{g}\Pi$

##  1\. Introduction

Extremal black holes are critical members of families of stationary black
holes. They sit exactly at a phase transition from black hole to “naked
singularity”.111In this context, naked singularities are spacetimes that, from
an evolutionary perspective, have geodesically incomplete future and past null
infinities and arise from _singular_ initial data. These are also called
“super-extremal” spacetimes. Criticality can be understood geometrically via
the vanishing of the surface gravity associated to Killing vector fields that
generate black hole event horizons.

In the context of the vacuum Einstein equations,

$\textnormal{Ric}[g]=0,$ (1.1)

such black holes are called _extremal Kerr black holes_ [Ker63] and
criticality can be interpreted as the black holes rotating at their maximally
allowed speed.

In this paper, we perform a mathematically rigorous investigation of the
dynamical properties of extremal black holes in the context of the geometric,
linear wave equation:

$\square_{g_{M,a}}\phi=0,$ (1.2)

on extremal Kerr spacetime backgrounds $(\mathcal{M},g_{M,a})$ with mass $M$
and specific angular momentum
$|a|=M$. A mathematical study of (1.2) constitutes a first step towards
addressing the dynamics of black holes that arise in the evolution of small
perturbations of extremal Kerr initial data for (1.1).

Even the most elementary dynamical question, the question of _stability_ ,
still remains widely open for small perturbations of extremal Kerr initial
data. This may be contrasted with the problem of stability of _sub_ -extremal
Kerr, corresponding to $|a|<M$ and describing black holes that rotate at
speeds strictly below the maximal speed limit, for which tremendous progress
has recently been achieved; see the discussion and references in §1.2.

Already at the level of the linear problem (1.2), the criticality of extremal
Kerr leads to a significant departure from the sub-extremal case. In [Are15],
Aretakis showed that extremal Kerr black holes feature an _axisymmetric,
asymptotic instability_ along their future event horizon $\mathcal{H}^{+}$:
higher-order transversal derivatives of axisymmetric $\phi$ generically blow
up polynomially as time
$\tau\to\infty$. The same type of instability is also present for linear wave
equations on the simpler, non-rotating but electromagnetically charged
(“extremal Reissner–Nordström”) black holes [Are11a, Are11b].

An asymptotic instability at the _linear_ level may lead to a genuine
instability at the _nonlinear level_ , i.e. the spacetimes arising from small
perturbations of Kerr data could deviate significantly from extremal Kerr at
late times. An asymptotic instability is, however, also compatible with
_stability_ at the nonlinear level. That is to say, while derivatives of the
metric might blow up as $\tau\to\infty$, the metric itself could stay close to
the unperturbed metric. Indeed, in the context of extremal Reissner–Nordström
black holes, an analysis of a nonlinear analogue of (1.2) with a similar
nonlinear structure to (1.1) has shown that the asymptotic instability remains
asymptotic in the nonlinear setting and the solutions decay to zero [AAG20b].
A similar result is expected to hold in the setting of _axisymmetric_
solutions to analogous nonlinear wave equations on extremal Kerr.

In this paper, we go _beyond axisymmetry_ and derive the existence of
_stronger_ asymptotic instabilities for non-axisymmetric solutions to (1.2).
These _azimuthal instabilities_ are derived via a sharp analysis of the
_stability_ properties of $\phi$ in the form of the precise leading-order
late-time behaviour (“late-time tails”) of azimuthal mode solutions $\phi_{m}$
to (1.2) along $\mathcal{H}^{+}$, with $m$ an azimuthal number.222These are
solutions with an azimuthal angular dependence $e^{im\varphi}$, where
$m\in\mathbb{Z}$. These tails feature _slower decay in time_ when compared to
the axisymmetric or sub-extremal settings.

Away from the event horizon and along future null infinity $\mathcal{I}^{+}$,
we moreover derive the existence of slowly-decaying, oscillating late-time
tails, which decay like $u^{-1}$ and oscillate to leading-order with a
frequency $m\upomega_{+}$, where $\upomega_{+}$ is the angular velocity of the
black hole and $u$ a standard Bondi time coordinate. This late-time behaviour
is markedly different from the faster-decaying, non-oscillating sub-extremal
late-time tails (“Price’s law”), as derived in [Hin22, AAG23].

While the nonlinear analysis of [AAG20b] suggests stability of extremal Kerr
within the restricted class of axisymmetric perturbations333Since the initial
data in this setting is necessarily geodesically incomplete and extremal Kerr
data has no trapped surfaces, “stability” in this context would correspond to
one of the following scenarios: the convergence of the perturbed metric in a
suitable choice of coordinates to the black hole exterior of 1) a nearby sub-
extremal Kerr spacetime, 2) a nearby extremal Kerr spacetime, with an
asymptotic instability at the level of metric derivatives along the event
horizon (“horizon hair”), or 3) the failure to collapse to a black hole: the
perturbed spacetime is close to the causal past of a finite subset of
$\mathcal{I}^{+}$ in extremal Kerr. See also [DHRT21, §IV.2] for a related
discussion on the formulations of nonlinear stability of extremal
Reissner–Nordström. , the slow decay and the stronger nature of the
instabilities derived in the present paper goes beyond the methods of [AAG20b]
and tantalizes the possibility of a _departure from the Kerr family_ in the
end-point of the nonlinear evolution of small perturbations of extremal Kerr
initial data.

The quantitative analysis of late-time tails along $\mathcal{I}^{+}$ in the
present paper also has observational implications, as it also implies the
existence of characteristic slowly decaying, oscillating late-time tails at
_late, but finite_ time intervals in sub-extremal Kerr black holes that are
sufficiently close to extremality (“near-extremal Kerr black holes”). One may
think of such _intermediate late time tails_ as carrying an imprint of the
azimuthal instability that would be occurring at the event horizon of an
extremal black hole.

###  1.1. Main theorems

We now give the first, rough formulations of the main two theorems that are
proved in the present paper. More precise versions are stated in §4, after an
introduction of the relevant geometric notation in §§2–3.

######  Theorem 1.1 (The existence of azimuthal instabilities on extremal
Kerr; rough version).

Non-axisymmetric solutions to (1.2), arising from suitably regular and
decaying initial data feature at least one of the following asymptotic
instabilities: generically,444The initial data for which none of these
instabilities occur can be thought of as having infinite codimension.

  1. _(a)_

first-order transversal derivatives of $\phi$ grow at least like
$\tau^{\frac{1}{2}}$ in time along a sequence of times $\tau_{n}\to\infty$
along the event horizon and the $H^{1}_{\rm space}$-norm (the non-degenerate
energy) is non-decaying in time, or

  2. _(b)_

$\phi\notin L^{2}_{\rm time}L^{2}_{\rm space}$, or

  3. _(c)_

$\partial\phi\notin L^{2}_{\rm time}L^{2}_{\rm space;loc}$ away from the event
horizon.

We refer to the instability (a) as an _azimuthal instability_ as it originates
from azimuthal modes $\phi_{m}$ with $m\neq 0$, which are solutions with an
azimuthal angular dependence $e^{im\varphi}$. In sub-extremal Kerr, or for
axisymmetric solutions on extremal Kerr, all the derivatives $\partial\phi$
can be uniformly bounded in time and the spacetime integrals of $|\phi|^{2}$
and $|\partial\phi|^{2}$ away from the horizon can be bounded via integrated
energy (Morawetz) estimates. Hence, none of the instabilities (a), (b) or (c)
occur in these settings.

The key ingredient for proving Theorem 1.1 is the following precise analysis
of the late-time behaviour of azimuthal modes $\phi_{m}$:

######  Theorem 1.2 (Late-time tails and stability for azimuthal modes on
extremal Kerr; rough version).

Let $|m|\geq 2$. Under a _qualitative_ global integrability assumption for
$\phi_{m}$, the $H^{1}_{\rm space}$-norm of $\phi_{m}$ is bounded and
$r\phi_{m}$ decays _quantitatively_ in time $\tau$ with the sharp rate
$\tau^{-\frac{1}{2}}$ along the future event horizon $\mathcal{H}^{+}$ and
$\tau^{-1}$ away from the event horizon.

More precisely, with respect to coordinates $(v,\theta,\widetilde{\varphi})$
along $\mathcal{H}^{+}$ that are adapted to the Killing vector field
generating the null hypersurface $\mathcal{H}^{+}$:

$\phi_{m}|_{\mathcal{H}^{+}}(v,\theta,\widetilde{\varphi})\sim
v^{-\frac{1}{2}}\sum_{\ell\>:\>\alpha_{m\ell}<0}\mathfrak{h}_{m\ell}\left(a_{m\ell}\cdot
e^{-i\left(\sqrt{-\alpha_{m\ell}}+m\right)\log v}+b_{m\ell}\cdot
e^{i\left(\sqrt{-\alpha_{m\ell}}-m\right)\log
v}\right)e^{im\widetilde{\varphi}}S_{m\ell}(\theta).$

With respect to Bondi coordinates $(u,\theta,\varphi)$ along future null
infinity,

$r\phi_{m}|_{\mathcal{I}^{+}}(u,\theta,\varphi)\sim
u^{-1}e^{-im\upomega_{+}u}\sum_{\ell\>:\>\alpha_{m\ell}<0}\mathfrak{h}_{m\ell}\left(c_{m\ell}\cdot
e^{-i\sqrt{-\alpha_{m\ell}}\log u}+d_{m\ell}\cdot
e^{i\sqrt{-\alpha_{m\ell}}\log u}\right)e^{im\varphi}S_{m\ell}(\theta),$

where

  * •

$\alpha_{m\ell}=\Lambda_{+,m\ell}-2m^{2}+\frac{1}{4}$,

  * •

$\Lambda_{+,m\ell}$ is the eigenvalue of a natural angular operator defined on
the 2-sphere $\mathcal{H}^{+}\cap\\{v=0\\}$, with eigenfunctions
$e^{im\widetilde{\varphi}}S_{m\ell}(\theta)$,

  * •

$\upomega_{+}$ is the angular velocity of the black hole,

  * •

$\mathfrak{h}_{m\ell}$ is an integral of the initial data, projected to
$e^{im\widetilde{\varphi}}S_{m\ell}(\theta)$, which is generically non-zero,

  * •

$a_{m\ell},b_{m\ell},c_{m\ell},d_{m\ell}\in\mathbb{C}$ are constants that are
independent of the initial data.

We now give several remarks regarding Theorems 1.1 and 1.2.

######  Remark 1.1.

The instabilities (b) and (c) in Theorem 1.1 can be excluded with the
assumption of appropriate, global, qualitative integrability assumptions for
$\phi$, as in the statement of Theorem 1.2. These assumptions can be be
verified if one extends the analysis in the present paper from exactly
extremal Kerr backgrounds to near-extremal Kerr backgrounds; see also §1.4.
This will be explored in future work.

######  Remark 1.2.

By combining the azimuthal instability (a) from Theorem 1.1 with the decay
results in Theorem 1.2, we can infer that the energy of azimuthal modes
$\phi_{m}$ with $|m|\geq 2$ (as measured by $H^{1}$-norms along a foliation of
spacelike constant-$\tau$ hypersurfaces) concentrates at the horizon as
$\tau\to\infty$, since it is non-decaying by Theorem 1.1, but decays when
restricted away from the horizon by Theorem 1.2.

######  Remark 1.3.

The azimuthal instability (a) differs from the axisymmetric instability
following from [Are15, Are12, Are13b] (see Theorem A in §1.2) and the
analogous extremal Reissner–Nordström instabilities from [Are11a, Are11b] in
the following ways:

  1. 1)

Axisymmetric solutions have bounded, but non-decaying first-order derivatives
along $\mathcal{H}^{+}$, as shown in [Are12] and, in view of the related
analysis in extremal Reissner–Nordström in [AAG20a], non-degenerate energies,
or $H^{1}_{\rm space}$ norms, are expected to decay in time.

  2. 2)

The axisymmetric instability mechanism relies on the existence of axisymmetric
conservation laws along $\mathcal{H}^{+}$. In that case, the coefficient
$H_{0}$ appearing in front of the terms that blow up along $\mathcal{H}^{+}$
is the value of the corresponding conserved charge (the “horizon hair”) and is
therefore defined at the horizon. These conserved charges along
$\mathcal{H}^{+}$ form a close analogue of the Newman–Penrose constants [NP68]
along $\mathcal{I}^{+}$. In contrast, the azimuthal instability in Theorem 1.1
does not follow from conservation laws, but instead follows from the precise
knowledge of the late-time tails of azimuthal modes along $\mathcal{H}^{+}$.

  3. 3)

In the non-axisymmetric setting, the role of $H_{0}$ is taken on by the
constants $\mathfrak{h}_{m\ell}$ appearing in Theorem 1.2, which are not
defined at the horizon but are integrals along the entire initial data
hypersurface. As a result, the azimuthal instability does not become weaker by
simply considering initial data supported away from $\mathcal{H}^{+}$, as is
the case in axisymmetry; see [Are13b]. Instead, it becomes weaker for data
which has vanishing constants $\mathfrak{h}_{m\ell}=0$.

######  Remark 1.4.

The constants $\mathfrak{h}_{m\ell}$, which are responsible for the azimuthal
instability (see Remark 1.3), can be observed in the late-time behaviour of
radiation fields $r\phi_{m}|_{\mathcal{I}^{+}}$, which also carries
information about the rotation of the black hole via the oscillating factor
$\frac{e^{-im\upomega_{+}u}(1+\log u\,O(u^{-1}))}{u}$. Since this late-time
behaviour differs significantly from the more rapid, non-oscillating inverse
polynomial decay in the sub-extremal setting [Hin22, AAG23], it provides an
_observational signature_ of extremality and the existence of azimuthal
instabilities.

For sub-extremal black holes that are close to extremality, the azimuthal
instability will instead become a “transient instability” by Cauchy stability,
i.e. quantities that blow up in extremal Kerr will instead become very large,
but they will eventually decay asymptotically in time. This transient
instability will leave an _imprint_ along $\mathcal{I}^{+}$ in the form of a
transient, oscillating late-time tail at late, but finite time intervals.

This observational signature may be compared with the setting of extremal
Reissner–Nordström, where (near)-extremality cannot immediately be inferred
from the late-time decay rate along a single far-away causal curve, but
requires additionally the consideration of integrals along the spheres
foliating $\mathcal{I}^{+}$ [AAG18a]. See also [BKS19, BKS21] for additional
numerical analysis on the observational signature in this setting.

###  1.2. Previous work

In this section, we give an overview of previous works relevant to Theorems
1.1 and 1.2.

##### Extremal black holes

A mathematical study of the dynamics of axisymmetric solutions to (1.2) was
pioneered in the works of Aretakis [Are12, Are15, Are13b], where the following
theorem was proved:

######  Theorem A (Axisymmetric instability on extremal Kerr; [Are12, Are15,
Are13b]).

Consider suitably regular and decaying initial data for (1.2). Then

  1. _(i)_

Generically, second-order derivatives of $\phi$ grow at least with the rate
$\tau$ as $\tau\to\infty$ along the future event horizon $\mathcal{H}^{+}$.

  2. _(ii)_

If the initial data are supported away from the future event horizon
$\mathcal{H}^{+}$, then the growth in _(i)_ occurs instead at the level of
third-order derivatives.

  3. _(iii)_

Axisymmetric solutions $\phi_{0}$ decay at least as fast as
$\tau^{-\frac{1}{2}}$ along $\mathcal{H}^{+}$ and future null infinity
$\mathcal{I}^{+}$, and at least as fast as $\tau^{-1}$ in bounded regions of
space away from $\mathcal{H}^{+}$, and their first-order derivatives are
bounded in time.

Theorem A demonstrates the existence of _axisymmetric instabilities_ for
higher-order transversal derivatives along $\mathcal{H}^{+}$. The underlying
mechanism for instabilities is based on the existence of conservation laws
along $\mathcal{H}^{+}$, together with the assumption of a _qualitative_ decay
statement for $\phi$ and its tangential derivatives along $\mathcal{H}^{+}$.
Under the global assumption of qualitative decay for $\phi$, the axisymmetric
instabilities have been shown to hold in more general settings [LMRT13,
Mur13]. See Remark 1.3 for a comparison of Theorem A with Theorem 1.1.

In view of the existence of a bijection between the behaviour of $\phi_{m=0}$
along $\mathcal{H}^{+}$ and $r\phi_{m=0}$ in $\mathcal{I}^{+}$ (valid only in
axisymmetry) [CT84, BF13], the conservation law of Aretakis along
$\mathcal{H}^{+}$ may be interpreted as a manifestation of the Newman–Penrose
conservation law along $\mathcal{I}^{+}$ that is present in general
asymptotically flat settings [NP68]. The relation between conservation laws
and characteristic initial data gluing problems for wave equations was
investigated further in [Are17] and has recently been analyzed in the context
of the Einstein equations in [ACR21a, ACR21c, ACR21b, CR22].

The first analysis of non-axisymmetric solutions to (1.2) was performed by
Teixeira da Costa in [TdC20], where it was shown that there are no
exponentially growing mode solutions on extremal Kerr, i.e. there is mode
stability on extremal Kerr.

A related problem is the study of the geometric wave equation on extremal
Reissner–Nordström black holes, which are charged but non-rotating analogues
of extremal Kerr black holes. Due to the lack of rotation, the analysis on
these spacetimes is greatly simplified and it mirrors the analysis of
axisymmetric solutions to (1.2). In particular, these spacetimes also feature
an instability mechanism based on conservation laws, and it was shown in
[Are11a, Are11b] that analogues of (i) and (ii) from Theorem A also hold in
this setting. Via a gluing analysis along the event horizon, spherically
symmetric black hole spacetimes have also been constructed that are isometric
to extremal Reissner–Nordström in the black hole exterior for sufficiently
late times, but arise from regular, geodesically complete initial data [KU22].

In the setting of extremal Reissner–Nordström, precise late-time decay
properties and the existence of late-time tails were obtained in [AAG20a]:

######  Theorem B (Late-time tails on extremal Reissner–Nordström; [AAG20a]).

Consider suitably regular and decaying initial data for the geometric wave
equation on extremal Reissner–Nordström spacetimes. Then the corresponding
solutions $\phi$ display the following late-time behaviour:

$\displaystyle\phi|_{\mathcal{H}^{+}}(v,\theta,\varphi)\sim$
$\displaystyle\>H_{0}v^{-1},$ $\displaystyle
r\phi|_{\mathcal{I}^{+}}(u,\theta,\varphi)\sim$
$\displaystyle\>(H_{0}+Q)u^{-2},$

with $H_{0}$ the value of the conserved quantity at $\mathcal{H}^{+}$ and $Q$
proportional to the time-inverted Newman–Penrose constant.

Theorem B demonstrates that despite the presence of asymptotic instabilities
along $\mathcal{H}^{+}$, precise late-time decay results can be obtained. In
particular, the late-time tails along $\mathcal{I}^{+}$ can be split into two
parts: a part related to the conservation law and instability along
$\mathcal{H}^{+}$, which comes with a constant $H_{0}$, and a part which is
also present in the sub-extremal setting and comes with a constant $Q$ that
may be interpreted as the Newman–Penrose conserved quantity at
$\mathcal{I}^{+}$ associated to the _integral in time_ of the spherically
symmetric part of $\phi$. Unlike $H_{0}$, the constant $Q$ can be expressed as
an integral along the entire initial data hypersurface.

Due to the strong similarity of the analysis of axisymmetric solutions to
(1.2) with general solutions to the geometric wave equation on extremal
Reissner–Nordström, it is expected that the results of Theorem B can be
extended to axisymmetric solutions to (1.2). The rates appearing in Theorem B
may be contrasted with the more intricate and slower decay in Theorem 1.2 and
the absence of the time-inverted Newman–Penrose constant $Q$ in the leading-
order asymptotics.

Recently, the analogue of the instability in (i) of Theorem A was also shown
to hold in the context of the linearized Einstein–Maxwell equations on
extremal Reissner–Nördstrom [Ape22]. In the nonlinear setting, the techniques
developed in the proof of Theorem B have been applied to show that the
instability remains asymptotic in the context of a nonlinear problem that
models the nonlinear structure present in the Einstein equations
[AAG20b].555The asymptotic instability can also lead to a finite-in-time
instability for nonlinear equations that have a different nonlinear structure;
see [Are13a].

The implications of the behaviour in Theorem B to the singularity properties
in the black hole interior of extremal Reissner–Nordström and axisymmetric
solutions to (1.2) was studied in [Gaj17a, Gaj17b, GL19]. In particular, it
was shown that the Cauchy horizon in this setting is _more stable_ than in the
sub-extremal setting and the regularity of solutions at the Cauchy horizon may
depend rather delicately on the leading and next-to-leading order late-time
tails along $\mathcal{H}^{+}$.

##### Sub-extremal black holes

The mathematical analysis of the geometric wave equation on sub-extremal black
holes has a rich history. Uniform boundedness and decay estimates for
solutions to the wave equation on the full sub-extremal range of Kerr
parameters were obtained by Dafermos–Rodnianski–Shlapentokh-Rothman in
[DRSR16]; see also the references therein for a comprehensive list of
intermediate and related results.

In [DRSR16], the following key mechanisms were exploited to establish weak,
integrated energy estimates or _Morawetz estimates_ in sub-extremal Kerr: 1)
the red-shift effect (first discussed as a decay mechanism in [DR09]), 2) the
decoupling of _null geodesic trapping_ and _superradiance_ in Fourier space,
and 3) quantitative mode stability, first established qualitatively in [Whi89]
and then quantitatively in [SR15].

Passing from integrated energy energy estimates to inverse-polynomial energy
decay in time involves an application of the $r$-weighted estimates of
Dafermos–Rodnianski [DR10], which are not restricted to Kerr spacetimes, but
hold on general asymptotically flat spacetimes, as shown in [Mos16]. The
analysis in [DRSR16] has recently also been extended to the setting of the
Teukolsky equations of linearized gravity [SRTdC20, SRTdC23].

The step from uniform upper bound decay estimates to precise leading-order
decay estimates and late-time tails (analogous to Theorems 1.2 and B) was
first made in [AAG18c, AAG18b] in the context of spherically symmetric, sub-
extremal black hole spacetimes and then extended to sub-extremal Kerr in
[AAG23], where it was moreover explored how the late-time tails corresponding
to different angular modes interact; see also [Hin22] for a different approach
that involves precise resolvent estimates near the zero time-frequency.

A useful subset of the sub-extremal Kerr family to consider first are the
slowly rotating Kerr spacetimes, which have a rotation parameter $a$ that
satisfies $|a|\ll M$. These include the Schwarzschild spacetimes for which
$a=0$. Due to the smallness of the parameter $a$, the issue of superradiance
and mode stability, which are absent when $a=0$, are easier to handle.
Significant advances have been made in this setting regarding decay and
stability at the level of the nonlinear Einstein equations. The nonlinear
stability of Schwarzschild was proved for polarized axisymmetric initial data
in [KS20]. Data without symmetry were considered in [DHRT21] and the full,
codimension 3 subset of initial data perturbations were identified for which
stability holds. More recently, a proof of nonlinear stability of slowly
rotating Kerr without symmetry restrictions was completed in a series of works
[KS21, KS22a, GKS22, KS22b, She22].

##### Inverse-square potentials

A setting that roughly models the situation “between” sub-extremal Kerr and
extremal Kerr are wave equations with negative inverse-square potentials on
Schwarzschild spacetimes. Here, future null infinity $\mathcal{I}^{+}$ plays
the role of the event horizon $\mathcal{H}^{+}$ in extremal Kerr. The absence
of conservation laws for non-axisymmetric solutions along $\mathcal{H}^{+}$ in
extremal Kerr is reflected in the breakdown of the Newman–Penrose conservation
law at $\mathcal{I}^{+}$. The proofs for deriving the precise late-time
behaviour of solutions to the wave equation without potential (see [AAG23] and
the earlier works [AAG18c, AAG18b]) rely crucially on the existence of this
conservation law, and they therefore break down. In [Gaj22], a new, more
robust, method was developed for dealing with the absence of conservation laws
and deriving late-time tails which depend on the coefficient in the leading-
order term of the inverse square potential. See also upcoming work on
establishing these tails in a different way for wave equations with inverse-
square potentials and electromagnetically charged wave equations on Minkowski
[GVdM23] and see [BGRM22] for recent work on late-time tails for wave
equations with inverse-square potentials on Minkowski.

##### Numerics and heuristics

The mathematical analysis described in the above paragraphs was preceded by a
large number of works with numerical and heuristic analysis in the physics
literature. In particular, the pointwise blow-up of transversal derivations in
the azimuthal instability of Theorem 1.1 was predicted from a heuristic
analysis of the Teukolsky equation in frequency space near the frequency
$\omega=m\upomega_{+}$ by Casals–Gralla–Zimmerman in [CGZ16], who moreover
predicted the decay rates of late-time tails along $\mathcal{H}^{+}$
consistent with Theorem 1.2. Growth along $\mathcal{H}^{+}$ with consistent
rates was also observed numerically in [HBB13]. See also the related
heuristics in [HR17] on instabilities in the near-horizon spacetimes
corresponding to extremal Kerr.

The late-time behaviour in Theorem 1.2 was first predicted (in bounded regions
in space, away from $\mathcal{H}^{+}$) by Andersson–Glampedakis [AG00] via a
heuristic analysis of slowly decaying _quasinormal modes_ (“zero-damped
modes”; see for example [Hod08] for a discussion) on near-extremal Kerr black
holes with complex time-frequencies that are expected to accumulate to the
real frequency $\omega=m\upomega_{+}$. While each quasinormal mode decays
exponentially in time, their decay rate goes to zero in the extremal limit.
The analysis in [AG00] suggested that the _sum_ over these modes should give
rise to an inverse-polynomial tail with decay rate $t^{-1}$ and an overall
oscillating factor $e^{-im\upomega_{+}t}$.

The analysis of [AG00] was later revisited, heuristically and numerically, in
[YZZ+13]. The numerics in [YZZ+13] predicted moreover that the slowly-decaying
late-time tail could constitute a dominant part of $\phi$ at relatively
_early_ times, for initial data that are relevant when modelling black hole
mergers. This stands in contrast with the expectation that the late-time tails
in sub-extremal Kerr become dominant only at relatively _late_ times. This
strengthens further the case for slowly decaying late-time tails in extremal
Kerr serving as a promising observational signature in the gravitational
radiation emitted by astrophysical systems settling down to near-extremal Kerr
black holes.

Note that _nonlinear_ aspects of the dynamics of extremal Reissner–Nordström
black holes were probed numerically in [MRT13].

###  1.3. The main difficulties compared to the axisymmetric and sub-extremal
settings

In this section, we give an overview of the key difficulties enountered when
proving Theorems 1.1 and 1.2. We will revisit these difficulties and outline
how they are resolved in the present paper in §5.

##### Integrated energy estimates: absence of (degenerate) red-shift

A key ingredient towards establishing energy decay estimate on sub-extremal
black holes is the red-shift estimate, which, schematically, gives the
following integrated energy estimate for solutions $\phi$ to the wave
equation:

$\kappa\int_{0}^{T}\left[\int_{\Sigma_{\tau}\cap\\{r_{+}\leq r\leq
r_{+}+\epsilon\\}}|\phi|^{2}+|\partial\phi|^{2}d\mu_{\Sigma_{\tau}}\right]\,d\tau\lesssim\int_{\Sigma_{0}\cap\\{r_{+}\leq
r\leq r_{+}+2\epsilon\\}}|\phi|^{2}+|\partial\phi|^{2}d\mu_{\Sigma_{\tau}}\\\
+\int_{0}^{T}\left[\int_{\Sigma_{\tau}\cap\\{r_{+}+\epsilon\leq r\leq
r_{+}+2\epsilon\\}}|\phi|^{2}+|\partial\phi|^{2}d\mu_{\Sigma_{\tau}}\right]\,d\tau.$

Here $r_{+}$ is the radius of the event horizon $\mathcal{H}^{+}$,
$\epsilon\ll r_{+}$ and $d\mu_{\Sigma_{\tau}}$ is the induced volume form on
the spacelike hypersurface or time slice $\Sigma_{\tau}$. The constant
$\kappa>0$ denotes the surface gravity associated to the future event horizon
$\mathcal{H}^{+}$, which appears as a proportionality constant in the
following equation:

$\nabla_{K}K=\kappa K,$

where $K$ is a Killing vector field which is a null generator of the event
horizon. The above integrated estimate allows one to extend integrated energy
estimates valid in $\\{r>r_{+}+\epsilon\\}$ up to $\\{r\geq r_{+}\\}$.

In extremal Kerr, $\kappa=0$, so the red-shift estimate breaks down.
Nevertheless, degenerate versions of the red-shift estimate remain valid for
axisymmetric solutions to (1.2). For example,

$\kappa\int_{0}^{T}\left[\int_{\Sigma_{\tau}\cap\\{r_{+}\leq r\leq
r_{+}+\epsilon\\}}|\phi|^{2}+(r-r_{+})|\partial\phi|^{2}d\mu_{\Sigma_{\tau}}\right]\,d\tau\lesssim\int_{\Sigma_{0}\cap\\{r_{+}\leq
r\leq r_{+}+2\epsilon\\}}|\phi|^{2}+|\partial\phi|^{2}d\mu_{\Sigma_{\tau}}\\\
+\int_{0}^{T}\left[\int_{\Sigma_{\tau}\cap\\{r_{+}+\epsilon\leq r\leq
r_{+}+2\epsilon\\}}|\phi|^{2}+|\partial\phi|^{2}d\mu_{\Sigma_{\tau}}\right]\,d\tau.$

For non-axisymmetric solutions, even the degenerate versions of the red-shift
estimate break down. Furthermore, while in the case of axisymmetric solutions
$\phi_{0}$, the energy current corresponding to $K$ is non-negative near
$\mathcal{H}^{+}$, i.e.

$\mathbb{T}[\phi_{0}](K,\mathbf{n}_{\Sigma_{\tau}})\geq 0,$

with $\mathbb{T}$ the energy momentum tensor and $\mathbf{n}_{\Sigma_{\tau}}$
the normal to the spacelike hypersurfaces $\Sigma_{\tau}$, the non-negativity
property breaks down for non-axisymmetric solutions, since $K$ is not causal
in any open neighbourhood of $\mathcal{H}^{+}$.

##### Integrated energy estimates: superradiance

Let $T$ denote the Killing vector field that is timelike and normalized at
$r=\infty$. In all rotating ($a\neq 0$) spacetimes, $T$ fails to be causal in
the _ergoregion_

$r<M\left(1+\sqrt{1+a^{2}\cos\theta}\right),$

with $r$ and $\theta$ Boyer–Lindquist coordinates. The existence of an
ergoregion leads to _superradiance_ , which is an energy amplification effect
that may be interpreted as “energy extracted from the black hole”. At the
level of the Fourier transform of azimuthal modes $\phi_{m}$ along integral
curves of $T$, superradiance affects the following frequency range:

$0<m\omega<m^{2}\upomega_{+}.$

Note in particular that for axisymmetric $\phi$, $m=0$, so the above frequency
range is empty and superradiance is not present. For bounded $m$, we can
divide solutions into those with _high_ angular frequencies $\Lambda$ and
those with bounded _angular_ frequencies $\Lambda\lesssim m^{2}$ (see §6 for a
precise definition of the angular frequency $\Lambda$). In both cases, the
analysis from the sub-extremal setting _breaks down_ and new techniques are
required to obtain the desired estimates in Fourier space.

##### Instabilities: absence of conservation laws

The instability mechanism underlying the proof of (i) of Theorem A relies on
the existence of conservation laws along $\mathcal{H}^{+}$ for axisymmetric
$\phi$ in extremal Kerr. For this reason, asymptotic instabilities for
transversal derivatives $\partial^{2}\phi_{0}$ along $\mathcal{H}^{+}$ follow
immediately from the conservation of an appropriate linear combination of
$\phi$ and $\partial\phi$, together with a _qualitative_ decay statement for
$\phi$ and its tangential derivatives along $\mathcal{H}^{+}$.

This stands in stark contrast with non-axisymmetric solutions, for which there
is no conservation law available along $\mathcal{H}^{+}$. Deriving an
instability in this setting requires instead sharp, _quantitative_ decay
statements for $\phi$ and its tangential derivatives along $\mathcal{H}^{+}$.
That is to say, one needs to derive the precise late-time tails for $\phi$
before deriving the existence of instabilities.

##### Late-time tails and time integrals

Once weak, integrated estimates have been established for $\phi$, one needs a
decay mechanism to derive sharp decay estimates and to derive the precise
leading-order behaviour (late-time tails) of $\phi$. In the sub-extremal
setting, such a mechanism was developed in [AAG18c, AAG18b, AAG23] and it
revolved around _time integrals of $\phi$_, which are integrals of $\phi$
along the integral curves of the vector field $T$, together with appropriate
hierarchies of $r$-weighted energy estimates far away from the black hole,
first derived in [DR10], and an exploitation of conservation laws along future
null infinity $\mathcal{I}^{+}$.

In the case of axisymmetric $\phi$ and, analogously, solutions to the wave
equation on extremal Reissner–Nordström, a similar mechanism can be exploited,
which depends also on the existence of hierarchies of degenerate energy
estimates near $\mathcal{H}^{+}$ and conservation laws along
$\mathcal{H}^{+}$.

In view of the aforementioned breakdown of the degenerate red-shift estimates
and the absence of conservation laws for non-axisymmetric $\phi$, the sub-
extremal decay mechanism breaks down. Furthermore, whereas there is an
unambiguous notion of “time integral” for axisymmetric $\phi$ in extremal
Kerr, associated to the vector field $T$, which agrees with $K$ when acting on
axisymmetric $\phi$, there are multiple possible notions of time integrals in
the non-axisymmetric, extremal setting.

###  1.4. Remaining questions

We list here some remaining questions and problems, which are not addressed in
the present paper, but will be explored in future investigations.

  * •

_Uniform frequency-space analysis on near-extremal Kerr._ Extending the
frequency-space analysis in the present paper to $|a|\leq M$ would allow one
to prove the qualitative integrability assumptions of Theorem 1.2 and to rule
out instabilities (b) and (c) in Theorem 1.1. While most of the frequency-
space analysis in the present paper includes near-extremal spacetimes, it
remains to revisit the estimates in the bounded, frequency regime, with
$|\omega-m\upomega_{+}|\ll 1$ and extend them to the $|a|\leq M$ setting.

  * •

_Uniform estimates in $m$._ The analysis in the present paper is concerned
with fixed azimuthal modes, or bounded $m$. This means that we do not keep
careful track of the $m$-dependence in all the estimates. Extending the
results of the present paper uniformly in $m$ would require revisiting the
analysis in the frequency regime $|\omega-m\upomega_{+}|\ll|m|$, which is then
no longer a bounded frequency regime. In view of the existence of trapped null
geodesics arbitrarily close to $\mathcal{H}^{+}$, one would expect any
extension of the integrated energy estimates to unbounded $m$ to come with an
additional loss of azimuthal $\partial_{\varphi}$ derivatives. As this
frequency regime is both superradiant and is affected by trapping, it could
also lead to more severe deviations from the bounded-$m$ setting and perhaps
even to stronger instabilities that are not visible for bounded $m$.

  * •

_Nonlinear analysis._ In order to address the effects on nonlinearities in the
Einstein equations to the presence of azimuthal instabilities in the
linearized setting, one could consider model nonlinear wave equations of the
form:

$\square_{g}\phi=A(\phi)\cdot g^{-1}(d\phi,d\phi),$

which satisfy the _null condition_ and are therefore consistent with the
nonlinear structure present in the Einstein equations. Even when considering
initial data supported on a fixed azimuthal mode $\phi_{m}$, all the other
modes are excited due to the presence of the nonlinearity and one needs to
first understand uniformly what happens to the estimates in the present paper
when $m\to\infty$.

The additional difficulties faced when considering $m\to\infty$
notwithstanding, we can make the optimistic supposition for a moment that the
growth and decay estimates in Theorems 1.1 and 1.2 remain valid uniformly in
$m$.

Since the nonlinearity is of the form $g^{-1}(d\phi,d\phi)$, growing
transversal derivatives along the event horizon along $\mathcal{H}^{+}$ are
not immediately fatal as will be multiplied by decaying tangential
derivatives, so it is _in principle_ possible to obtain global existence and
uniqueness of solutions and show that the behaviour of the linear problem
persists. Indeed, the above null structure of the nonlinearity was used in
[AAG20b] to show that the instability in extremal Reissner–Nordström remains
asymptotic, just like in the linear problem.

Given the slower decay and stronger azimuthal instabilities in the present
paper, the methods of [AAG20b] break down and new methods need to be developed
to perform an analysis of nonlinear wave equations satisfying the null
condition. The expected loss of derivatives in integrated energy estimates
when considering the limit $m\to\infty$ and the resulting loss of derivatives
in (degenerate) energy boundedness estimates would form additional
difficulties for the nonlinear problem.

###  1.5. Outline of the remainder of the paper

We provide here an outline of the structure of the remainder of the paper.

  * •

In §2 and §3 we introduce the necessary geometric preliminaries and basic
facts regarding linear wave equations on Kerr spacetimes, as well as the main
notation in the rest of the paper. We moreover derive some properties of the
coefficients $\alpha_{\ell m}$ from Theorem 1.2 and discuss stationary, or
$K$-invariant solutions to the wave equation on extremal Kerr.

  * •

Then, we state precise versions of Theorems 1.1 and 1.2 in §4 and sketch how
they are proved in §5.

  * •

The frequency-space analysis of the paper is fully contained in §6, resulting
in integrated energy estimates in physical space, which are derived in §7.

  * •

The key hierarchies of $r$-weighted (in far-away regions) and
$(r-r_{+})$-weighted (in near-horizon regions) energy estimates are derived in
§8. Together with the integrated energy estimates in §7, these form the key
ingredient for deriving energy decay estimates for $K$-derivatives in §9.

  * •

In §10, we construct time integrals of $\phi_{m}$ along integral curves of the
vector field $K$ and we derive the necessary elliptic-type estimates. Then we
combine these with the energy decay estimates from §9 in §11 to obtain decay
(energy and pointwise) estimates for $\phi_{m}$.

  * •

Finally, we use the precise energy and pointwise decay estimates from §11 to
derive azimuthal instabilities in §12.

###  1.6. Acknowledgements

The author would like to thank Rita Teixeira da Costa and Yakov Shlapentokh-
Rothman for useful discussions.

##  2\. Preliminaries: geometry and foliations

In this section, we introduce the geometric properties that are required for
the analysis in the paper.

###  2.1. The spacetime metric

In this section, we introduce the 2-parameter family of _Kerr spacetimes_
$(\mathcal{M}_{M,a},g_{M,a})$ and the corresponding geometric notation.

Let $M>0$ and $a\in\mathbb{R}$ with $|a|\leq M$. Then $\mathcal{M}_{M,a}$ is
the following manifold-with-boundary:

$\displaystyle\mathcal{M}_{M,a}=$
$\displaystyle\>\mathbb{R}\times[r_{+},\infty)\times\mathbb{S}^{2},$
$\displaystyle r_{+}=$ $\displaystyle\>M+\sqrt{M^{2}-a^{2}}.$

The constant $M$ is the _mass_ of the Kerr black hole and $a\cdot M$ can be
considered its _specific angular momentum_ , or its angular momentum/mass.

We can cover $\mathcal{M}_{M,a}$ with coordinates $(v,r,\theta,\varphi_{*})$,
where $v\in\mathbb{R}$, $r\in[r_{+},\infty)$ and $(\theta,\varphi_{*})$ denote
standard spherical coordinates on the unit round 2-sphere $\mathbb{S}^{2}$
(which degenerate at a half circle connecting the north and south pole). In
these coordinates, the Lorentzian metric $g_{M,a}$ takes the following form:

$g_{M,a}=-\rho^{-2}(\Delta-a^{2}\sin^{2}\theta)dv^{2}+2dvdr-4Mar\rho^{-2}\sin^{2}\theta
dvd\varphi_{*}-2a\sin^{2}\theta drd\varphi_{*}\\\
+\rho^{2}d\theta^{2}+\rho^{-2}((r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta)\sin^{2}\theta
d\varphi_{*}^{2},$ (2.1)

with

$\displaystyle\rho^{2}=$ $\displaystyle\>r^{2}+a^{2}\cos^{2}\theta,$
$\displaystyle\Delta=$ $\displaystyle\>r^{2}-2Mr+a^{2}=(r-r_{+})(r-r_{-}),$
$\displaystyle r_{-}=$ $\displaystyle\>M-\sqrt{M^{2}-a^{2}}.$

The above local expression for $g_{M,a}$ is called the _ingoing
Eddington–Finkelstein_ form of the Kerr metric.

The 1-parameter subfamily $g_{|a|,a}$ of Kerr spacetimes for which $M=|a|$ are
known as the _extremal Kerr spacetimes_.

The inverse metric $g^{-1}_{M,a}$ corresponding to $g_{M,a}$ then takes the
following form:

$g^{-1}_{M,a}=a^{2}\rho^{-2}\sin\theta^{2}\theta\partial_{v}\otimes\partial_{v}+\rho^{-2}(r^{2}+a^{2})[\partial_{v}\otimes\partial_{r}+\partial_{r}\otimes\partial_{v}]+\rho^{-2}\Delta\partial_{r}\otimes\partial_{r}\\\
+a\rho^{-2}[(\partial_{v}+\partial_{r})\otimes\partial_{\varphi_{*}}+\partial_{\varphi_{*}}\otimes(\partial_{v}+\partial_{r})]+\rho^{-2}[\partial_{\theta}\otimes\partial_{\theta}+\sin^{-2}\theta\partial_{\varphi_{*}}\otimes\partial_{\varphi_{*}}].$
(2.2)

We introduce the following notation for submanifolds of $\mathcal{M}_{M,a}$:

$\displaystyle\mathcal{H}^{+}:=$
$\displaystyle\>\\{r=r_{+}\\}\quad\textnormal{is called the (future) event
horizon},$ $\displaystyle\mathring{\mathcal{M}_{M,a}}:=$
$\displaystyle\>\mathcal{M}_{M,a}\setminus\mathcal{H}^{+},$ $\displaystyle
S^{2}_{v^{\prime},r^{\prime}}:=$
$\displaystyle\>\\{r=r^{\prime}\\}\cap\\{v=v^{\prime}\\}\>\>\textnormal{with
$v^{\prime}\in\mathbb{R}$ and $r^{\prime}\in[r_{+},\infty)$ are diffeomorphic
to $\mathbb{S}^{2}$}.$

We will moreover consider the following functions:

$\displaystyle r_{*}=$ $\displaystyle\>\int_{3M}^{r}\frac{r^{\prime
2}+a^{2}}{\Delta}\,dr^{\prime}\quad\textnormal{on
$\mathring{\mathcal{M}}_{M,a}$},$ $\displaystyle t=$
$\displaystyle\>v-r_{*}\quad\textnormal{on $\mathring{\mathcal{M}}_{M,a}$},$
$\displaystyle u=$ $\displaystyle\>v-2r_{*}=t-r_{*}\quad\textnormal{on
$\mathring{\mathcal{M}}_{M,a}$},$ $\displaystyle\varphi=$
$\displaystyle\>\varphi_{*}+\int_{r}^{\infty}\frac{a}{\Delta}\,dr^{\prime}\>\>\textnormal{mod}\,2\pi\quad\textnormal{on
$\mathring{\mathcal{M}}_{M,a}$},$ $\displaystyle\widetilde{\varphi}=$
$\displaystyle\>\varphi_{*}-\upomega_{+}v\>\>\textnormal{mod}\,2\pi\quad\textnormal{on
$\mathcal{M}_{M,a}$},$

where $\upomega_{+}=\frac{a}{r_{+}^{2}+a^{2}}$ can be interpreted as the
“angular velocity of the Kerr black hole”, assigned by stationary observers at
infinity. The coordinates $(v,r,\theta,\widetilde{\varphi})$ may be
interpreted as “rotating with the Kerr black hole”, because the integral
curves of the null generator of $\mathcal{H}^{+}$ correspond to the level sets
$\\{(v,r,\theta,\widetilde{\varphi})=(v,r_{+},\theta_{0},\widetilde{\varphi}_{0})\\}$,
with $(\theta_{0},\widetilde{\varphi}_{0})$ constant.

The functions $(t,r,\theta,\varphi)$ define coordinates on the manifold
$\mathring{\mathcal{M}}_{M,a}$ and are called _Boyer–Lindquist coordinates_.

To simplify the notation, we will frequently omit one parameter in the
2-parameter family of spacetimes by fixing $r_{+}=1$, which implies that:

$\displaystyle r_{-}=$ $\displaystyle\>a^{2},$ $\displaystyle M=$
$\displaystyle\>\frac{1}{2}(1+a^{2}).$

Then extremal Kerr spacetimes correspond simply to the choices $a=\pm 1$.

###  2.2. Foliations and conformal coordinates

In this section, we introduce a foliation of $\mathcal{M}_{M,a}$ by
appropriate isometric spacelike hypersurfaces, i.e. a “time-slicing”.

Let $\mathbbm{h}:[r_{+},\infty)\to\mathbb{R}_{+}$ be a smooth function, such
that for all $r\in[r_{+},\infty)$:

$\displaystyle\frac{2(r^{2}+a^{2})}{\Delta}-\mathbbm{h}(r)=$
$\displaystyle\>O_{\infty}(r^{-2}),$
$\displaystyle\mathbbm{h}(r)\Delta\left(\frac{2(r^{2}+a^{2})}{\Delta}-\mathbbm{h}(r)\right)>$
$\displaystyle\>a^{2}\sin^{2}\theta,$

Then we define the _time function_ $\tau$ on $\mathcal{M}_{M,a}$ as follows:

$\tau=v-\int_{r_{+}}^{r}\mathbbm{h}(r^{\prime})\,dr^{\prime}-v_{0},$

with $v_{0}\in\mathbb{R}_{+}$. Define

$\Sigma_{\tau^{\prime}}=\\{\tau=\tau^{\prime}\\}.$

The analysis in the present paper, will be restricted to the spacetime subset:

$\mathcal{R}:=\\{\tau\geq 0\\}.$

We also denote

$\Sigma:=\Sigma_{0}.$

See Figure 1 below for a picture.

Figure 1. A 2-dimensional representation of the region under consideration in
the spacetime $\mathcal{M}_{M,a}$, with the hypersurfaces $\Sigma_{\tau}$ and
the shaded region depicting $\mathcal{R}$. Each point in the picture
represents a 2-sphere, composed of all points with $(v,r)$ constant.

###  2.3. Key vector fields

Kerr spacetimes are stationary and axisymmetric, which is reflected in the
$v$\- and $\varphi_{*}$-independence of the metric components in (2.1). The
following vector fields are therefore Killing vector fields:

$\displaystyle T:=$ $\displaystyle\>\partial_{v},$ $\displaystyle\Phi:=$
$\displaystyle\>\partial_{\varphi_{*}},$

with respect to the coordinates $(v,r,\theta,\varphi_{*})$.

The linear combination:

$K:=T+\upomega_{+}\Phi$

plays a particularly important role, as it is a null generator of
$\mathcal{H}^{+}$. Note that $K=\partial_{v}$ in
$(v,r,\theta,\widetilde{\varphi})$ coordinates.

We moreover introduce:

$\displaystyle Y=$ $\displaystyle\>\partial_{r},$
$\displaystyle\underline{L}=$ $\displaystyle-\frac{\Delta}{2(r^{2}+a^{2})}Y,$
$\displaystyle L=$ $\displaystyle\>T+\frac{a}{r^{2}+a^{2}}\Phi-\underline{L},$

Note that $L(v)=-\underline{L}(u)=1$. It is straightforward to verify the
following causality properties for $r_{+}=1$:

$\displaystyle g(T,T)<$
$\displaystyle\>0\quad\textnormal{in}\quad\\{\Delta-a^{2}\sin^{2}\theta>0\\}\subset\\{r<2M\\},$
$\displaystyle g(L,L)=$ $\displaystyle\>g(\underline{L},\underline{L})=0,$
$\displaystyle g(K,K)=$
$\displaystyle\>-\frac{(2+a^{2}+a^{2}\cos(2\theta))^{2}}{4(1+a^{2})^{2}}(1-a^{2})(r-1)+\left(-1+\frac{a^{2}\sin^{2}\theta(6+2a^{2}-a^{2}\sin^{2}\theta)}{1+a^{2}}\right)(r-1)^{2}$
$\displaystyle+O((r-1)^{3})\quad\textnormal{when fixing $r_{+}=1$}.$

Note therefore that $K$ is null along $\mathcal{H}^{+}$ ($r=1$) and, in sub-
extremal Kerr spacetimes, it extends as a timelike vector field in a
neighbourhood of the form $\\{r-r_{+}<\delta_{a}\\}$, with $\delta_{a}>0$
suitably small (depending on $a$). In extremal Kerr spacetimes, on the other
hand, $K$ is not uniformly timelike or spacelike in a region of the form
$\\{r-r_{+}\leq\delta\\}$, no matter how small we take $\delta>0$.

We denote further:

$\displaystyle X:=$ $\displaystyle\>Y+\mathbbm{h}T,$
$\displaystyle\widetilde{X};=$
$\displaystyle\>Y+\mathbbm{h}K=X+\upomega_{+}\mathbbm{h}\Phi.$

The vector fields $X$ and $\widetilde{X}$ are tangential to $\Sigma_{\tau}$.
Note that $X=\partial_{r}$ in $(\tau,r,\theta,\varphi_{*})$ coordinates and
$\widetilde{X}=\partial_{r}$ in $(\tau,r,\theta,\widetilde{\varphi})$
coordinates.

Finally, we denote

${\mathfrak{X}}_{*}=\frac{\Delta}{r^{2}+a^{2}}Y+T+\frac{a}{r^{2}+a^{2}}\Phi.$

We can express ${\mathfrak{X}}_{*}=\partial_{r_{*}}$ in
$(t,r_{*},\theta,\varphi)$ coordinates.

We moreover introduce the Carter operator

$Q:=\not{\Delta}_{\mathbb{S}^{2}}-\Phi^{2}+\sin^{2}\theta T^{2}.$

From standard elliptic estimates on $\mathbb{S}^{2}$ we have the following
inequality for $f\in H^{2}(\mathbb{S}^{2})$:

$\int_{\mathbb{S}^{2}}|f|^{2}+|\not{\nabla}_{\mathbb{S}^{2}}f|^{2}+|\not{\nabla}_{\mathbb{S}^{2}}^{2}f|^{2}\,d\sigma\leq
C\int_{\mathbb{S}^{2}}|Qf|^{2}+|\Phi^{2}f|^{2}+|K\Phi
f|^{2}+|K^{2}f|^{2}\,d\sigma.$

###  2.4. Differential operators on $\mathbb{S}^{2}$

We introduce the following second-order differential operator on the unit
round 1-sphere $\mathbb{S}^{2}$:

$\not{\mathcal{D}}(\cdot):=\not{\Delta}_{\mathbb{S}^{2}}(\cdot)+a^{2}\upomega_{+}^{2}\sin^{2}\theta\Phi^{2}(\cdot),$
(2.3)

where $\not{\Delta}_{\mathbb{S}^{2}}$ is the Laplacian on the unit 1-sphere.

The operator $\not{\mathcal{D}}$ appears in the purely angular part of the
wave operator $\square_{g_{M,a}}$. This follows by writing down
$\square_{g}(\cdot)$ in terms of the vector fields $(K,Y,\Phi)$; see (iv) of
Lemma 3.1.

We will now relate $\not{\mathcal{D}}$ to the differential operator
corresponding to _oblate spheroidal harmonics_.

Let $\nu\in\mathbb{R}$ and consider the following operator on
$L^{2}(\mathbb{S}^{2})$:

$P(\nu)=-\frac{1}{\sin\theta}\partial_{\theta}(\sin\theta\partial_{\theta}(\cdot))-\frac{1}{\sin^{2}\theta}\partial_{\varphi}^{2}(\cdot)-\nu^{2}\cos^{2}\theta(\cdot).$

Note that $P(\nu)$ is self-adjoint, so there exists an orthonormal basis of
eigenfunctions. Since $[P(\nu),\partial_{\varphi}]=0$, we can decompose these
eigenfunctions by employing eigenfunctions
$\\{e^{im\varphi}\\}_{m\in\mathbb{Z}}$ of $\partial_{\varphi}$:

$S_{m\ell}(\theta;\nu)e^{im\varphi},$

with $\ell\in\mathbb{N}_{0}$ and $m\in\mathbb{Z}$ labels, such that $-\ell\leq
m\leq\ell$. We denote the corresponding eigenvalues of $P(\nu)$ by
$\lambda_{m\ell}^{(\nu)}$, with
$\lambda_{m\ell}^{(\nu)}<\lambda_{m(\ell+1)}^{(\nu)}$, and note that they
satisfy the inequalities:

$\displaystyle\lambda_{m\ell}^{(\nu)}+\nu^{2}\geq$
$\displaystyle\>|m|(|m|+1),$ (2.4)
$\displaystyle\lambda_{m\ell}^{(\nu)}+\nu^{2}\geq$ $\displaystyle\>2|m\nu|.$
(2.5)

We will moreover use the following shorthand notation: let $|a|\leq M$, then

$\displaystyle P:=$
$\displaystyle\>P\left(a\upomega_{+}\right)=-\frac{1}{\sin\theta}\partial_{\theta}(\sin\theta\partial_{\theta}(\cdot))-\frac{1}{\sin^{2}\theta}\partial_{\varphi}^{2}(\cdot)-a^{2}\upomega_{+}^{2}m^{2}\cos^{2}\theta(\cdot),$
$\displaystyle S_{m\ell}(\theta):=$
$\displaystyle\>S_{m\ell}\left(\theta;am\upomega_{+}\right),$
$\displaystyle\lambda_{m\ell}:=$
$\displaystyle\>\lambda_{m\ell}^{\left(a\upomega_{+}\right)}.$

We can relate $P$ as follows to $\not{\mathcal{D}}$

$-\not{\mathcal{D}}(\cdot)=P(\cdot)-a^{2}\upomega_{+}^{2}\Phi^{2},$

so by (2.4), the following inequality holds:

$-\int_{\mathbb{S}^{2}}\Re(\not{\mathcal{D}}(S_{m\ell}e^{im\widetilde{\varphi}})\overline{S}_{m\ell}e^{-im\widetilde{\varphi}})\,d\sigma\geq|m|(|m|+1)\int_{\mathbb{S}^{2}}|S_{m\ell}|^{2}\,d\sigma.$

We moreover have that for all $\theta\in(0,\pi)$

$\frac{1}{\sin^{2}\theta}+a^{2}\upomega_{+}^{2}\sin^{2}\theta\geq
1+a^{2}\upomega_{+}^{2},$

so we also have that

$\begin{split}-\int_{\mathbb{S}^{2}}\Re(\not{\mathcal{D}}S_{m\ell}e^{im\widetilde{\varphi}}\overline{S}_{m\ell}e^{-im\widetilde{\varphi}})\,d\sigma=&\>\int_{\mathbb{S}^{2}}|\partial_{\theta}S_{m\ell}|^{2}+\left(\frac{1}{\sin^{2}\theta}+a^{2}\upomega_{+}^{2}\sin^{2}\theta\right)m^{2}|S_{m\ell}|^{2}\,d\sigma\\\
\geq&\>\left(1+a^{2}\upomega_{+}^{2}\right)m^{2}\int_{\mathbb{S}^{2}}|S_{m\ell}|^{2}\,d\sigma.\end{split}$

Denote $\Lambda_{+,m\ell}=\lambda_{m\ell}+a^{2}\upomega_{+}^{2}m^{2}$. Then

$\not{\mathcal{D}}(e^{im\varphi}S_{m\ell}(\theta))=-\Lambda_{+,m\ell}e^{im\widetilde{\varphi}}S_{m\ell}(\theta).$

and, by the above,

$\Lambda_{+,m\ell}\geq\max\\{(1+a^{2}\upomega_{+}^{2})|m|,|m|+1\\}|m|.$ (2.6)

In particular, for $r_{+}=1$ and $|a|=1$,

$\Lambda_{+,m\ell}\geq\max\left\\{\frac{5}{4}|m|,|m|+1\right\\}|m|.$

Define

$|\widetilde{\not{\nabla}}_{\mathbb{S}^{2}}(\cdot)|^{2}:=|{\not{\nabla}}_{\mathbb{S}^{2}}(\cdot)|^{2}+a^{2}\upomega_{+}^{2}\sin^{2}\theta|\Phi(\cdot)|^{2},$

with $\not{\nabla}_{\mathbb{S}^{2}}$ denoting the covariant derivative with
respect to the metric on the unit 1-sphere.

Let $r_{+}=1$ and $|a|=1$. Then the following quantity will play a key role in
the paper:

$\alpha_{m\ell}:=\Lambda_{+,m\ell}-2m^{2}+\frac{1}{4}.$

The late-time asymptotics and sharp decay rates that we will derive follow
from the fact that $\alpha_{m|m|}>0$ for $|m|\geq 2$. This is proved in the
lemma below.

######  Lemma 2.1.

For $|m|=1$, $\alpha_{|m|m}\geq\frac{1}{4}$ and for $|m|\geq 2$,
$\alpha_{m|m|}<0$.

###### Proof.

The inequality for $|m|=1$ follows immediately from (2.6). Now let $|m|\geq
2$. We appeal to the Rayleigh–Ritz quotient: let $f:[0,\pi]\to\mathbb{R}$,
then

$\Lambda_{m|m|}=\inf_{e^{im\widetilde{\varphi}}f\in
L^{2}(\mathbb{S}^{2})\setminus\\{0\\}}\frac{\int_{\mathbb{S}^{2}}|\partial_{\theta}f|^{2}\sin\theta+(\frac{1}{4}\sin^{2}\theta+\sin^{-2}\theta)\sin\theta|f|^{2}\,d\theta
d\widetilde{\varphi}}{\int_{\mathbb{S}^{2}}|f|^{2}\,\sin\theta\,d\theta
d\widetilde{\varphi}}$

Let $f(\theta)=\sin\theta$, then we obtain via the above formula,

$m^{-2}\Lambda_{m|m|}\leq\frac{17}{10}+\frac{1}{2m^{2}}.$

Hence,

$m^{-2}\alpha_{m|m|}\leq-\frac{3}{10}+\frac{3}{4m^{2}}.$

For $|m|\geq 2$, the above expression therefore implies that

$m^{-2}\alpha_{m|m|}\leq-\frac{9}{80}.$

∎

We will denote $d\sigma=\sin\theta d\theta d\varphi_{*}$ and
$d\sigma=\sin\theta d\theta d\widetilde{\varphi}$ when integrating functions
along constant $(\tau,r)$ spheres with respect to
$(\tau,r,\theta,\varphi_{*})$ and $(\tau,r,\theta,\widetilde{\varphi})$
coordinates, respectively.

Let $f\in L^{2}(\mathbb{S}^{2})$. Then we can write
$f=\sum_{m\in\mathbb{Z}}f_{m}$, with

$f_{m}(\theta,\widetilde{\varphi})=\left[\int_{\mathbb{S}^{2}}f(\theta,\widetilde{\varphi}^{\prime})e^{-im\widetilde{\varphi}^{\prime}}\,\sin\theta
d\theta d\widetilde{\varphi}^{\prime}\right]e^{im\widetilde{\varphi}}.$

We refer to $f_{m}$ as the $m$-th azimuthal mode of $f$. We can
straightforwardly extend the definition of $f_{m}$ to functions on
$\mathcal{R}$.

Note that the eigenfunctions $S_{m\ell}(\theta)$ and the eigenvalues
$\Lambda_{+,m\ell}$ may be interpreted as the oblate spheroidal harmonics and
angular frequencies evaluated at the frequency $\omega=m\upomega_{+}$
corresponding to a Carter separation of the Fourier transform of $\phi$ in
time $t$; see §6.1.

###  2.5. Notation

We will make use of the notation $O(|f|)$ to indicate functions on
$\mathcal{M}$ that can be bounded uniformly by $|f|$ with a constant that
depends only on $M$ and $\mathbbm{h}$. We denote with $O_{n}(|f|)$ functions
$g$ on $\mathcal{M}$ such that $\partial_{r}^{k}g$ is bounded uniformly by
$|\partial_{r}^{k}f|$ for all $k\leq n\in\mathbb{N}_{0}$.

We use the notation $O^{m}_{k}(|f|)$ for functions with azimuthal angular
dependence $e^{im\widetilde{\varphi}}$, when the uniform constant is allowed
to depend additionally on $m$.

We will use the notation $c$ and $C$ to indicate constants that depend only on
$M,a$ and $\mathbbm{h}$. If the constants depends on an additional set of
parameters $\beta_{1},\ldots,\beta_{s}\in\mathbb{C}$, we will instead use the
notation $c_{\beta_{1},\ldots,\beta_{s}}$ and
$C_{\beta_{1},\ldots,\beta_{s}}$. To simplify notation, we make use the
following “algebra of constants”:

$C\cdot C=C+C=C,\quad c\cdot c=c+c=c.$

##  3\. Preliminaries: wave equation

###  3.1. The wave operator

In this section, we express the wave operator $\square_{g}$ using different
sets of vector fields and we state the relevant local-in-time weighted energy
estimates.

######  Lemma 3.1.

Let $\phi$ be a solution to the inhomogeneous wave equation:

$\square_{g_{M,a}}\phi=G.$ (3.1)

Then, we can express (3.1) in the following different ways (interpreted in a
distributional sense):

  1. _(i)_

In terms of the vector fields $(T,Y)$, we can express:

$\rho^{2}G=Y(\Delta
Y\phi)+2aY\Phi\phi+\not{\Delta}_{\mathbb{S}^{2}}\phi+2(r^{2}+a^{2})TY\phi+a^{2}\sin^{2}\theta
T^{2}\phi+2aT\Phi\phi+2rT\phi.$ (3.2)

  2. _(ii)_

In terms of the vector fields $(T,X)$, we can express:

$\begin{split}\rho^{2}G=&\>X(\Delta
X\phi)+2aX\Phi\phi+\not{\Delta}_{\mathbb{S}^{2}}\phi+2(r^{2}+a^{2}-\mathbbm{h}\Delta)TX\phi+[a^{2}\sin^{2}\theta+\mathbbm{h}(\Delta\mathbbm{h}-2(r^{2}+a^{2}))]T^{2}\phi\\\
&\>+2a(1-\mathbbm{h})T\Phi\phi+(2r-\frac{d}{dr}(\Delta\mathbbm{h}))T\phi\\\
=&\>X(\Delta
X\phi)+2aX\Phi\phi+\not{\Delta}_{\mathbb{S}^{2}}\phi-2(r^{2}+a^{2})^{\frac{1}{2}}(1+O(r^{-1}))TX\psi\\\
&\>+[a^{2}\sin^{2}\theta+\mathbbm{h}(\Delta\mathbbm{h}-2(r^{2}+a^{2}))]T^{2}\phi+2a(1-\mathbbm{h})T\Phi\phi+O(1)T\phi.\end{split}$
(3.3)

  3. _(iii)_

In terms of the vector fields $(K,Y)$, we can express:

$\rho^{2}G=Y(\Delta
Y\phi)-2\upomega_{+}(r^{2}-r_{+}^{2})Y\Phi\phi+(\not{\mathcal{D}}-2a\upomega_{+}\Phi^{2})\phi-2\upomega_{+}r\Phi\phi+2(r^{2}+a^{2})KY\phi\\\
+a^{2}\sin^{2}\theta(K^{2}\phi-2\upomega_{+}K\Phi\phi)+2aK\Phi\phi+2rK\phi.$
(3.4)

  4. _(iv)_

In terms of the vector fields $(K,\widetilde{X})$ and the rescaled quantity
$\psi:=(r^{2}+a^{2})^{\frac{1}{2}}\phi$, we can express:

$\rho^{2}G=\widetilde{X}(\Delta\widetilde{X}\phi)-2\upomega_{+}(r^{2}-r_{+}^{2})\widetilde{X}\Phi\phi+(\not{\mathcal{D}}-2a\upomega_{+}\Phi^{2})\phi-2\upomega_{+}r\Phi\phi+2(r^{2}+a^{2}-\mathbbm{h}\Delta)K\widetilde{X}\phi\\\
+[a^{2}\sin^{2}\theta+\mathbbm{h}(\Delta\mathbbm{h}-2(r^{2}+a^{2}))]K^{2}\phi+2(a+\mathbbm{h}\upomega_{+}(r^{2}-r_{+}^{2})-a^{2}\upomega_{+}\sin^{2}\theta)K\Phi\phi+\left(2r-\frac{d}{dr}(\Delta\mathbbm{h})\right)K\phi\\\
=\widetilde{X}(\Delta\widetilde{X}\phi)-2\upomega_{+}(r^{2}-r_{+}^{2})\widetilde{X}\Phi\phi+(\not{\mathcal{D}}-2a\upomega_{+}\Phi^{2})\phi-2\upomega_{+}r\Phi\phi-2(r+O(r^{-1}))KX\psi\\\
+[a^{2}\sin^{2}\theta+O(1)]K^{2}\phi+2(O(1)-a^{2}\upomega_{+}\sin^{2}\theta)K\Phi\phi+O(1)K\phi.$
(3.5)

  5. _(v)_

In terms of the principal null fields $(L,\underline{L})$, we can express:

$(r^{2}+a^{2})^{\frac{1}{2}}\rho^{2}G=-4\frac{(r^{2}+a^{2})^{2}}{\Delta}L\underline{L}\psi+a^{2}\sin^{2}\theta
K^{2}\psi+2a(1-a\upomega_{+}\sin^{2}\theta)K\Phi\psi+(\not{\mathcal{D}}-2a\upomega_{+}\Phi^{2})\psi\\\
-2\frac{ar}{r^{2}+a^{2}}\Phi\psi-\frac{(r^{2}+a^{2})^{2}}{2r\Delta}\frac{d}{dr}\left(r^{2}(r^{2}+a^{2})^{-3}\Delta^{2}\right)\psi.$
(3.6)

###### Proof.

We write:

$\square_{g}\phi=\frac{1}{\sqrt{-\det g_{M,a}}}\sum_{0\leq\alpha,\beta\leq
3}\partial_{\alpha}(\sqrt{-\det
g_{M,a}}(g^{-1})^{\alpha\beta}\partial_{\beta}\phi)$

and evaluate this expression in $(v,r,\theta,\varphi_{*})$, using (2.2), in
order to obtain (i). The remaining equations follow in a straightforward
manner by using the relations between $T,K,Y,Y,\widetilde{X}$ and $\Phi$ from
§2.3. ∎

By standard local-in-time energy estimates for linear wave equations, we
obtain global existence and uniqueness for solutions to the initial value
problem corresponding to (3.1); see also [AAG23, Proposition 3.4].

######  Theorem 3.2.

Let $k\in\mathbb{N}_{0}$ and $G\in
L^{1}_{\tau}H^{k}(\Sigma_{\tau};\mathbb{C})$ and $(\phi_{\rm data},\phi_{\rm
data}^{\prime})\in H^{k+1}_{\rm loc}(\Sigma_{0};\mathbb{C})\times H^{k}_{\rm
loc}(\Sigma_{0};\mathbb{C})$. Then there exists a solution $\phi$ to (3.1) (in
the distributional sense), such that for all $\tau\geq 0$:

$\displaystyle\phi\in$ $\displaystyle\>C^{0}([0,\tau],H^{k+1}_{\rm
loc}(\Sigma_{0};\mathbb{C})\cap C^{1}([0,\tau],H^{k}_{\rm
loc}(\Sigma_{0};\mathbb{C}),$ $\displaystyle\phi|_{\Sigma_{0}}=$
$\displaystyle\>\phi_{\rm data},$ $\displaystyle K\phi|_{\Sigma_{0}}=$
$\displaystyle\>\phi_{\rm data}^{\prime}.$

Furthermore, if $(\phi_{\rm data},\phi_{\rm data}^{\prime})\in
C^{\infty}_{c}(\Sigma_{0};\mathbb{C})\times
C^{\infty}_{c}(\Sigma_{0};\mathbb{C})$, then $\phi\in
C^{\infty}(\mathcal{R};\mathbb{C})$ and for all $j,k,l\in\mathbb{N}_{0}$

$\lim_{R\to\infty}\int_{\Sigma_{\tau}\cap\\{r=R\\}}(r^{2}X)^{j}K^{k}\Phi^{l}\psi
d\sigma<\infty,$

with $\psi$ defined in _(v)_ of Lemma 3.1.

In the $a=|M|$ case, we define the following _energy densities_ , with weights
near $r=r_{+}=M$:

$\mathcal{E}_{p}[\phi]:=(1-Mr^{-1})^{2-p}|X\phi|^{2}+|T\phi|^{2}+r^{-2}|\not{\nabla}_{\mathbb{S}^{2}}\phi|^{2}\quad\textnormal{with
$0\leq p\leq 2$}.$ (3.7)

######  Proposition 3.3.

Assume $a=|M|$. Let $T_{0}\geq 0$ and let $\phi$ be a solution to (3.1) with
$G\in H^{k}(\Sigma_{\tau};\mathbb{C})$ for all $\tau\in[0,T_{0}]$, such that
$\int_{\Sigma_{0}}\mathcal{E}_{p}[\phi]\,r^{2}d\sigma dr<\infty$ for some
$0\leq p\leq 2$. Then there exists a constant $C_{T_{0}}>0$ such that for all
$\tau\in[0,T_{0}]$

$\int_{\Sigma_{\tau}}\mathcal{E}_{p}[\phi]\,r^{2}d\sigma
dr+\int_{\mathcal{I}^{+}\cap\\{0\leq\tau^{\prime}\leq\tau\\}}|T\psi_{m}|^{2}\,d\sigma
d\tau+\int_{\mathcal{H}^{+}\cap\\{0\leq\tau^{\prime}\leq\tau\\}}|K\psi_{m}|^{2}\,d\sigma
d\tau\\\ \leq
C_{T_{0}}\left[\int_{\Sigma_{0}}\mathcal{E}_{p}[\phi]\,r^{2}d\sigma
dr+\int_{0}^{\tau}\int_{\Sigma_{\tau}}|G|^{2}r^{2}\,d\sigma drd\tau\right].$

###### Proof.

Let $V_{p}$ be a smooth vector field such that $V=L+(r-M)^{-p}\underline{L}$
in $r\leq 3M$ and $N=T$ in $r\geq 4M$. Then

$\begin{split}g_{M,\pm|M|}(V_{p},V_{p})=&\>2(r-M)^{-p}g(L,\underline{L})=(r-M)^{-p}g(L+\underline{L},L+\underline{L})\\\
=&\>(r-M)^{-p}g\left(T+\frac{M}{r^{2}+M^{2}}\Phi,T+\frac{a}{r^{2}+a^{2}}\Phi\right)\\\
=&\>\rho^{-2}(r-M)^{-p}(r^{2}+M^{2})^{-1}\big{[}M^{2}(r^{2}+M^{2})\sin^{2}\theta-\Delta(r^{2}+M^{2})-4M^{3}r\sin^{2}\theta\\\
&\>+M^{2}(r^{2}+M^{2})\sin^{2}\theta-M^{4}(r^{2}+M^{2})^{-1}\Delta\sin^{4}\theta\big{]}\\\
=&\>-\rho^{-2}(r-M)^{2-p}(r^{2}+M^{2}\cos(2\theta)+M^{4}(r^{2}+M^{2})^{-1}\sin^{4}\theta).\end{split}$

Hence, $V_{p}$ is timelike in $\\{r>M\\}$ for all $0\leq p\leq 2$, timelike
along $\mathcal{H}^{+}$ if $p=2$ and null along $\mathcal{H}^{+}$ if $p<2$.

It then follows that:

$\mathbb{T}[\phi](V_{p},\mathbf{n}_{\tau})\sim\mathcal{E}_{p}[\phi],$

with $\mathbb{T}[\phi]:=\Re(d\phi\otimes
d\overline{\phi})-\frac{1}{2}|\nabla\phi|^{2}g$ the energy-momentum tensor
corresponding to $\square_{g}\phi$ and $\mathbf{n}_{\tau}$ the unit normal to
$\Sigma_{\tau}$. ∎

The main theorems concern solutions to:

$\square_{g_{M,a}}\phi=0$ (3.8)

with $|a|=M$.

###  3.2. Stationary solutions in extremal Kerr

A key role in the analysis will be played by _stationary_ , $K$-invariant
solutions to to (3.1) with $G=0$ and $|a|=M$. If we express these with respect
to the coordinates $(t,r,\theta,\varphi)$, they take the form

$\mathfrak{w}_{m\ell}(r)e^{-i\upomega_{+}t+im\varphi}S_{m\ell}(\theta).$

Assume for the sake of convenience that $r_{+}=1$, then

$K(\mathfrak{w}_{m\ell}e^{-i\upomega_{+}t+im\varphi}S_{m\ell})=0\quad\textnormal{and}\quad\square_{g_{M,a}}(\mathfrak{w}_{m\ell}e^{-i\upomega_{+}t+im\varphi}S_{m\ell})=0.$

is equivalent to the following ODE for $\mathfrak{w}_{m\ell}(r(r_{*}))$:

$\displaystyle\mathfrak{w}^{\prime\prime}-\widetilde{V}_{\rm
stat}(r(r_{*}))\mathfrak{w}=$ $\displaystyle\>0,$ (3.9)
$\displaystyle\widetilde{V}_{\rm stat}(r):=$
$\displaystyle\>-\frac{1}{4}m^{2}+(r^{2}+1)^{-2}\left[m^{2}a^{2}(2r-1)+\Lambda_{+,m\ell}\Delta\right]+(r^{2}+1)^{-3}\Delta(3r^{2}-4r+1)$
$\displaystyle-3(r^{2}+1)^{-4}\Delta^{2}r^{2}.$

Denote $\alpha_{m\ell}=\Lambda_{+,m\ell}-2m^{2}+\frac{1}{4}$.

######  Lemma 3.4.

Let $r_{+}=1$ and $a=1$. Assume that there exist $b,B>0$ such that
$\Lambda_{+,m\ell}\leq Bm^{2}$ and $\alpha_{m\ell}\neq 0$ for all
$m\in\mathbb{Z}\setminus\\{0\\}$. Then for all
$m\in\mathbb{Z}\setminus\\{0\\}$, there exists two unique solutions
$\mathfrak{w}_{\pm,m\ell}$ to (6.5.8) that satisfy:

$\displaystyle\mathfrak{w}_{\pm,m\ell}(r)=$
$\displaystyle\>(r-1)^{-\frac{1}{2}\pm
i\sqrt{-\alpha_{m\ell}}}\left[1+O_{\infty}((r-1))\right].$ (3.10)

Furthermore,

$\displaystyle\mathfrak{w}_{\pm,m\ell}^{-1}\frac{d\mathfrak{w}_{\pm,m\ell}}{dr}=$
$\displaystyle\>-\frac{1}{2}(r-1)^{-1}\pm
i\sqrt{-\alpha_{m\ell}}(r-1)^{-1}+O(1).$ (3.11)

When $\alpha_{m\ell}=0$, there exist again two unique solutions
$\mathfrak{w}_{\pm}$, satisfying:

$\displaystyle\mathfrak{w}_{-,m\ell}(r)=$
$\displaystyle\>(r-1)^{-\frac{1}{2}}\left[1+O_{\infty}^{m}((r-1))\right],$
(3.12) $\displaystyle\mathfrak{w}_{+,m\ell}(r)=$
$\displaystyle\>(r-1)^{-\frac{1}{2}}\log\left(\frac{1}{r-1}\right)[1+O_{\infty}^{m}((r-1)]$
(3.13)
$\displaystyle\mathfrak{w}_{-,m\ell}^{-1}\frac{d\mathfrak{w}_{-,m\ell}}{dr}=$
$\displaystyle\>-\frac{1}{2}(r-1)^{-1}(r-1)^{-1}+O^{m}(1),$ (3.14)
$\displaystyle\mathfrak{w}_{+,m\ell}^{-1}\frac{d\mathfrak{w}_{+,m\ell}}{dr}=$
$\displaystyle\>-\left(\frac{1}{2}+\frac{1}{\log(r-1)^{-1}}\right)(r-1)^{-1}+\log(r-1)O^{m}(1).$
(3.15)

###### Proof.

We will suppress the subscript $\pm,m\ell$ in $\mathfrak{w}_{\pm,m\ell}$
below. Note first that we can express (3.10) as follows:

$\displaystyle\frac{d^{2}\mathfrak{w}}{dr^{2}}+2(r-1)^{-1}[1+O_{\infty}((r-1))]\frac{d\mathfrak{w}}{dr}-(r-1)^{-4}(r^{2}+a^{2})^{2}\widetilde{V}_{\rm
stat}\mathfrak{w}=0.$

By applying (6.5.8) and expanding $\widetilde{V}_{\rm stat}$ in $r-1$ around
$r=1$ (see (6.28)), it follows that:

$\frac{d^{2}\mathfrak{w}}{dr^{2}}+2(r-1)^{-1}[1+O_{\infty}((r-1))]\frac{d\mathfrak{w}}{dr}+(r-1)^{2}\left[2m^{2}-\Lambda_{+,m\ell}+O_{\infty}(r-1)\right]\mathfrak{w}=0.$

so $r=1$ is a regular singular point for (6.5.8) and we can apply the
Liouville–Green or WKB approximation in [Olv74, §6.2, Theorem 2.2] to obtain
(3.10) and (3.11). Here we make use of the condition $\alpha_{m\ell}\neq 0$ to
ensure that the Liouville–Green approximation does not break down.

In the case $\alpha_{m\ell}\neq 0$, we instead apply standard ODE theory for
expansions of solutions around regular singular points [Olv74, §5.4, Theorem
4.1] which introduces $m$ dependence in the expansions (3.12)–(3.15). ∎

We now introduce the functions $w_{\pm,m\ell}$, which are related to
$\mathfrak{w}_{\pm,m\ell}$, defined in Lemma 3.4, as follows:

$w_{\pm,m\ell}(r)e^{-i\upomega_{+}\tau+im\tilde{\varphi}}=\frac{\sqrt{2}}{(r^{2}+1)^{\frac{1}{2}}}\mathfrak{w}_{\pm,m\ell}e^{-i\upomega_{+}t+im\varphi}.$

By Lemma 3.4, we can expand uniformly in $m$ for $\alpha_{m\ell}\neq 0$ :

$\displaystyle w_{\pm}(r)=$ $\displaystyle\>(r-1)^{-\frac{1}{2}-im\pm
i\sqrt{2m^{2}-\Lambda_{+,m\ell}-\frac{1}{4}}}\left[1+O_{\infty}((r-1))\right],$
(3.16) $\displaystyle w_{\pm,m\ell}^{-1}\frac{dw_{\pm,m\ell}}{dr}=$
$\displaystyle\>-\frac{1}{2}(r-1)^{-1}+i\left(-m\pm\sqrt{2m^{2}-\Lambda_{+,m\ell}-\frac{1}{4}}\right)(r-1)^{-1}+O_{\infty}(1).$
(3.17)

Furthermore, there exist constants
$A_{\pm,m\ell},\widetilde{A}_{\pm,m\ell}\in\mathbb{C}$ such that the following
expansions hold uniformly in $m$ in $r\geq 2$:

$w_{\pm,m\ell}=\frac{\widetilde{A}_{\pm,m\ell}}{r}+\frac{{A}_{\pm,m\ell}e^{imr_{*}}}{r}+O_{\infty}(r^{-2}).$
(3.18)

When $2m^{2}-\Lambda_{+,m\ell}-\frac{1}{4}=0$, there exist again two unique
solutions $w_{\pm,m\ell}$, satisfying (3.18) and:

$\displaystyle w_{-,m\ell}(r)=$
$\displaystyle\>(r-1)^{-\frac{1}{2}-im}\left[1+O_{\infty}^{m}((r-1))\right],$
(3.19) $\displaystyle w_{+,m\ell}(r)=$
$\displaystyle\>(r-1)^{-\frac{1}{2}-im}\log\left(\frac{1}{r-1}\right)[1+O_{\infty}^{m}((r-1)],$
(3.20) $\displaystyle w_{-,m\ell}^{-1}\frac{dw_{-,m\ell}}{dr}=$
$\displaystyle\>-\frac{1}{2}(1+2im)(r-1)^{-1}+O_{\infty}^{m}(1),$ (3.21)
$\displaystyle w_{+,m\ell}^{-1}\frac{dw_{+,m\ell}}{dr}=$
$\displaystyle\>-\left(\frac{1}{2}+im+(\log(r-1)^{-1})^{-1})\right)(r-1)^{-1}+\log(r-1)O_{\infty}^{m}(1).$
(3.22)

Finally, we denote with $w_{\infty}(r)$ and $\widetilde{w}_{\infty}(r)$ the
linear combinations of $w_{\pm,m\ell}$ such that
$w_{\infty,m\ell}(r)=r^{-1}e^{imr_{*}}+O(r^{-2})$ and
$\widetilde{w}_{\infty,m\ell}(r)=r^{-1}+O(r^{-2})$. We can express:

$\displaystyle w_{\infty,m\ell}(r)=$
$\displaystyle\>B_{+}w_{+,m\ell}(r)+B_{-}w_{-,m\ell}(r),$
$\displaystyle\widetilde{w}_{\infty,m\ell}(r)=$
$\displaystyle\>\widetilde{B}_{+}w_{+,m\ell}(r)+\widetilde{B}_{-}w_{-,m\ell}(r),$
$\displaystyle w_{\pm,m\ell}(r)=$
$\displaystyle\>\widetilde{A}_{\pm,m\ell}\widetilde{w}_{\infty,m\ell}(r)+{A}_{\pm,m\ell}w_{\infty,m\ell}(r),$

with $\widetilde{B}_{\pm}=\widetilde{A}_{\pm}(1+A_{\pm}B_{\pm})$ or
$\widetilde{B}_{\mp}=\widetilde{A}_{\pm}^{-1}A_{\pm}B_{\mp}$.

##  4\. Precise statements of the main theorems

In this section, we will state precisely the main theorems in the present
paper. To be able to state a precise version of Theorem 1.2, we will introduce
the function $\Pi$ that will give the leading-order late-time asymptotics of
$\phi$ globally. We will assume without loss of generality that $a>0$. The
$a<0$ can be included in the analysis by a redefinition
$\varphi_{*}\to-\varphi_{*}$. Then we set (without loss of generality)
$r_{+}=1$ and $a=1$.

Recall the definition of $\alpha_{m\ell}$:
$\alpha_{m\ell}=\Lambda_{+,m\ell}-2m^{2}+\frac{1}{4}$. We will consider in
this section $m\in\mathbb{Z}\setminus\\{-1,0,1\\}$ and
$\ell\in\mathbb{N}_{0}$, $l\geq|m|$, such that

$\alpha_{m\ell}\neq 0.$ (4.1)

Then we define the following function, which will describe the global, late-
time asymptotic behaviour of $\phi$:

$\displaystyle\Pi(\tau,r,\theta,\varphi)=$
$\displaystyle\>\sum_{\begin{subarray}{c}|m|\leq\ell\leq l_{m}\\\
\alpha_{m\ell}\neq
0\end{subarray}}\Pi_{m\ell}(\tau,r)e^{im\widetilde{\varphi}}S_{m\ell}(\theta),$

where $l_{m}$ is the largest value of $\ell$ such that $\Lambda_{+,m\ell}\leq
2m^{2}$ (or equivalently, $\alpha_{m\ell}\leq\frac{1}{4}$) and $\Pi_{m\ell}$
is defined as follows: let ${\mathfrak{h}}_{m\ell}\in\mathbb{C}$ and let
$A_{\pm},\widetilde{A}_{\pm},B_{\pm},\widetilde{B}_{\pm}$ be the constants
appearing in the expansions of the stationary solutions defined in §3.2, then

  1. (1)

If $\alpha_{m\ell}<0$, $\widetilde{B}_{-}\neq 0$ and $\widetilde{B}_{+}\neq
0$, we define:

$\Pi_{m\ell}(\tau,r):={\mathfrak{h}}_{m\ell}w_{\infty,m\ell}(r)\left[f_{+}^{m\ell}(\tau,r)+f_{-}^{m\ell}(\tau,r)\right]+{\mathfrak{h}}_{m\ell}\widetilde{w}_{\infty,m\ell}(r)\left[E_{+}f_{+,{\rm
exp}}^{m\ell}(\tau,r)+E_{-}f_{-,{\rm exp}}^{m\ell}(\tau,r)\right],$ (4.2)

with $E_{+}=-\widetilde{B}_{-}^{-1}B_{-}$ and
$E_{-}=-\widetilde{B}_{+}^{-1}B_{+}$, ${\mathfrak{h}}_{m\ell}\in\mathbb{C}$,
and

$\displaystyle
f_{\pm}^{m\ell}(\tau,r):=\left(\tau+\frac{4}{r-1}+1\right)^{-\frac{1}{2}\mp\sqrt{\alpha_{m\ell}}+im}(\tau+1)^{-\frac{1}{2}\mp\sqrt{\alpha_{m\ell}}-im},$
$\displaystyle f_{\pm,{\rm
exp}}^{m\ell}(\tau,r):=\left(\tau+\frac{4}{r-1}+e^{r-1}\right)^{-\frac{1}{2}\mp\sqrt{\alpha_{m\ell}}+im}(\tau+1)^{-\frac{1}{2}\mp\sqrt{\alpha_{m\ell}}-im}.$

  2. (2)

If $\alpha_{m\ell}<0$, $\widetilde{B}_{+}=0$, then $\widetilde{A}_{+}=0$, and
we define:

$\Pi_{m\ell}(\tau,r):={\mathfrak{h}}_{m\ell}w_{\infty,m\ell}(r)f_{+}^{m\ell}(\tau,r)={\mathfrak{h}}_{m\ell}A_{+}^{-1}w_{+,m\ell}(r)f_{+}^{m\ell}(\tau,r).$
(4.3)

  3. (3)

If $\alpha_{m\ell}<0$, $\widetilde{B}_{-}=0$, then $\widetilde{A}_{-}=0$ and
we define:

$\Pi_{m\ell}(\tau,r):={\mathfrak{h}}_{m\ell}w_{\infty,m\ell}(r)f_{-}^{m\ell}(\tau,r)={\mathfrak{h}}_{m\ell}A_{-}^{-1}w_{-,m\ell}(r)f_{-}^{m\ell}(\tau,r).$
(4.4)

  4. (4)

If $\alpha_{m\ell}>0$, then we define:

$\Pi_{m\ell}(\tau,r):={\mathfrak{h}}_{m\ell}w_{\infty,m\ell}(r)f_{+}^{m\ell}(\tau,r).$
(4.5)

Denote

$F_{\Pi}:=2(r^{2}+1-\mathbbm{h}\Delta)\widetilde{X}\Pi|_{\Sigma_{0}}+2\left(1+\frac{1}{2}\mathbbm{h}(r^{2}-1)-\frac{1}{2}\sin^{2}\theta\right)\Phi\Pi|_{\Sigma_{0}}\\\
+\left(\mathbbm{h}(\Delta\mathbbm{h}-2(r^{2}+1))-\sin^{2}\theta\right)K\Pi|_{\Sigma_{0}}-\left(2r-\frac{d}{dr}(\Delta\mathbbm{h})\right)\Pi|_{\Sigma_{0}}-\int_{0}^{\infty}\square_{g}\Pi(\tau^{\prime},r,\theta,\widetilde{\varphi})\,d\tau^{\prime}.$

We consider the following initial data quantity

$F_{\rm
hom}[\phi]=-2(r^{2}+1-\mathbbm{h}\Delta)\widetilde{X}\phi|_{\Sigma_{0}}-2\left(1+\frac{1}{2}\mathbbm{h}(r^{2}-1)-\frac{1}{2}\sin^{2}\theta\right)\Phi\phi|_{\Sigma_{0}}\\\
-\left(\mathbbm{h}(\Delta\mathbbm{h}-2(r^{2}+1))-\sin^{2}\theta\right)K\phi|_{\Sigma_{0}}+\left(2r-\frac{d}{dr}(\Delta\mathbbm{h})\right)\phi|_{\Sigma_{0}}$

and the following initial data energy norms along $\Sigma_{0}$.

$D^{(1)}[\phi]=\sum_{|m|\leq\ell\leq
l_{m}}|{\mathfrak{h}}_{m\ell}|^{2}+\sum_{k+l\leq 4,l\leq
2}\int_{\Sigma_{0}}\mathcal{E}_{1+\epsilon}[K^{k+l}\phi]\,r^{2}d\sigma dr\\\
+\int_{\Sigma_{0}\cap\\{r\geq 2\\}}\sum_{k\leq
2}r^{2-\delta}|LK^{k}\psi|^{2}+\sum_{k\leq
1}r^{2-\delta}|L((r^{2}L)K^{k}\psi)|^{2}\,d\sigma dr\\\
+\sum_{\begin{subarray}{c}k\leq 2\\\ j\leq
1\end{subarray}}\sup_{r^{\prime}\leq
2}\int_{\Sigma_{0}\cap\\{r=r^{\prime}\\}}|\widetilde{X}^{j}((r-1)\widetilde{X})^{k}F_{\rm
hom}[\phi])|^{2}\,d\sigma+\sup_{r^{\prime}\geq
2}\int_{\Sigma_{0}\cap\\{r=r^{\prime}\\}}|rF_{\rm hom}[\phi]|^{2}\,d\sigma\\\
+\int_{\Sigma_{0}\cap\\{r\geq 2\\}}r^{2}|\widetilde{X}F_{\rm
hom}[\phi]|^{2}\,d\sigma dr$

and

$D_{2}^{(1)}[\phi]=\int_{\Sigma_{0}\cap\\{r\geq
r_{I}\\}}r|L(r^{2}L)^{2}K\widehat{\psi}_{|m|\geq 2}|^{2}\,d\sigma
dr+D^{(1)}[\phi]+D^{(1)}[K\phi].$

Let $r_{0}>1$ and let $\mathfrak{h}_{m\ell}$ be chosen such that the following
cancellation occurs:

$\int_{1}^{\infty}w_{\infty,m\ell}(r)e^{-im\int_{r_{0}}^{r}\frac{x+1}{x-1}\,dx}((F_{\Pi})_{m\ell}+(F_{\rm
hom})_{m\ell}[\phi])\,dr\neq 0.$

To guarantee this can be done for general initial data, we need that (for
$\mathfrak{h}_{m\ell}\neq 0$):

$\int_{1}^{\infty}w_{\infty,m\ell}(r)e^{-im\int_{r_{0}}^{r}\frac{x+1}{x-1}\,dx}(F_{\Pi})_{m\ell}\,dr\neq
0.$

We will assume, without loss of generality, that the above condition holds for
all $\ell\leq l_{m}$. If it did not, we could modify the functions
$f_{\pm}^{m\ell}$ and $f_{\pm,{\rm exp}}^{m\ell}$ by replacing
$\left(\tau+\frac{4}{r-1}+1\right)$ with
$\left(\tau+\frac{4}{r-1}+\beta\right)$ and
$\left(\tau+\frac{4}{r-1}+e^{r-1}\right)$ with $\left(\tau+\frac{4}{r-1}+\beta
e^{r-1}\right)$, respectively, and choose $\beta\in\mathbb{R}$ appropriately,
as this modification does not affect the leading-order asymptotic behaviour of
$f_{\pm}^{m\ell}$ and $f_{\pm,{\rm exp}}^{m\ell}$.

Observe that $\mathfrak{h}_{m\ell}$ is therefore a constant multiple of
$\int_{1}^{\infty}w_{\infty,m\ell}(r)e^{-im\int_{r_{0}}^{r}\frac{x+1}{x-1}\,dx}(F_{\rm
hom})_{m\ell}[\phi]\,dr$.

Define

$\mathfrak{h}_{m}:=\sqrt{\sum_{\begin{subarray}{c}|m|\leq\ell\leq l_{m}\\\
\alpha_{m\ell}<0\end{subarray}}|{\mathfrak{h}}_{m\ell}|^{2}}.$

By Lemma 2.1, the set of $\ell$ for which $\alpha_{m\ell}<0$ is non-empty for
all $m$, so $\mathfrak{h}_{m}\neq 0$ for generic data (i.e. when restricted to
the azimuthal number $m$, this will be the complement of a codimension
$l_{m}+1-|m|$ set of data).

We first state the following precise version of Theorem 1.2:

######  Theorem 4.1 (Late-time tails and stability for azimuthal modes on
extremal Kerr; precise version).

Let $|m|\geq 2$ and consider solutions $\phi_{m}$ to (3.8) arising from
initial data satisfying
$\sum_{2k+l\leq 2}D_{2}^{(1)}[Q^{l}K^{k}\phi_{m}]<\infty$. Assume that
$\alpha_{m\ell}\neq 0$ for all $\ell$ and define

$\nu_{m}:=\min\left(\\{\alpha_{m\ell}\,|\,\alpha_{m\ell}>0\\}\cup\\{1\\}\right).$

Assume that for all $1<r_{0}<r_{1}$

$\displaystyle\int_{0}^{\infty}\int_{\Sigma_{\tau}\cap\\{r\leq
r_{1}\\}}|\phi_{m}|^{2}\,d\sigma drd\tau<$ $\displaystyle\>\infty,$ (4.6)
$\displaystyle\int_{0}^{\infty}\int_{\Sigma_{\tau}\cap\\{r_{0}\leq r\leq
r_{1}\\}}|X\phi_{m}|^{2}+|T\phi_{m}|^{2}+|\not{\nabla}_{\mathbb{S}^{2}}\phi_{m}|^{2}\,d\sigma
drd\tau<$ $\displaystyle\>\infty.$ (4.7)

Let $0<\eta<\frac{\nu_{m}}{4}$ be arbitrarily small. Then there exists a
constant $C_{m}>0$, such that for all $\tau\geq 0$:

$\displaystyle\left|\phi_{m}-\Pi\right|(\tau,r,\theta,\widetilde{\varphi})$
$\displaystyle\leq$
$\displaystyle\>C_{m}r^{-1}((\tau+1)(1-r^{-1})+1)^{-\frac{1}{2}}(\tau+1)^{-\frac{1}{2}-\frac{\nu_{m}}{4}+\eta}\sqrt{\sum_{2k+l\leq
2}D_{2}^{(1)}[Q^{l}K^{k}\phi]},$
$\displaystyle\int_{\Sigma_{\tau}}\mathcal{E}_{2}[\phi-\Pi]\,r^{2}d\sigma
dr\leq$ $\displaystyle\>C_{m}(1+\tau)^{-\nu_{m}+\eta}D^{(1)}[\phi].$

###### Proof.

The theorem is proved by combining Proposition 11.2 and Corollary 11.6, which
are proved in §11. ∎

We will now arrive at a precise version of Theorem 1.1. First, Theorem 4.2
below shows that if the global assumptions (4.6) and (4.7) hold, the azimuthal
instability (a) must hold.

######  Theorem 4.2 (The existence of azimuthal instabilities on extremal
Kerr; precise version).

Let $|m|\geq 2$ and consider solutions $\phi_{m}$ satisfying the assumptions
from Theorem 4.1. If there exists an $|m|<\ell_{*}\leq l_{m}$ such that
$\alpha_{m\ell_{*}}=0$, restrict to initial data for which correspondingly
$\mathfrak{h}_{m\ell_{*}}=0$.

  1. _(i)_

Let $\eta>0$ be suitably small. Then there exist constants $c_{m}>0$ and
$C_{m}>0$, such that

$\int_{\Sigma_{\tau}}\mathcal{E}_{2}[\phi_{m}]\,r^{2}d\sigma dr\geq
c_{m}\mathfrak{h}_{m}^{2}-C_{m}(1+\tau)^{-\nu_{m}+\eta}D^{(1)}[\phi_{m}].$

  2. _(ii)_

Let $k\in\mathbb{N}$. There exists a constant $c_{m}=c_{m}(m,k)>0$ and a
monotonically increasing sequence $\\{\tau_{n}\\}$ in $\mathbb{R}_{+}$ with
$\tau_{n}\to\infty$ along which

$||Y^{k}\phi_{m}||^{2}_{L^{\infty}(\Sigma_{\tau_{n}}\cap\mathcal{H}^{+})}\geq\frac{1}{4\pi}\int_{\Sigma_{\tau_{n}}\cap\mathcal{H}^{+}}|Y^{k}\phi_{m}|^{2}\,d\sigma\geq
c_{m}\mathfrak{h}_{m}^{2}(1+\tau_{n})^{1+2k}.$

###### Proof.

The statements (i) and (ii) follow from a combination of Propositions 12.1,
12.2 and 12.3. ∎

From Theorem 4.2, it follows easily deduce a general instability result, which
includes the instability scenarios (b) and (c) of Theorem 4.1.

######  Corollary 4.3.

There exist smooth, compactly supported initial data
$(\phi_{*}|_{\Sigma},K\phi_{*}|_{\Sigma})$, such that the corresponding
solution $\phi_{*}$ satisfies at least one of the following instabilities:

  1. _(a)_

Let $k\in\mathbb{N}$. There exists a constant $b>0$, depending on initial
data, and two sequences of times $\\{\tau_{n}\\}$ and
$\\{\tau_{n}^{\prime}\\}$ with $\tau_{n},\tau_{n}^{\prime}\to\infty$, such
that

$\displaystyle\int_{\Sigma_{\tau_{n}^{\prime}}}\mathcal{E}_{2}[\phi_{*}]\,r^{2}d\sigma
dr$ $\displaystyle\>\geq b^{2},$ (4.8)
$\displaystyle||Y^{k}\phi_{*}||_{L^{\infty}(\Sigma_{\tau_{n}}\cap\mathcal{H}^{+})}$
$\displaystyle\>\geq b\tau_{n}^{\frac{1}{2}+k}.$ (4.9)

  2. _(b)_

There exists an $|m|\geq 2$ for which

$(\phi_{*})_{m}\notin L^{2}_{\tau}L^{2}(\Sigma_{\tau}).$ (4.10)

  3. _(c)_

There exists an $|m|\geq 2$ and radii $1<r_{0}\leq r_{1}<\infty$ for which

$|(Y\phi_{*})_{m}|+|(K\phi_{*})_{m}|+|(\not{\nabla}_{\mathbb{S}^{2}}\phi_{*})_{m}|\notin
L^{2}_{\tau}L^{2}(\Sigma_{\tau}\cap\\{r_{0}\leq r\leq r_{1}\\}).$ (4.11)

Furthermore, the subset of smooth compactly supported data which does not
satisfy any of the instabilities (a), (b) and (c) has infinite codimension.

###### Proof.

Suppose that there do not exist smooth, compactly supported data for which the
instabilities (b) and (c) hold. Then for all $|m|\geq 2$, smooth compactly
supported data supported on the $m$th azimuthal mode must satisfy (4.6) and
(4.7). We then take $(\phi_{*}|_{\Sigma},K\phi_{*}|_{\Sigma})$ to be any
smooth and compactly supported data with $\mathfrak{h}_{m|m|}\neq 0$. If
$\alpha_{m\ell_{*}}=0$ for some $\ell_{*}>|m|$, we invoke the additional
restriction $\mathfrak{h}_{m\ell_{*}}=0$. Then the instability (a) follows
from Theorem 4.2, with $b$ proportional to $\mathfrak{h}_{m}$.

To conclude infinite codimensionality, we first observe that $|m|\geq 2$ above
was arbitrary, so we can in fact find an infinite linearly independent set of
data, supported on different $m$, with each element resulting in instability
(a).

Consider now a solution $\phi$ arising from smooth, compactly supported data
$(\phi|_{\Sigma},K\phi|_{\Sigma})$, for which none of the instabilities
(a)–(c) hold. By linearity of the wave equation, the data
$(\phi|_{\Sigma}+\lambda\phi_{*}|_{\Sigma},K\phi|_{\Sigma}+\lambda
K\phi_{*}|_{\Sigma})$, with $\lambda\in\mathbb{R}$, correspond to the solution
$\phi+\lambda\phi_{*}$. If $\phi_{*}$ satisfies (b) or (c), then so will
$\phi+\lambda\phi_{*}$ for any $\lambda\neq 0$.

Suppose that $\phi_{*}$ satisfies (a) with the sequences $\\{\tau_{n}\\}$ and
$\\{\tau_{n}^{\prime}\\}$. If there exists a $\lambda_{*}\in\mathbb{R}$ such
that $\phi+\lambda_{*}\phi_{*}$ does not satisfy (4.8) (along any subsequence
of $\\{\tau_{n}^{\prime}\\}$), then there exists a subsequence
$\\{\tau_{n_{l}}^{\prime}\\}$ of $\\{\tau_{n}^{\prime}\\}$ along which
$\int_{\Sigma_{\tau_{n_{l}}^{\prime}}}\mathcal{E}_{2}[\phi+\lambda_{*}\phi_{*}]\,r^{2}d\sigma
dr\to 0$ as $l\to\infty$. By taking a further subsequence, it follows that
$\int_{\Sigma_{\tau}}\mathcal{E}_{2}[\phi+\lambda\phi_{*}]\,r^{2}d\sigma
dr\geq\frac{b^{2}(\lambda-\lambda_{*})^{2}}{2}$ along a subsequence of
$\\{\tau_{n}^{\prime}\\}$.

Similarly, we can conclude that if for some $k\in\mathbb{N}$ and
$\lambda_{*}\in\mathbb{R}$, $\phi+\lambda_{*}\phi_{*}$ does not satisfy (4.9)
(along any subsequence), then there exist a sequence of times along which
(4.9) does hold for $\phi+\lambda\phi_{*}$ with $b$ replaced by
$\frac{b}{2}|\lambda-\lambda_{*}|$. Hence, the subset of initial data for
which (a), (b) and (c) all do not hold has codimension at least 1. Since we
can repeat this argument with $\phi_{*}$ supported on different azimuthal
modes, this subset must have infinite codimension. ∎

##  5\. A sketch of the proofs and an overview of the main new ideas

In this section, we give an overview of the main steps in the proofs of
Theorems 4.1 and 4.2. We will highlight the new techniques that are introduced
in the present paper and sketch how the novel difficulties listed in §1.3 are
overcome. We will again assume for the sake of convenience that $r_{+}=1$ and
$a=1$.

###  5.1. Part I: Integrated energy estimates in frequency space

The first step towards proving energy decay is to control spacetime integrals
of the form

$\int_{0}^{\infty}\int_{\Sigma_{\tau}}r^{-p}(1-r^{-1})^{q}[|\phi_{m}|^{2}+|\partial\phi_{m}|^{2}]\,r^{2}d\sigma
dr,$

with $p,q>0$ appropriate exponents, in terms of a weighted energy flux along
$\Sigma_{0}$. In view of the presence of superradiance and trapped null
geodesics that cannot be easily localized in physical space, the strategy for
controlling the above integrals involves a Fourier transform in time $t$,
which introduces a time frequency $\omega$. Here, we make use of the
qualitative assumptions (4.6) and (4.7), which guarantee sufficient
integrability of $\phi$ and its derivatives to make sense of the Fourier
transform. The main estimates in Part I therefore take place in _Fourier
space_.

##### Carter separation and the frequency triple $(\omega,m,\Lambda)$

To analyze the Fourier transform, we make use of the _Carter separation
property_ for (3.8), which implies that if we decompose the Fourier transform
of $\phi$ in $t$ into appropriate angular functions
$e^{im\varphi}S_{m\ell}(\theta;a\omega)$, we can separate variables and the
wave equation becomes equivalent to an ODE for the coefficients
$u_{m\ell}(r_{*};\omega)$ in the expansion of the Fourier transform, which
takes the form of a time-independent Schrödinger equation:

$u^{\prime\prime}+(\omega^{2}-V)u=H,$ (5.1)

with $V=V_{\ell m}(r_{*};\omega)$ a potential function and the inhomogeneity
$H=H_{m\ell}(r;\omega)$ determined by the choice of initial data for $\phi$
and $(\cdot)^{\prime}=\frac{d}{dr*}(\cdot)$. The angular functions
$e^{im\varphi}S_{m\ell}$ are eigenfunctions of oblate spheroidal operators
with eigenvalues $\lambda_{m\ell}$, with $\lambda_{m\ell}=\ell(\ell+1)$ in the
special case $a=0$; see §2.4. The translated angular frequency
$\Lambda_{m\ell}=\lambda_{m\ell}+a^{2}\omega^{2}$ forms part of the frequency
triple $(\omega,m,\Lambda)$ that can be used to characterize the relevant
phenomena like superradiance and trapping, which are reflected in the
different shapse of the potential $V$, depending on $m,\omega,\Lambda$.

Integrated energy estimates are obtained at the level of $u$ by multiplying
the equation (5.1) with $f(r;\omega)u+g(r;\omega)u^{\prime}$, for appropriate
(frequency dependent) choices of $f,g$ and then integrating by parts in $r$.
Different frequency ranges are dealt with using different multipliers that are
adapted to the monotonicity properties of $V$ in the frequency regimes under
consideration.

##### Key frequency regimes

The main differences with the frequency-space analysis in the sub-extremal
setting from [DRSR16] appear for _near-superradiant frequencies_ :

$|\omega-m\upomega_{+}|\leq\epsilon\sqrt{\Lambda}$

with $\epsilon\ll 1$. These frequencies can be separated into two sub-regimes:
1) $\Lambda\gg m^{2}$ and 2) $\Lambda\lesssim m^{2}$. An important fact about
regime 1) is that it does not feature the difficulties associated to the
trapping of null geodesics. The main difference with the $|a|<M$ case is the
need for _additional degeneracies_ in the estimates at the event horizon,
related to the higher degeneracy of $V-\omega^{2}$ at $\omega=m\upomega_{+}$
in the $|a|=M$ case. The relevant estimates are proved in §6.5.4.

In general, an important new difficulty in the $|a|=M$ case is that the
phenomenon of trapping and superradiance are coupled in regime 2). However,
since we are assuming $m$ to be bounded or fixed, regime 2) is a bounded
frequency regime. Trapping of null geodesics affects unbounded $\Lambda$ and
it is therefore not an issue in regime 2) in the present paper. See the
comments on the unbounded $m$ case in §1.4.

##### Bounded, near-superradiant frequencies

The main new techniques introduced in the frequency-space analysis in the
present paper are concerned with the regime 2) for bounded $m$. The strategy
here is to express:

$u(r)=W^{-1}\left[u_{\rm inf}(r)\int_{1}^{r}u_{\rm
hor}(r^{\prime})H(r^{\prime})\,\frac{r^{\prime
2}+a^{2}}{\Delta}dr^{\prime}+u_{\rm hor}(r)\int_{r}^{\infty}u_{\rm
inf}(r^{\prime})H(r^{\prime})\,\frac{r^{\prime
2}+a^{2}}{\Delta}dr^{\prime}\right],$ (5.2)

with $u_{\rm hor}$ and $u_{\rm inf}$ solutions to (5.1) with $H=0$, with
appropriate boundary conditions at $r=r_{+}$ and $r=\infty$, respectively.
Estimates for $u$ follow from estimates for $u_{\rm hor}$ and $u_{\rm inf}$,
combined with an estimate for $W^{-1}$. The main difficulties, when compared
with the sub-extremal setting, are the following:

  1. (1)

As part of the proof of mode stability, [TdC20] derived uniform bounds for
$W^{-1}$. In contrast with the analogous estimate for sub-extremal Kerr in
[SR15], these bounds feature constants that blow up as $\omega\to
m\upomega_{+}$. Indeed, in [TdC20, Proposition 6.3] it is shown that

$|W^{-1}|\lesssim\frac{1}{\sqrt{|\omega-m\upomega_{+}|}}.$

  2. (2)

In the limit $\omega\to m\upomega_{+}$, $(V-\omega^{2})(r)$ has a quadratic
zero at $r=r_{+}$, whereas it is linear in the sub-extremal case. For this
reason, $r=r_{+}$ changes character from an _irregular_ singular point to a
_regular_ singular point of (5.1).

Since the singular point $r=r_{+}$ or $r_{*}=-\infty$ of the ODE (5.1) changes
character at $m\upomega_{+}$, one can not appeal directly to standard ODE
estimates to derive pointwise bounds for $u_{\rm inf}$ and $u_{\rm hor}$ for
$r\in(r_{+},\infty)$. By considering instead the resulting ODE with respect to
the rescaled variable $x:=\frac{|\omega-m\upomega_{+}|r_{*}}{|m|}$, it is
possible to obtain uniform pointwise estimates in the near-horizon region
$|r_{*}|\gtrsim\frac{|m|}{|\omega-m\upomega_{+}|}$ that do not breakdown in
the limit $\omega\to m\upomega_{+}$ (however, the corresponding region in
space shrinks in the limit).

We are left with obtaining estimates for the homogeneous solutions $u_{\rm
inf}$ in the region $|r_{*}|\lesssim\frac{|m|}{|\omega-m\upomega_{+}|}$. Here
we face the difficulty that $V-\omega^{2}$ changes sign arbitrarily close to
$r=r_{+}$ for $|\omega-m\upomega_{+}|$ sufficiently small. For this reason, it
is significantly more difficult to derive integrated estimates in the bounded
frequency regime with $|\omega-m\upomega_{+}|\ll 1$, compared to the bounded
frequency regime with $|\omega|\ll 1$. Indeed at $\omega=0$, the singular
point at $r=\infty$ similarly changes character from an irregular to a regular
singular point. However, in that case, $V(r)\sim\Lambda r^{-2}$ as
$r\to\infty$, so one can exploit that $V-\omega^{2}$ is positive for large but
finite $r$-intervals and $\omega^{2}$ suitably small.

The key idea for resolving the difficulty of the change in sign of
$V-\omega^{2}$ is to rescale $u$ by considering

$\check{u}_{\pm,\rm inf}=\frac{u_{\rm inf}}{\mathfrak{w}_{\pm}},$

where $\mathfrak{w}_{\pm}$ give the radial part of the $K$-independent
solutions to (3.8) that were introduced in §3.2. In frequency space,
$\mathfrak{w}_{\pm}$ can be considered solutions to (5.1) with
$\omega=m\upomega_{+}$, with $H\equiv 0$ and appropriate boundary conditions
at $r=r_{+}$. The effect of this rescaling is that the ODE satisfied by
$\check{u}_{\pm,\rm inf}$ features a potential which factors of
$\omega-m\upomega_{+}$ that has a good sign in the region
$|r_{*}|\lesssim\frac{|m|}{|\omega-m\upomega_{+}|}$. A physical space analogue
of this rescaling trick was used in the integrated energy estimates of
[Gaj22]. The analysis of $\check{u}_{\pm,\rm hor}$ and $\check{u}_{\pm,\rm
inf}$ is carried out in §6.5.8.

Employing the identity (5.2), together with estimates for $u_{\rm inf}$ in the
region $|r_{*}|\gtrsim\frac{|m|}{|\omega-m\upomega_{+}|}$ and global estimates
for $u_{\rm inf}$, obtained via the strategy above, then results in estimates
for $u(r)$ that are also valid in the region of the form
$|r_{*}|\gtrsim\frac{|m|}{|\omega-m\upomega_{+}|}$. Extending these to
$|r_{*}|\lesssim\frac{|m|}{|\omega-m\upomega_{+}|}$ involves considering again
the rescaled variables $\check{u}_{\pm}=\frac{u}{\mathfrak{w}_{\pm}}$, as
discussed in the estimates for $\check{u}_{\pm,\rm inf}$.

The main conclusion of the above estimates in frequency space, when combined
with Plancherel’s theorem, is the following integrated energy estimate and
energy boundedness estimate: for all $\epsilon>0$, there exists a constant
$C_{m}>0$, such that

$\int_{\Sigma_{\tau_{2}}}\mathcal{E}_{1-\epsilon}[\phi]\,r^{2}d\sigma
dr+\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}\cap\\{r\leq
2M\\}}(1-Mr^{-1})^{\epsilon}\mathcal{E}_{0}[\phi]+(1-Mr^{-1})^{-1+\epsilon}|K\phi|^{2}\,d\sigma
drd\tau\\\ \leq
C_{m}\int_{\Sigma_{\tau_{1}}}\mathcal{E}_{1+\epsilon}[\phi]\,r^{2}d\sigma dr,$
(5.3)

which is derived in §7.

Note in particular that the energy controlled in the spacetime integral on the
LHS degenerates at $r=M$ and the energy flux term on the LHS features an
energy with an additional factor $(1-Mr^{-1})^{2\epsilon}$, when compared to
the energy flux term on the RHS. There is no loss of derivatives on the RHS
due to the presence of trapped null geodesics, as for bounded $m$, there is no
trapping in the region $r\leq 2M$. We complement the above estimate with an
integrated energy estimate that does feature the region containing trapped
null geodesics and hence loses a derivative on the RHS: let $r_{I}>0$ be
arbitrarily large, then

$\int_{\tau_{1}}^{\tau_{2}}\int_{\Sigma_{\tau}\cap\\{2M\leq r\leq
r_{I}\\}}\mathcal{E}_{2}[\phi]\,d\sigma drd\tau\leq C\sum_{k\leq
1}\int_{\Sigma_{\tau_{1}}}\mathcal{E}_{1+\epsilon}[K^{k}\phi]\,r^{2}d\sigma
dr.$ (5.4)

###  5.2. Part II: Energy decay, $K$-derivatives and $K$-integrals

The remaining part of the proof concerns energy estimates in physical space,
starting from the integrated and energy boundedness estimates (5.3) and (5.4).

##### More $K$-derivatives lead to more decay

The first new difficulty one faces when trying to convert the integrated
energy estimates (5.3) and (5.4) into inverse-polynomial decay estimates for
(weighted) energies along $\Sigma_{\tau}$ is the fact that there is no
hierarchy of estimates with weights $(r-M)^{-p}$ near $r=M$, as is the case in
the region $r\gg M$ for energies with $r^{p}$-weights, or in the extremal
Reissner–Nordström setting near $r=M$.

Indeed, by applying the mean-value theorem to (5.3) and (5.4), combined with
an $r^{p}$-estimate in $r\gg M$, the best that one can get is $\tau^{-1}$
decay for the strongly degenerate energy with energy density
$(1-Mr^{-1})^{\epsilon}\mathcal{E}_{0}[\phi]$ along some sequence
$\\{\tau_{n}\\}$ of times. To get the desired pointwise estimates, one would
need decay for all times $\tau\geq 0$, also for less degenerate energies, with
better decay rates.

The key idea here is that there is an alternative way to obtain a hierarchy of
additional, less degenerate estimates near $r=M$, namely, by replacing $\phi$
with the time derivative $K\phi$. Indeed, from (3.8) if follows that,
schematically,

$\mathcal{E}_{p}[K\phi]\lesssim\mathcal{E}_{p-2}[(r-M)Y\phi]+\mathcal{E}_{p-2}[\not{\nabla}_{\mathbb{S}^{2}}\phi]+\mathcal{E}_{p-2}[K\phi].$

Hence, $K$ derivatives can be traded for additional degenerate factors of
$r-M$, at the expense of requiring estimates additionally estimates for the
derivatives $(r-M)Y\phi$ and $\not{\nabla}_{\mathbb{S}^{2}}\phi$. Fortunately,
the estimates for $\phi$ extend naturally to $(r-M)Y\phi$ and
$\not{\nabla}_{\mathbb{S}^{2}}\phi$. In [Gaj22], energy decay with the desired
rates is similarly obtained by replacing $\phi$ with time derivatives
$T^{k}\phi$. The additional difficulty in the present setting is that, in
contrast with [Gaj22], there is no (degenerate) energy boundedness estimate
available without additional degeneracies on the LHS.

Note that the decay rate obtained by repeatedly applying $K$ is limited by the
length of the hierarchy of $r^{p}$-estimates near infinity, so it is not
possible to obtain arbitrarily fast decay for a suitable number of
$K$-derivatives. The energy decay estimates for $K$-derivatives of $\phi$ are
derived in §9.

##### The difference function $\phi-\Pi$

It remains to apply the energy decay estimates for $K\phi$ to obtain energy
decay estimates for $\phi$. In view of the analogous steps in [Gaj22], it
would be natural to think of $\phi$ as a $K$-derivative: $\phi=K(K^{-1}\phi)$,
where, formally,

$K^{-1}\phi(\tau,r,\theta,\widetilde{\varphi})=-\int_{\tau}^{\infty}\phi(\tau^{\prime},r,\theta,\widetilde{\varphi})\,d\tau^{\prime}$

and use the energy decay estimates for $K$-derivatives from the previous step.

This strategy does not quite work. Indeed, in view of the decay rates in
Theorem 1.2, we will prove $\tau^{-1}$-decay for $\phi$ away from $r=M$, but
this is not integrable, so the time integral does not make sense.

Instead, we consider

$\widehat{\phi}=\phi-\Pi$

with $\Pi$ the function defined in §4, which will encode the precise, global
leading-order late-time behaviour of $\phi$. As can be seen in Theorem 4.1,
$\widehat{\phi}$ _does_ decay with an integrable rate in time and we are able
to make sense of $K^{-1}\widehat{\phi}$. Since $\widehat{\phi}$ satisfies an
inhomogeneous wave equation (3.1) with $G=-\square_{g}\Pi$, the inhomogeneity
has to decay suitably fast in $\tau$ and $r$ in order to be able to repeat the
integrated energy estimates and energy decay estimates for $\widehat{\phi}$
instead of $\phi$.

The consideration of $\widehat{\phi}$ moreover allows us to go beyond (sharp)
upper bound decay estimates for $\phi$ and to derive the _precise_ leading-
order late-time behaviour of $\phi$. Indeed, if we can show that
$\widehat{\phi}$ decays _faster_ than $\Pi$, it means that the leading-order
late-time asymptotics or $\phi$ coincide with the leading-order late-time
asymptotics of $\Pi$.

#####  $K$-inversion and elliptic estimates

Since we do not know _a priori_ that $\widehat{\phi}$ decays suitably fast to
be able to make sense of $K^{-1}\phi$, defined as an integral in time, we
instead construct the initial data for $K^{-1}\phi$ by solving an elliptic
type equation:

$\mathcal{L}K^{-1}\widehat{\phi}|_{\Sigma_{0}}=F[\phi|_{\Sigma_{0}}],$

where $F[\phi|_{\Sigma_{0}}]$ can be split as follows:

$F[\phi|_{\Sigma_{0}}]=F_{\rm hom}[\phi]+F_{\Pi},$

with $F_{\rm hom}[\phi|_{\Sigma_{0}}]$ depending on the initial data for
$\phi$ along $\Sigma_{0}$ and $F_{\Pi}$ depending only on $\Pi$.

More concretely, we construct the projections $(K^{-1}\widehat{\phi})_{\ell
m}$ to the angular functions $S_{m\ell}(\theta)$, defined in §2.4, and derive
uniform elliptic estimates for the operator $\mathcal{L}$ for large values of
$\ell$ to be able to sum $(K^{-1}\widehat{\phi})_{\ell m}$ over $\ell$. These
elliptic estimates are adapted to the vector fields $(K,\widetilde{X})$ and
require more decay of the initial data for $\phi$ as $r\to\infty$ than
elliptic estimates adapted to the $(T,X)$ vector fields.

The construction of $(K^{-1}\widehat{\phi})_{\ell m}$ amounts to solving an
ODE. This ODE can easily be solved by considering instead the rescaled
variables

$\frac{(K^{-1}\widehat{\phi})_{\ell m}}{w_{\infty,m\ell}},$

with $w_{\infty,m\ell}(r)$ the radial parts of the $K$-invariant stationary
solutions introduced in to (3.8) that were introduced in §3.2, which are
closely related to the functions $\mathfrak{w}_{\pm}(r)$ described in Part I
above; see §3.2. The ODE for these rescaled variables can be put in the
folllowing form for $a=M=1$:

$\widetilde{X}\left(\Delta
w_{\infty,m\ell}^{2}e^{-im\int_{r_{0}}^{r}\frac{x+1}{x-1}\,dx}\widetilde{X}\left(\frac{(K^{-1}\widehat{\phi})_{\ell
m}}{w_{\infty,m\ell}}\right)\right)=e^{-im\int_{r_{0}}^{r}\frac{x+1}{x-1}\,dx}w_{\infty,m\ell}F_{m\ell},$

and can therefore be solved by simply integrating twice in the
$\widetilde{X}$-direction. The requirement that the integral of the RHS over
$r\in(M,\infty)$ vanishes for $\alpha_{m\ell}\leq\frac{1}{4}$, fixes the
constants $\mathfrak{h}_{m\ell}$ appearing in the definition of $\Pi$ and
therefore also the initial-data-dependent factors in the coefficients in front
of the late-time tails of $\phi$. The decay rates in the late-time tails for
$\phi$ are then encoded in the behaviour of $(K^{-1}\widehat{\phi})$ as
$r\downarrow M$. The data corresponding to the $K$-integral of $\phi_{m}$ and
the corresponding elliptic estimates are derived in §10.

###  5.3. Part III: Instability follows from stability

Part I and Part II above lead to a proof of Theorem 4.1, which is the main
ingredient in the proof of Theorem 4.2.

First, non-decay of the non-degenerate energy
$\int_{\Sigma_{\tau}}\mathcal{E}_{2}[\phi]\,r^{2}d\sigma dr$ follows from
decay of the energy of the difference function
$\int_{\Sigma_{\tau}}\mathcal{E}_{2}[\widehat{\phi}]\,r^{2}d\sigma dr$,
combined with non-decay of

$\int_{\Sigma_{\tau}}\mathcal{E}_{2}[\Pi]\,r^{2}d\sigma dr,$

which in turn follows by plugging in the explicit expressions for $\Pi$.

To obtain pointwise blow-up of $Y\phi_{m}$ we use that $Y\phi_{m}$ satisfies
the following equation along $\mathcal{H}^{+}$, obtained by simply evaluating
(3.8) along $\mathcal{H}^{+}$:

$K(-4Y\phi_{m\ell})=(\Lambda_{m\ell}+m^{2}-im)\phi_{m\ell}+(\sin^{2}\theta(K^{2}\phi-
imK\phi))_{m\ell}+2(1+im)K\phi_{m\ell}.$

Since Theorem 4.1 gives the precise late-time tails of $\phi_{m\ell}$ and
implies that $K^{2}\phi$ and $K\phi$ decay faster, we integrate the above
equation to obtain the precise late-time behaviour of $Y\phi_{m}$. Due to the
possible oscillation present in the late-time behaviour of $|Y\phi_{m}|$, it
will not satisfy a uniform lower bound, but we can identify a sequence of
times $\\{\tau_{n}\\}$ along which blow-up occurs. Stronger blow-up of higher-
order $Y$-derivatives can similarly be obtained (along a sequence of times) by
plugging in the late-time asymptotics of lower-order $Y$-derivatives into the
equation along $\mathcal{H}^{+}$ that are obtained by commuting (3.8) with
$Y^{k}$ and evaluating the resulting equation along $\mathcal{H}^{+}$.

The instability results in the paper are all derived in §12.

##  6\. Frequency-space analysis

In this section, we derive the estimates in frequency space that are required
to obtain integrated energy estimates. In order to be able to take a Fourier
transform, we will introduce a _global assumption_ on solutions $\phi$ arising
from smooth and compactly supported initial data on $\Sigma_{0}$. Without loss
of generality, we will assume in this section that $r_{+}=1$ and $a\geq 0$.
Indeed, the $a<0$ can be included in the analysis by transforming
$\varphi_{*}\to-\varphi_{*}$.

First we make the following integrability assumption on the inhomogeneity $G$:

######  Assumption 1.

Let $m\in\mathbb{Z}$ and let $G_{m}$ denote the projection of $G$ to the
$m$-th azimuthal mode. Then for all $n_{1},n_{2},n_{3}\in\mathbb{N}_{0}$:

$\displaystyle\int_{0}^{\infty}\int_{\Sigma_{\tau}}|\rho^{2}G_{m}|^{2}\,d\sigma
drd\tau<$ $\displaystyle\>\infty,$ (6.1)
$\displaystyle\sup_{r\in[r_{0},r_{1}]}\int_{0}^{\infty}\int_{\mathbb{S}^{2}}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}X^{n_{2}}T^{n_{3}}G_{m}|^{2}(\tau,r,\theta,\varphi)\,d\sigma
d\tau<$ $\displaystyle\>\infty$ (6.2)

We make the following global assumption on solutions $\phi$ to (3.1):

######  Assumption 2.

Let $\phi$ be a solution to (3.1) with $G=0$, arising from initial data
$(\phi_{0},\phi_{0}^{\prime})\in C_{c}^{\infty}(\Sigma_{0})\times
C_{c}^{\infty}(\Sigma_{0})$. Then for all $m\in\mathbb{Z}$ and
$r_{1}>r_{0}>r_{+}$:

$\displaystyle\int_{0}^{\infty}\int_{\Sigma_{\tau}\cap\\{r\leq
r_{1}\\}}|\phi_{m}|^{2}\,d\sigma drd\tau<$ $\displaystyle\>\infty,$ (6.3)
$\displaystyle\int_{0}^{\infty}\int_{\Sigma_{\tau}\cap\\{r_{0}\leq r\leq
r_{1}\\}}|X\phi_{m}|^{2}+|T\phi_{m}|^{2}+|\not{\nabla}_{\mathbb{S}^{2}}\phi_{m}|^{2}\,d\sigma
drd\tau<$ $\displaystyle\>\infty.$ (6.4)

Note that for $m=0$ and $|a|=M$, the Assumptions 6.3 and 6.4 hold by the decay
results in [Are12]. For $|a|<M$, the Assumptions 6.3 and 6.4 hold for all
$m\in\mathbb{Z}$ by the decay results in [DRSR16].

###  6.1. Carter separation

We first introduce the class of _sufficiently integrable functions_ :

######  Definition 6.1.

A function $f:\mathring{\mathcal{M}}_{M,a}\to\mathbb{C}$ is _sufficiently
integrable_ if: for all $n_{1},n_{2},n_{3}\in\mathbb{N}_{0}$ and
$r_{1}>r_{0}>r_{+}$:

$\displaystyle\sup_{r\in[r_{0},r_{1}]}\int_{\mathbb{R}}\int_{\mathbb{S}^{2}}\left[|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}{\mathfrak{X}}_{*}^{n_{2}}T^{n_{3}}f|^{2}+|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}{\mathfrak{X}}_{*}^{n_{2}}T^{n_{3}}\square_{g_{M,a}}f|^{2}\right](t,r_{*},\theta,\varphi)\,d\sigma
dt<$ $\displaystyle\>\infty,$ (6.5)
$\displaystyle\int_{r_{+}}^{r_{0}}\int_{\mathbb{R}}\int_{\mathbb{S}^{2}}|({\mathfrak{X}}_{*}-K)f|^{2}\,\frac{r^{2}+a^{2}}{\Delta}d\sigma
dtdr+\int_{r_{1}}^{\infty}\int_{\mathbb{R}}\int_{\mathbb{S}^{2}}|({\mathfrak{X}}_{*}+T)f|^{2}\,d\sigma
dtdr<$ $\displaystyle\>\infty.$ (6.6)

Now, let $\phi:\mathcal{M}_{M,a}\to\mathbb{C}$ be a solution to (3.1) with
$G=0$ arising from smooth and compactly supported initial data, and let
$\Pi:\mathcal{M}_{M,a}\to\mathbb{C}$ be a function that will be fixed later in
§10.5. We denote $\widehat{\phi}=\phi-\Pi$ and $G=-\square_{g}\Pi$. Note that
$\widehat{\phi}$ is a solution to (3.1) with this choice of $G$.

Let $\xi:\mathcal{M}_{M,a}\to\mathbb{R}$ be a smooth cut-off function, such
that $\xi\equiv 0$ in $\\{\tau\leq 0\\}$ and $\xi\equiv 1$ in $\\{\tau\geq
1\\}$. Define:

$\upphi=\xi\cdot\widehat{\phi}.$

Then

$\square_{g_{M,a}}\upphi=F_{\xi}+\xi G=:F,$ (6.7)

with

$\rho^{2}F_{\xi}=2(r^{2}+a^{2}-\mathbbm{h}\Delta)T(\xi)X\phi+[a^{2}\sin^{2}\theta+\mathbbm{h}(\Delta\mathbbm{h}-2(r^{2}+a^{2}))](2T(\xi)T\phi+T^{2}(\xi)\phi)+2a(1-\mathbbm{h})T(\xi)\Phi\phi\\\
+(2r-(\Delta\mathbbm{h})^{\prime})T(\xi)\phi\\\
=-2T(\xi)(r^{2}+a^{2})^{\frac{1}{2}}(1+O(r^{-1}))X\psi+O(1)T(\xi)\phi+[a^{2}\sin^{2}\theta+\mathbbm{h}(\Delta\mathbbm{h}-2(r^{2}+a^{2}))](2T(\xi)T\phi+T^{2}(\xi)\phi)\\\
+2a(1-\mathbbm{h})T(\xi)\Phi\phi\ ,$ (6.8)

which is supported in $\\{0\leq\tau\leq 1\\}$.

######  Proposition 6.1.

Assuming (6.3) and (6.4), the function $\upphi_{m}$ satisfying (6.7) is
sufficiently integrable if $\xi\cdot\Pi$ is sufficiently integrable and
$G=-\square_{g}\Pi$ satisfies (6.1).

###### Proof.

By standard elliptic estimates in the bounded region $1<r_{0}<r_{1}\leq\infty$
(away from the event horizon and for bounded $m$) and the fundamental theorem
of calculus, we apply the assumption on $\Pi$ and (6.1) to conclude that

$\sum_{n_{1}+n_{2}+n_{3}\leq
N}\sup_{r\in[r_{0},r_{1}]}\int_{\mathbb{R}}\int_{\mathbb{S}^{2}}|\not{\nabla}_{\mathbb{S}^{2}}^{n_{1}}\partial_{r_{*}}^{n_{2}}\partial_{t}^{n_{3}}\upphi_{m}|^{2}(t,r_{*},\theta,\varphi)\,dt<\infty$

holds if (6.4) holds with $\phi_{m}$ replaced by $K^{n}\phi_{m}$, where $n\leq
N$. Since $K^{n}\phi_{m}$ is also a solution to (3.1) with $G=0$ arising from
smooth, compactly supported data, (6.4) applies also to $K^{n}\phi_{m}$ and,
applying (6.1) once more to estimate $\square_{g}\upphi_{m}$, we conclude that
(6.5) holds for $f=\upphi_{m}$.

We have additionally that

$\int_{r_{1}}^{\infty}\int_{\mathbb{R}}\int_{\mathbb{S}^{2}}|({\mathfrak{X}}_{*}+T)\upphi|^{2}\,d\sigma
dtdr<\infty$

by the sufficient integrability assumption on $\Pi$ together with a standard
$r^{p}$-estimate with $p=1$ in the region $r\geq r_{I}$, for $r_{I}$
sufficiently large.

We similarly obtain

$\int_{r_{1}}^{\infty}\int_{\mathbb{R}}\int_{\mathbb{S}^{2}}|({\mathfrak{X}}_{*}-K)\upphi|^{2}\,\frac{r^{2}+a^{2}}{\Delta}d\sigma
dtdr<\infty$

by applying Corollary 8.2 together with Assumption 6.3.

∎

###  6.2. The frequency-localized ODE

For sufficiently integrable $\upphi$, the Fourier transform in $t$ along
constant $(r,\theta,\varphi)$ of $\upphi$ is well-defined, and we will denote
it by $\widehat{\upphi}$. Then we can express:

$\upphi(t,r,\theta,\varphi)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{i\omega
t}\widehat{\upphi}(\omega,r,\theta,\varphi)\,d\omega,$

where the above identity is to be interpreted in the function space
$L^{2}_{t}L^{2}_{\mathbb{S}^{2}}$.

We will moreover define:

$\widetilde{\omega}:=\omega-m\upomega_{+}.$

The frequency $\omega$ is the time-frequency with respect to the flow along
$T$, whereas the shifted frequency $\widetilde{\omega}$ may be interpreted as
the time-frequency with respect to the flow along $K$.

Subsequently, we can decompose $\widehat{\upphi}$ as follows in terms of the
orthonormal basis of oblate spheroidal harmonics in $L^{2}(\mathbb{S}^{2})$
(see §2.4):

$\widehat{\upphi}(\omega,r,\theta,\varphi)=\sum_{\begin{subarray}{c}\ell\in\mathbb{N}_{0}\\\
|m|\leq\ell\end{subarray}}\widehat{\upphi}^{(a\omega)}_{m\ell}(r)S_{m\ell}(\theta;a\omega)e^{im\varphi},$

which is to be interpreted as an identity in the function space
$L^{2}_{\omega}L^{2}_{\mathbb{S}^{2}}$. To simplify the notation, we introduce
the shorthand notation:

$\sum_{\begin{subarray}{c}\ell\in\mathbb{N}_{0}\\\
|m|\leq\ell\end{subarray}}\to\sum_{m\ell}.$

Hence, we obtain the following identity in the space
$L^{2}_{t}L^{2}_{\mathbb{S}^{2}}$:

$\upphi(t,r,\theta,\varphi)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\sum_{m\ell}\widehat{\upphi}^{(a\omega)}_{m\ell}(r)S_{m\ell}(\theta;a\omega)e^{i(m\varphi+\omega
t)}\,d\omega.$

A remarkable property of the Kerr spacetime is that the equation (6.7) reduces
to an ODE for the variable $\widehat{\upphi}^{(a\omega)}_{m\ell}(r)$. This is
known as _Carter’s separation_.

######  Proposition 6.2 (Carter’s separation).

Let $\upphi$ be a sufficiently integrable solution to (6.7). Then
$\widehat{\upphi}^{(a\omega)}_{m\ell}$ satisfies the following equality in
$L^{2}_{\omega}l^{2}_{ml}$:

$\Delta(\rho^{2}F)^{(a\omega)}_{m\ell}=\Delta\frac{d}{dr}\left(\Delta\frac{d\widehat{\upphi}^{(a\omega)}_{m\ell}}{dr}\right)+[a^{2}m^{2}+(r^{2}+a^{2})^{2}\omega^{2}-4Mra\omega
m-\Delta(\lambda^{(a\omega)}_{m\ell}+a^{2}\omega^{2})]\widehat{\upphi}^{(a\omega)}.$

Furthermore, if we rescale

$\displaystyle u^{a\omega}_{m\ell}:=$
$\displaystyle\>\sqrt{r^{2}+a^{2}}\widehat{\upphi}^{a\omega}_{m\ell},$
$\displaystyle H^{(a\omega)}_{m\ell}:=$
$\displaystyle\>\Delta(r^{2}+a^{2})^{-\frac{3}{2}}(\rho^{2}F)^{(a\omega)}_{m\ell},$

then $u^{a\omega}_{m\ell}$ satisfies the following in
$L^{2}_{\omega}l^{2}_{ml}$:

$\frac{d^{2}u^{(a\omega)}_{m\ell}}{dr_{*}^{2}}+(\omega^{2}-V_{ml}^{(a\omega)})u^{a\omega}_{m\ell}=H^{(a\omega)}_{m\ell},$
(6.9)

with

$V_{m\ell}^{(a\omega)}=(r^{2}+a^{2})^{-2}\left[4Ma\omega
r-a^{2}m^{2}+\Delta(\lambda^{(a\omega)}_{m\ell}+a^{2}\omega^{2})\right]+(r^{2}+a^{2})^{-3}\Delta\left[3r^{2}-4Mr+a^{2}\right]\\\
-3r^{2}\Delta^{2}(r^{2}+a^{2})^{-4}.$

We denote:

$\Lambda_{m\ell}^{(a\omega)}=\lambda^{(a\omega)}_{m\ell}+a^{2}\omega^{2}.$

From §2.4, it follows that for $\omega=m\upomega_{+}$:

$\Lambda_{m\ell}^{(am\upomega_{+})}=\Lambda_{+,m\ell}.$

We moreover have that

$\displaystyle\lim_{m\widetilde{\omega}\to
0}\frac{1}{a^{2}m\widetilde{\omega}}(\Lambda_{m\ell}^{(a\omega)}-\Lambda_{m\ell}^{(am\upomega_{+})})<$
$\displaystyle\>\infty,$ (6.10) $\displaystyle\lim_{a\to
M}\frac{1}{(a^{2}-M^{2})m^{2}\omega^{2}}\left(\Lambda_{m\ell}^{(a\omega)}-\Lambda_{m\ell}^{(\omega)}\right)<$
$\displaystyle\>\infty,$ (6.11)

by the smooth dependence of $\Lambda_{m\ell}^{(a\omega)}$ on $a\omega$; see
for example [MS54, §3.22, Proposition 1].

###  6.3. Boundary conditions

In this section, we will make use of the regularity assumptions at
$\mathcal{H}^{+}$ and $\mathcal{I}^{+}$ encoded in the definition of
sufficient integrability in Definition 6.1, in order to derive boundary
conditions for the ODE (6.9).

######  Lemma 6.3.

Let $\upphi$ be a sufficiently integrable solution to (6.7). Then
$u^{(a\omega)}_{m\ell}$ satisfies the following boundary conditions: there
exists sequences $\\{r_{i}\\}$ and $\\{\tilde{r}_{i}\\}$, with
$r_{i}\to\infty$ and $\tilde{r_{i}}\downarrow 1$ as $i\to\infty$, such that:

$\displaystyle\lim_{i\to\infty}\left|\frac{du^{(a\omega)}_{m\ell}}{dr_{*}}-i\omega
u^{(a\omega)}_{m\ell}\right|(r_{i})=$ $\displaystyle\>0,$ (6.12)
$\displaystyle\lim_{i\to\infty}\left|\frac{du^{(a\omega)}_{m\ell}}{dr_{*}}+i\widetilde{\omega}u^{(a\omega)}_{m\ell}\right|(\tilde{r}_{i})=$
$\displaystyle\>0,$ (6.13)

for almost every $\omega\in\mathbb{R}$.

###### Proof.

By (6.6) combined with Plancherel, it follows that: for $r_{0}>r_{+}$ and
$r_{1}<\infty$

$\int_{r_{+}}^{r_{0}}\int_{\mathbb{R}}\left|\frac{du^{(a\omega)}_{m\ell}}{dr_{*}}+i\widetilde{\omega}u^{(a\omega)}_{m\ell}\right|^{2}\,d\omega
dr+\int_{r_{1}}^{\infty}\int_{\mathbb{R}}\left|\frac{du^{(a\omega)}_{m\ell}}{dr_{*}}-i\omega
u^{(a\omega)}_{m\ell}\right|^{2}\,d\omega dr<\infty.$

Let $\rho_{n}$, $i\geq 1$ be a strict monotonically increasing sequence with
$\rho_{1}=r_{1}$ and $\rho_{i}\to\infty$ as $i\to\infty$. By applying the
mean-value theorem with $r\in[\rho_{i},\rho_{i+1}]$, we can find a sequence
$\rho^{\prime}_{i}$ such that $\rho_{i}^{\prime}\to\infty$ and

$\lim_{i\to\infty}\int_{\mathbb{R}}\left|\frac{du^{(a\omega)}_{m\ell}}{dr_{*}}-i\omega
u^{(a\omega)}_{m\ell}\right|^{2}(\rho_{i}^{\prime})\,d\omega=0.$

Convergence of a sequence with respect to the
$L^{2}_{\omega}(\mathbb{R})$-norm implies that there exists a subsequence
along which the sequence convergence pointwise for almost every $\omega$. This
concludes (6.12).

The limit (6.13) follows similarly, by taking instead $\rho_{n}$, $i\geq 1$ to
be a strict monotonically decreasing sequence with $\rho_{1}=r_{0}$ and
$\rho_{i}\to 1$ as $i\to\infty$. ∎

As the goal of Carter’s separation is to arrive at integrated energy
estimates, it is sufficient to restrict the consideration to a subset of
frequencies $\omega$ with full Lebesgue measure. In the proposition below, we
show that the limits in Lemma 6.3 can be interpreted as boundary conditions
for smooth solutions to the (6.9), for almost every $\omega\in\mathbb{R}$.

######  Proposition 6.4.

Let $\upphi$ be a sufficiently integrable solution to (6.7). Then, for all
$m,\ell$, with $\ell\in\mathbb{N}_{0}$ and
$m\in\mathbb{Z}_{\geq-\ell}\cap\mathbb{Z}_{\leq\ell}$, and for almost every
$\omega\in\mathbb{R}$:

  1. _(i)_

$H^{(a\omega)}_{m\ell}$ is smooth on $(\infty,\infty)_{r_{*}}$ and the
corresponding solution $u^{(a\omega)}_{m\ell}$ to (6.9) is smooth and
satisfies the following boundary conditions:

$\displaystyle\lim_{r_{*}\to\infty}\left|\frac{du^{(a\omega)}_{m\ell}}{dr_{*}}-i\omega
u^{(a\omega)}_{m\ell}\right|(r_{*})=$ $\displaystyle\>0,$ (6.14)
$\displaystyle\lim_{r_{*}\to-\infty}\left|\frac{du^{(a\omega)}_{m\ell}}{dr_{*}}+i\widetilde{\omega}u^{(a\omega)}_{m\ell}\right|(r_{*})=$
$\displaystyle\>0.$ (6.15)

  2. _(ii)_

$H^{(a\omega)}_{m\ell}$ satisfies:

$\int_{r_{+}}^{\infty}\Delta^{-2}r^{4}|H^{(a\omega)}_{m\ell}|^{2}\,dr<\infty.$
(6.16)

###### Proof.

We refer to [DRSR16, Lemma 5.4.1] for a proof of (i), with the minor
modification that we can restrict to the region $r\in[r_{0},\infty)$, since we
are proving smoothness in $(-\infty,\infty)_{r_{*}}$.

Note that by Plancherel

$\lim_{n\to\infty}\sum_{m\ell}\int_{\mathbb{R}}\int_{r_{+}+\frac{1}{n}}^{n}\Delta^{-2}r^{4}|H^{(a\omega)}_{m\ell}|^{2}\,drd\omega=\int_{0}^{\infty}\int_{\Sigma_{\tau}}r^{-2}|\rho^{2}F|^{2}\,d\sigma
drd\tau.$

By the local-in-time weighted energy estimates from Proposition 3.3 and the
assumption (6.1), we can further estimate

$\begin{split}\int_{0}^{\infty}\int_{\Sigma_{\tau}}r^{-2}|\rho^{2}F|^{2}\,d\sigma
drd\tau\leq&\>\int_{0}^{1}\int_{\Sigma_{\tau}}r^{-2}|\rho^{2}F_{\xi}|^{2}\,d\sigma
drd\tau+C\int_{0}^{\infty}\int_{\Sigma_{\tau}}|rG|^{2}\,d\sigma drd\tau\\\
\leq&\>C\int_{\Sigma_{0}}\mathcal{E}_{2}[\widehat{\phi}]\,r^{2}d\sigma
dr+C\int_{0}^{\infty}\int_{\Sigma_{\tau}}|rG|^{2}\,d\sigma
drd\tau<\infty.\end{split}$

Hence,
$\sum_{m\ell}\int_{r_{+}+\frac{1}{n}}^{n}\Delta^{-2}r^{4}|H^{(a\omega)}_{m\ell}|^{2}\,dr$
is Cauchy in $L^{1}_{\omega}(\mathbb{R})$ and therefore has a subsequence that
converges pointwise for almost every $\omega$. As the original sequence is
monotonically increasing in $n$, it must also converge pointwise for almost
every $\omega$, which concludes (6.16). ∎

###  6.4. The potential $V_{ml}^{(a\omega)}$

In this section, we derive the main properties and decompositions of the
potential $V_{ml}^{(a\omega)}$ that play a role in the rest of the analysis in
this section.

####  6.4.1. The main potentials

With the choice $r_{+}=1$, we can express:

$\displaystyle M=$ $\displaystyle\>\frac{1}{2}(1+a),$
$\displaystyle\upomega_{+}=$ $\displaystyle\>\frac{a}{1+a^{2}},$
$\displaystyle\Delta=$ $\displaystyle\>(r-1)(r-a^{2}).$

and $0\leq a\leq 1$ remains our only free parameter. In the sections below we
will suppress the super- and subscripts in the notation
$V_{m\ell}^{(a\omega)}$ and $\lambda_{m\ell}^{(a\omega)}$; we will simply
write $V$ and $\Lambda$, respectively.

##### The potentials $V_{0}$ and $V_{1}$

The potential $V(r)$ can be split as follows:

$\displaystyle V(r)=$
$\displaystyle\>V_{0}(r)+V_{1}(r),\quad\textnormal{with}$ (6.17)
$\displaystyle V_{0}(r)=$
$\displaystyle\>(r^{2}+a^{2})^{-2}\left[2ma(1+a^{2})\omega
r-a^{2}m^{2}+\Lambda\Delta\right],$ (6.18) $\displaystyle V_{1}(r)=$
$\displaystyle\>(r^{2}+a^{2})^{-3}\Delta(3r^{2}-2(1+a^{2})r+a^{2})-3(r^{2}+a^{2})^{-4}\Delta^{2}r^{2},$
(6.19) $\displaystyle\Lambda:=$ $\displaystyle\>\lambda+a^{2}\omega^{2}.$

Note that $V_{1}\geq 0$.

##### The potential $\widetilde{V}$

We define the shifted potential:

$\widetilde{V}-\widetilde{\omega}^{2}:=V-\omega^{2}.$

We can express:

$\displaystyle\widetilde{V}(r)=$
$\displaystyle\>\widetilde{V}_{0}(r)+\widetilde{V}_{1}(r),\quad\textnormal{with}$
$\displaystyle\widetilde{V}_{0}(r)=$
$\displaystyle\>-2(1+a^{2})^{-1}ma\widetilde{\omega}-a^{2}m^{2}(1+a^{2})^{-2}+(r^{2}+a^{2})^{-2}\left[2ma(1+a^{2})\widetilde{\omega}r+m^{2}a^{2}(2r-1)+\Lambda\Delta\right],$
$\displaystyle\widetilde{V}_{1}(r)=$ $\displaystyle\>V_{1}(r).$

##### The potential $V_{q,\varpi}$

Let $q,\varpi\in\mathbb{R}$. Consider the following modified potential that
arises when considering the ODE for the rescaled variable
$v^{(a\omega)}_{m\ell}=e^{i(\varpi+\widetilde{\omega})r_{*}}\Delta^{iq}u^{(a\omega)}_{m\ell}$
(see §6.5.1 for the motivation for considering such a rescaling):

$V_{q,\varpi}(r):=\widetilde{V}-\widetilde{\omega}^{2}+\left[(\widetilde{\omega}+\varpi)^{2}+2q(\widetilde{\omega}+\varpi)\frac{d\Delta}{dr}(r^{2}+a^{2})^{-1}+q^{2}\left(\frac{d\Delta}{dr}\right)^{2}(r^{2}+a^{2})^{-2}\right]$

In order for the leading-order terms in $r-1$ of $V_{q,\varpi}(r)$ to cancel,
we will need to make the following choices for the constants $q$ and $\varpi$:

$\displaystyle q=$ $\displaystyle\>\frac{(3-a^{2})m}{4a},$ (6.20)
$\displaystyle\varpi=$
$\displaystyle\>-\frac{1-a^{2}}{1+a^{2}}q=-\frac{(1-a^{2})(3-a^{2})m}{4a(1+a^{2})^{2}}.$
(6.21)

##### The potentials $\widetilde{V}_{\rm stat}$ and $\check{V}$

We introduce the shorthand notation:

$\Lambda_{+}=\Lambda_{+,m\ell}$

and split $\widetilde{V}$ as follows:

$\widetilde{V}=\widetilde{V}_{\rm stat}+\check{V},$

with

$(1+a^{2})^{2}(r^{2}+a^{2})^{2}\widetilde{V}_{\rm
stat}(r):=\sum_{k=2}^{4}\frac{1}{k!}\frac{d^{k}((r^{2}+a^{2})^{2}(1+a^{2})^{2}\widetilde{V_{0}})}{dr^{k}}\Big{|}_{r=1,\widetilde{\omega}=0,\Lambda=\Lambda_{+}}\cdot(r-1)^{k}+V_{1}(r)|_{a=1}$
(6.22)

When $a=1$, the potential $V_{\rm stat}$ may be interpreted as the potential
corresponding to stationary solutions with respect to the vector field $K$
($\widetilde{\omega}=0$); see §3.2.

####  6.4.2. Properties of the potentials

In this section, we will analyze the main quantitative and qualitative
properties of the potentials introduced in §6.4.1.

The lemma below concerns precise quantitiative properties of the potentials in
the form of their asymptotic behaviour in the limits $r\downarrow 1$ and
$r\to\infty$.

######  Lemma 6.5.

The potentials defined in §6.4.1 have the following asymptotic behaviour:

$\displaystyle\widetilde{V}_{0}(r)=$
$\displaystyle\>(1+a^{2})^{-3}\left[-8am\widetilde{\omega}+2(\Lambda-m^{2})(1-a^{2})+(2m^{2}-\Lambda+2am\widetilde{\omega})(1-a^{2})^{2}\right](r-1)+$
(6.23)
$\displaystyle+\>(1+a^{2})^{-4}\Big{[}4(\Lambda-2m^{2})+6(3m^{2}-2\Lambda+4am\widetilde{\omega})(1-a^{2})$
$\displaystyle+(5\Lambda-10m^{2}-12am\widetilde{\omega})(1-a^{2})^{2}\Big{]}(r-1)^{2}$
$\displaystyle+\Lambda O((r-1)^{3}),$ $\displaystyle\widetilde{V}_{0}(r)=$
$\displaystyle\>(1+a^{2})^{-3}\left[-8am\widetilde{\omega}-2m^{2}(1-a^{2})+2(m^{2}+am\widetilde{\omega})(1-a^{2})^{2}\right](r-1)$
(6.24)
$\displaystyle+\>(1+a^{2})^{-4}\left[-8m^{2}+2(9m^{2}+12ma\widetilde{\omega})(1-a^{2})-2(5m^{2}+6am\widetilde{\omega})(1-a^{2})^{2}\right](r-1)^{2}$
$\displaystyle+\>\Lambda(r^{2}+a^{2})^{-2}\Delta+(m^{2}+|m\widetilde{\omega}|)O((r-1)^{3}),$
$\displaystyle\widetilde{V}_{1}(r)=$
$\displaystyle\>(1+a^{2})^{-3}(1-a^{2})^{2}(r-1)+2(1+a^{2})^{-4}\left[3(1-a^{2})-4(1-a^{2})^{2}+(1-a^{2})^{3}\right](r-1)^{2}$
(6.25) $\displaystyle+O((r-1)^{3}),$ $\displaystyle V_{q,\varpi}(r)=$
$\displaystyle\>\frac{\left(\Lambda-\frac{2a^{2}m^{2}}{(1+a^{2})}\right)\Delta}{(r^{2}+a^{2})^{2}}-m\widetilde{\omega}(r-1)^{2}+(r-1)\Delta
O(m^{2}+|m\widetilde{\omega}|),$ (6.26) $\displaystyle\check{V}=$
$\displaystyle\>\left[-m\widetilde{\omega}+\frac{1}{4}(1-a^{2})(\Lambda-m^{2}+O(|m\widetilde{\omega}|))+(1-a^{2})^{2}O(|\Lambda|)\right](r-1)$
(6.27)
$\displaystyle+O(|m\widetilde{\omega}|+\Lambda(1-a^{2})+|\Lambda-\Lambda_{+}|)(r-1)^{2},$
$\displaystyle\widetilde{V}_{\rm stat}(r)=$
$\displaystyle\>\left[\frac{1}{4}(\Lambda_{+}-2m^{2})+\Lambda_{+}O((1-a^{2}))\right](r-1)^{2}+\Lambda_{+}O((r-1)^{3}),$
(6.28) $\displaystyle V(r)=$ $\displaystyle\>\Lambda(r^{-2}+O_{1}(r^{-3})).$
(6.29)

###### Proof.

To obtain (6.23)–(6.27), we perform a Taylor expansion in $r-1$.

Similarly, (6.29) follows by Taylor expanding in $\frac{1}{r}$ at $r=\infty$.
The term $O(r^{-3})$ does not depend on $m,\omega,\Lambda$ because we are
using that $\frac{m^{2}+|m\omega|+1}{\Lambda}$ is uniformly bounded, since
$\Lambda\geq\max\\{|m|(|m|+1),2|m\omega|\\}$ by (2.4) and (2.5). ∎

The following lemma concerns the qualitative behaviour of the potential
$V_{0}$, via an investigation of the possible shapes of its graph. This lemma
forms the analogue of [DRSR16, Lemma 6.3.1] and [DRSR16, Lemma 6.3.2] and
includes a more refined analysis of the $a=1$ case.

######  Lemma 6.6.

The following properties hold:

  1. _(i)_

The potential $V_{0}$ is either:

    1. _(a)_

strictly decreasing;

    2. _(b)_

has a unique critical point $r^{0}_{\rm max}\geq 1$ corresponding to a global
maximum, with equality iff $a=1$ and $m\omega=m\upomega_{+}=\frac{m^{2}}{2}$
and $\Lambda\leq 2m^{2}$;

    3. _(c)_

has exactly two critical points $1\leq r^{0}_{\rm min}<r^{0}_{\rm max}$,
corresponding to a local minimum and local maximum, respectively, with
equality iff $a=1$ and $m\omega=m\upomega_{+}=\frac{m^{2}}{2}$ and
$\Lambda>2m^{2}$.

  2. _(ii)_

There exists a numerical constant $B>0$ (independent of any parameters), such
that

$1\leq r^{0}_{\rm max}<B.$

  3. _(iii)_

If we restrict to $m\omega<m\upomega_{+}$, then

$\frac{dV_{0}}{dr}(1)=\frac{dV_{0}}{dr}(1)\geq 0,$

so $V_{0}(r)$ has exactly one critical point $r_{\max}^{0}$.

(a) $V_{0}$ strictly decreasing (Case (i)(a)).

(b) $V_{0}$ with one maximum and no minima (Case (i)(b)).

(c) $V_{0}$ with one minimum away from $r=1$ and one maximum (Case (i)(c)).

(d) $V_{0}$ with one maximum at
$r=1$ and no minima (Case (i)(b)).

(e) $V_{0}$ with one minimum at $r=1$ and one maximum (Case (i)(c)).

Figure 2. The different shapes of the graph of $V_{0}(r)$ corresponding to the
cases outlined in Lemma 6.6.

###### Proof.

We can write

$\displaystyle(r^{2}+a^{2})^{3}\frac{dV_{0}}{dr}=$
$\displaystyle\>2(1+a^{2})am\omega(a^{2}-3r^{2})+4m^{2}a^{2}r+\Lambda[(3-2r)r^{2}-a^{4}+a^{2}(r-1)(3r+1)],$
$\displaystyle\frac{d}{dr}\left((r^{2}+a^{2})^{3}\frac{dV_{0}}{dr}\right)=$
$\displaystyle\>-12(1+a^{2})am\omega r-6\Lambda
r(r-1)+a^{2}\Lambda(6r-2)+4a^{2}m^{2}.$

Let $r_{1}<r_{2}$ be the solutions of
$\frac{d}{dr}\left((r^{2}+a^{2})^{3}\frac{dV_{0}}{dr}\right))(r_{i})=0$. Then

$r_{1,2}=\frac{1}{2}(1+a^{2})(1-2ma\omega\Lambda^{-1})\mp\sqrt{\frac{1}{4}(1+a^{2})^{2}(2ma\omega\Lambda^{-1}-1)^{2}-\frac{a^{2}}{3}(1-2m^{2}\Lambda^{-1})}.$

Suppose $m\omega>0$. Then clearly $r_{1}<1$ if it is real. Suppose
$m\omega\leq 0$. Then we can express:

$r_{1}=\frac{1}{2}(1+a^{2})(1+2a|m\omega|\Lambda^{-1})-\sqrt{\frac{1}{4}(1+a^{2})^{2}(2a|m\omega|\Lambda^{-1}+1)^{2}-\frac{a^{2}}{3}(1-2m^{2}\Lambda^{-1})}$

so, using that

$\begin{split}\frac{1}{4}(1+a^{2})^{2}(2a|m\omega|\Lambda^{-1}+1)^{2}-\frac{a^{2}}{3}(1-2m^{2}\Lambda^{-1})\geq&\>\frac{1}{4}(1+a^{2})^{2}(2a|m\omega|\Lambda^{-1})^{2}+\frac{1}{4}(1+a^{2})^{2}-\frac{1}{3}a^{2}\\\
\geq&\>\frac{1}{4}(1+a^{2})^{2}(2a|m\omega|\Lambda^{-1})^{2}\end{split}$

we once again have that $r_{1}<\frac{1}{2}(1+a^{2})<1$.

In both cases, $(r^{2}+a^{2})^{3}\frac{dV_{0}}{dr}$ can therefore have at most
one critical point in $[1,\infty)$. We moreover have that

$\lim_{r\to\infty}(r^{2}+a^{2})^{3}\frac{dV_{0}}{dr}=-\infty.$

Hence, $\frac{dV_{0}}{dr}$ either: A) strictly negative, B) vanishes at a
unique point $r^{0}_{\rm max}$, C) vanishes at two points $r^{0}_{\rm
min}<r^{0}_{\rm max}$. By the above observations together with the fact that
$\lim_{r\to\infty}r^{2}V_{0}(r)=\Lambda$, we can moreover infer that in the
case: $V_{0}(r)$ is 1) strictly decreasing, 2) $V_{0}(r)$ has a unique
critical point $r^{0}_{\rm max}$ which is a global maximum, or 3) has two
critical points, a local minimum with agrees with $r^{0}_{\rm min}$ and a
local maximum which agrees with $r^{0}_{\rm max}$.

From (6.23) it follows that in the special case $a=1$ and
$\omega=\upomega_{+}$: $r=1$ is either: A) a local minimum if
$\Lambda>2m^{2}$, B) a global maximum if $\Lambda<2m^{2}$, C) a global maximum
if $\Lambda=2m^{2}$.

Finally, we observe that there exists a numerical constant $A>0$, such that

$|(r^{2}+a^{2})^{3}\frac{dV_{0}}{dr}+2\Lambda
r^{3}-3(1+a^{2})(\Lambda-2am\omega)r^{2}|\leq
A(\Lambda+|m\omega|+m^{2}r^{-1})r$

Hence,

$|\Lambda r^{0}_{\rm max}-(1+a^{2})(\Lambda-2am\omega)|\leq
A(\Lambda+|m\omega|+m^{2}).$

We divide the above inequality with $\Lambda$ and use that
$\Lambda\geq|m|(|m|+1)$ and $\Lambda\geq 2|m\omega|$ to conclude that there
exists a numerical constant $B>0$ such that $r^{0}_{\rm max}<B$.

It follows from (6.23) and (6.25) that when $m\widetilde{\omega}<0$:

$\frac{dV_{0}}{dr}(1)=\frac{d\widetilde{V}_{0}}{dr}(1)\geq 0,$

so it immediately follows that $V_{0}$ cannot have a minimum. ∎

The lemma below concerns the quantitative behaviour of $V$ around its extrema.

######  Lemma 6.7.

The potential $V$ has at most two critical points $r_{\min}<r_{\max}$,
corresponding to a local minimum and a global maximum. Furthermore:

  1. _(i)_

There exists a numerical constant $C>0$, such that

$\displaystyle|r_{\max}-(1+\sqrt{2})|\leq$
$\displaystyle\>C(|m\omega|+m^{2})\Lambda^{-1}+C(1-a^{2}),$ $\displaystyle
0\leq r_{\min}-1\leq$
$\displaystyle\>C(|m\omega|+m^{2})\Lambda^{-1}+C(1-a^{2})\qquad\textnormal{if
$m\omega\geq m\upomega_{+}$},$ $\displaystyle\textnormal{$V(r)$ has no minimum
in $[1,\infty)$ if $m\omega<m\upomega_{+}$}.$

  2. _(ii)_

For sufficiently small $\frac{|m\omega|+m^{2}+1}{\Lambda}$ and
$\frac{\omega^{2}}{\Lambda}$, there exist constants $b>0$ and $\delta>0$, such
that

$V(r)-\omega^{2}\geq b\Lambda\qquad\textnormal{for all
$r\in(r_{\max}-\delta,r_{\max}+\delta)$}.$

  3. _(iii)_

For sufficiently small $\frac{|m\omega|+m^{2}+1}{\Lambda}$, there exist
constants $b,B>0$, such that for all $r\geq 1$:

$\displaystyle(r_{\max}-r)\frac{dV}{dr}(r)\geq$ $\displaystyle\>b\Lambda
r^{-5}(r-1)(r-r_{\max})^{2}\qquad\textnormal{if $m\omega<m\upomega_{+}$},$
$\displaystyle(r_{\max}-r)\frac{dV}{dr}(r)\geq$ $\displaystyle\>b\Lambda
r^{-5}(r-1)(r-r_{\max})^{2}-B\left(m\omega-m\upomega_{+}\right)r^{-4}(r-r_{\max})^{2}\qquad\textnormal{if
$m\omega\geq m\upomega_{+}$}.$

  4. _(iv)_

Let $0<\eta<\frac{1}{4}$ and $m\neq 0$. Consider $m\omega\leq
m\upomega_{+}-\eta m^{2}$. Then there exists constants
$b_{\eta},\delta_{\eta}>0$ such that:

$\displaystyle(r_{\max}-r)\frac{dV}{dr}(r)\geq$
$\displaystyle\>b_{\eta}r^{-4}(r-r_{\max})^{2}m^{2}\quad\textnormal{for
$r\in[1,\infty)$},$ $\displaystyle V(r)-\omega^{2}\geq$
$\displaystyle\>b_{\eta}\Lambda\quad\textnormal{for
$r\in(r_{\max}-\delta_{\eta},r_{\max}+\delta_{\eta})$},$

with $r_{\rm max}-\delta_{\eta}>1$.

###### Proof.

First of all, since

$\begin{split}(r^{2}+a^{2})^{3}\frac{dV_{0}}{dr}=&\>2(1+a^{2})am\omega(a^{2}-3r^{2})+4m^{2}a^{2}r+\Lambda[(3-2r)r^{2}-a^{4}+a^{2}(r-1)(3r+1)]\end{split}$

and there exists a uniform constant $A>0$ such that

$(r^{2}+a^{2})^{3}\frac{dV_{1}}{dr}\leq Ar^{2},$

it follows that there exists a numerical constant $C>0$, such that

$\displaystyle|r^{0}_{\min}-r_{\min}|\leq C\frac{|m\omega|+m^{2}+1}{\Lambda},$