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\DOI{10.5802/crmeca.290}
\datereceived{2024-10-15}
\daterevised{2025-02-13}
\dateaccepted{2025-02-14}
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\dateposted{2025-04-16}
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\begin{noXML}

\CDRsetmeta{articletype}{review}

\title{The black hole stability problem}

\alttitle{Le probl\`{e}me de la stabilit\'{e} des trous noirs}

\author{\firstname{Sergiu} \lastname{Klainerman}}
\address{Mathematics Department, Princeton University, USA}
\email{seri@math.princeton.edu}

\thanks{NSF grant  220103}

\begin{abstract} 
This is a  short survey on the black hole stability problem written in
honor of Yvonne Choquet Bruhat's 100th birthday, subject very dear to her  and to which
she has  contributed  greatly, in particular  her foundational local 
existence  result \cite{Br} and the maximal globally hyperbolic development of a
given initial data set.    The main focus  is on the recent results  
on the stability of slowly rotating Kerr contained in  the  sequence of
works \cite{KS:Kerr,GKS-2022,KS-GCM1,KS-GCM2,Shen}.  
\end{abstract}

\begin{altabstract}
Ce texte donne un panorama succinct du probl\`{e}me de la stabilit\'{e} des
trous noirs, r\'{e}dig\'{e} en l'honneur d' Yvonne Choquet Bruhat, un sujet
qui lui est cher et auquel elle a grandement contribu\'{e}, notamment
gr\^{a}ce \`{a} son r\'{e}sultat fondamental d'existence locale \cite{Br} et au
d\'{e}veloppement maximal globalement hyperbolique d'un ensemble de
donn\'{e}es initiales. L'accent est mis principalement sur les r\'{e}sultats
r\'{e}cents concernant la stabilit\'{e} du Kerr en faible rotation,
tels qu'expos\'{e}s dans la s\'{e}rie de travaux
\cite{KS:Kerr,GKS-2022,KS-GCM1,KS-GCM2,Shen}.
\end{altabstract}

\keywords{\kwd{Black holes}
\kwd{Stability}
\kwd{GCM spheres}
\kwd{Null frames}
\kwd{Non-integrable  horizontal structures}}

\altkeywords{\kwd{Trous noirs}\kwd{Stabilit\'{e}}\kwd{Sph\`{e}res GCM}\kwd{Rep\`{e}res nuls}\kwd{Structures horizontales non int\'{e}grables}}

\maketitle
{\vspace*{-10pt}}

\end{noXML}

\let\widecheck\check

\section{Introduction}

The  Einstein Field  equations
(EFT)  $\R_{\a\b}-(1/2) \g_{\a\b}
\R= \T_{\a\b}$  possess two remarkable family of explicit  solutions:
\begin{itemize} 
\item The  Kerr family $\KK(a,m)$,   with  mass $m\ge 0$ and angular
momentum  $J=am$,    consists of   stationary, asymptotically flat,   
solutions  of  the  Einstein vacuum equations (EVE), $\R_{\a\b}=0$, 
that is  with vanishing energy--momentum tensor  $\T=0$.  The first
example, corresponding to $a=0$, $m&gt;0$, was discovered by Schwarzschild
in 2016, within months  after  Einstein  wrote the final version of 
his field equations. It took however a lot longer to understand its 
\textit{black hole}  implications.  The general  family was 
discovered  by   Kerr~\cite{Kerr} in  1963.  
 \item The  Kerr--Neumann family $\KN\!(a,m,e)$, with the
additional charge  parameter $e$,    consists, likewise, of  
stationary, asymptotically flat,    solutions  of  the 
Einstein--Maxwell  equations (EM) for  the Einstein--Maxwell equations,
corresponding to the  case when  $\T$ is the energy momentum tensor  of
an electromagnetic  Field $\F$, i.e.\  $\T_{\a\b}= \F_{\a\ga}
\F_{\b}\,^\ga -(1/4) \g_{\a\b}   \F_{\mu\nu} \F^{\mu\nu}$.  The new
extended   family was discovered by  Neumann~\cite{Neum} in{\unskip\break} 1964.
\end{itemize}

Similar families of  black hole solutions  have been discovered in  the
the case  the Einstein field equations  with a positive  cosmological
constant $\La$, i.e.\ $\R_{\a\b}-(1/2) \g_{\a\b} \R+\La \g_{a\b}=
\T_{\a\b}$.   The corresponding families  of  black holes are known
under the names \textit{Reissner--Nordstr\"om--de Sitter},
(corresponding  to   Schwarzschild), \textit{Kerr--de Sitter} and
\textit{Kerr--Newman--de Sitter}. 


  In this  survey I will  focus on   the  stability of the Kerr family,
considered the most important case,  and make   only  a few references 
on  results concerning  the stability of the   Kerr--Newman family  in
Section \ref{section:KN}.  In Section~\ref{section:Kerr-DS}  I also
make a     few remarks     on   the stability of the Kerr--de Sitter 
family.

  
\subsection{Relevance of the problem}


The discovery of the Kerr  black holes\footnote{According to
Chandrasekhar ``Black holes are macroscopic objects with masses varying
from a few solar masses to millions of solar masses. To the extent that
they may be considered as stationary and isolated, to that extent, they
are all, every single one of them, described exactly by the Kerr
solution. This is the only instance we have of an exact description of
a macroscopic object. Macroscopic objects, as we see them around us,
are governed by a variety of forces, derived from a variety of
approximations to a variety of physical theories. In contrast, the only
elements in the construction of black holes are our basic concepts of
space and time. They are, thus, almost by definition, the most perfect
macroscopic objects there are in the universe. And since  the general
theory of relativity provides a single two parameter family of
solutions for their description, they are the simplest as well.''}, 
first  as  explicit solutions of  the  Einstein Field Equations  (by
Roy  Kerr~\cite{Kerr})  and later as possible explanations of
astrophysical phenomena  has not only revolutionized our understanding
of the universe,   it  also gave mathematicians a monumental task: to
test the physical reality of these solutions. This may seem nonsensical
since physics tests the reality of its objects by experiments and
observations and, as such, needs mathematics to formulate the theory
and make quantitative predictions, not to test it. The problem, in this
case, is that black holes are by definition non-observable and thus no
direct experiments are possible.  Astrophysicists ascertain the
presence of such objects through indirect observations\footnote{The
physical reality of these objects was recently put to test  by 
LIGO-Viergo which is supposed to have detected the gravitational waves
generated in the final stage of the in-spiraling of two black holes.  
Rainer Weiss,    Barry C. Barish and  Kip  S. Thorne received the 2017
Nobel prize   for their ``decisive contributions'' in this respect. 
The 2020 Nobel  prize in Physics  was  awarded  to   R. Genzel and A.
Ghez    for providing  observational   evidence for the presence of
super massive black holes in the center of our galaxy, and  to R. 
Penrose  for his theoretical    foundational  work:   the  concept of a
trapped surface and the proof of  his famous singularity theorem.}  and
numerical experiments, but  both are limited in scope to the range of
possible observations or  the  specific initial conditions  in which
numerical  simulations are conducted.  One can rigorously check that
the Kerr solutions have vanishing Ricci curvature, that is, their
mathematical reality is undeniable.   But to be real in a physical
sense, they have to satisfy certain properties which, as it turns out, 
can  be neatly formulated in unambiguous mathematical language. Chief
among them\footnote{Other  such properties concern the  rigidity of the
Kerr family, see~\cite{IK:rigidity} for a  current survey,   or  the
dynamical formations  of black holes from regular configurations,  see 
the~\cite{Chr-BH,KlRod}  and the introduction to~\cite{An} 
for an  up to date account of more recent results.} is the problem of
stability, that is, to show that if the precise initial data
corresponding to a given  Kerr  are perturbed a bit, the basic features
of the corresponding solutions do not change much. Indeed, if   the
Kerr family would be  unstable   under  perturbations, black holes 
would be nothing more than mathematical artifacts,  ``mathematical 
ghosts''.  

 
\subsubsection{Final state conjecture} 

The  problem of  stability is intimately tied   the  far reaching
\textit{final state conjecture} (FSC)  according to which:   
\textit{The  evolution    of  general   (generic),   asymptotically  flat, 
initial   data sets for    the Einstein  vacuum field equations leads 
to  solutions  which behave, in the large,   like    a finite number of
Kerr black  holes   plus a  radiative  decaying term}\footnote{The
celebrated LIGO  experimental events are understood  in terms of a
final state consisting of just one large, spinning black hole (Kerr
black hole) plus a burst of gravitational radiation which propagated
through the universe   over an enormously large time   before reaching
us.}. The statement,   consistent  with  both astrophysical 
observations and numerical simulations,    is  rightly  regarded as 
the  holy grail of mathematical    General Relativity and is on a par 
in depth,  beauty and difficulty  with any other major  open problem in
Mathematics, such as the millennium Clay problems.

 Though totally out of reach in full generality,    FSC is   related to
a myriad   of     deep and challenging   problems  which are the focus
of active research.  To start with  the conjecture, if true,  implies
either that the initial data is too small to concentrate and thus 
perturbations  must asymptotically disperse to zero (\textit{stability
of Minkowski})\footnote{See discussion  in Section~\ref{section:stabilityMink}.}, 
or that it is large enough to produce
bound states, i.e.\ black holes (\textit{collapse
problem})\footnote{Important progress on the related problem  on the
formation  of trapped surfaces  was recently made  starting with the  
seminal work  of  Christodoulou~\cite{Chr-BH}. See also  the
introduction to~\cite{An2}   for   more  recent results.}. The fact
that final states  of   sufficiently large data  are  described, 
locally, by  Kerr black hole solutions  requires, in particular, that
any stationary solution of the equations must be a Kerr solution
(\textit{rigidity  problem}\footnote{See~\cite{IK:rigidity}  for a 
survey of recent results  on rigidity.}) and that the  Kerr family  is 
\textit{stable} under perturbations. Moreover, singularities are
expected not to appear outside black holes, at least from generic
initial conditions; a statement postulating this is known as the
\textit{weak cosmic censorship} conjecture\footnote{See~\cite{Chr-sing}
and the references therein.}. The Final State Conjecture also requires
us to rigorously  treat     interactions of black holes such as the
in-spiral  colliding of  black holes.  This  is a  problem of great
interest   for  astrophysicists  and numerical relativists, yet   we
have close to  nothing   in terms of rigorous mathematical results.

  
\section{Kerr stability}

The issue of the stability of the Kerr    family  has been at the
center of attention  of GR physics and  mathematical relativity for  
more than half a century, ever since their  discovery   by Kerr 
in~\cite{Kerr}.  In  this section I recall the main features of  the
Kerr  solutions,   describe the main conjecture  and  the recent
results   which settle it in the case of small angular momentum.
Sections~\ref{section:NotionsofLNL-stability}--\ref{subs:MainDiffic}  
introduce the reader to the main difficulties of the problem, 
Section~\ref{sect:LinearStab}   to  the   linearized problem and 
Section~\ref{sect:ModelProblems}  to the most relevant simplified 
model problems. These latter  are then discussed in more detail  in
Section~\ref{sect:ShortSurveyModel}.


  
\subsection{Kerr spacetime}
\lab{section:Kerr}

As mentioned above,  Einstein Vacuum Equations admit a remarkable   two
parameter family of explicit  solutions, the  Kerr spacetimes 
$\KK(a,m)$, $0\le |a|\le m$,   depending  on the  parameters $m$ 
(mass)  and $a$ (with $J=am$    angular momentum),     which are 
stationary and axisymmetric.  In the usual  Boyer--Lindquist coordinates
they take the form  Let  $(\KK(a, m), \g_{a,m})$  denote   the family
of  Kerr spacetimes

 In Boyer--Lindquist  coordinates  the Kerr metric is given by
{\begin{equation}\label{zq1}
\g_{a,m}=-\frac{q^2\Delta}{\Sigma^2}(\mathrm{d}t)^2+
\frac{\Sigma^2(\sin\theta)^2}{|q|^2}\left(\mathrm{d}\phi-\frac{2amr}{\Sigma^2}\mathrm{d}t\right)^2 
+\frac{|q|^2}{\Delta}(\mathrm{d}r)^2+|q|^2(\mathrm{d}\theta)^2,
\end{equation}}\unskip
where
{\begin{equation}\label{zq2}
\begin{cases}
\Delta=r^2+a^2-2mr, \quad  q=r+\mathrm{i} a \cos \th,\\
\Sigma^2=(r^2+a^2)|q|^2+2mra^2(\sin\theta)^2=(r^2+a^2)^2-a^2(\sin\theta)^2\Delta.
\end{cases}
\end{equation}}\unskip
The asymptotically flat\footnote{That is they approach the Minkowski
metric for large $r$.}  metrics    $\g_{a,m}$ verify   the Einstein
vacuum equations (EVE)
{\begin{equation}
  \label{eq:EVE-intro}
  \Ric(\g)=0,
  \end{equation}}\unskip  
are   stationary and axially symmetric\footnote{That is  $\KK(a,m)$
possess two Killing vectorfields: the stationary  vectorfield   
$\T=\partial_t$,  which is time-like   in the  asymptotic region, away  from
the horizon, and  the  axial symmetric Killing field $\Z=\partial_{\phi}$.},
possess  well-defined  event  horizon $r=r_+$ (the largest  root  of
$\De(r)=0$), domain of outer communication $r&gt;r_+$ and  smooth  future 
null infinity $\II^+$  where  $r=+\infty$.   The metric can be extended
smoothly inside  the black hole region, see  Figure~\ref{fig:penrosediagramofkerr}.   The boundary  $r=r_-$  (the smallest
root  of $\De(r)=0$) inside the black hole region is a Cauchy horizon
across which predictability fails\footnote{Infinitely many smooth
extensions are possible beyond the  boundary.}.


\begin{figure}
\includegraphics{fig01}
\caption{\label{fig:penrosediagramofkerr}
Penrose diagram of Kerr for $0&lt;|a|&lt;m$. The surface  $r=r_+$, the larger
root  of $\De=0$,  is the event horizon of the black hole, $r&gt;r_+$ the
domain of outer communication,   $\II^+$  is  the future null infinity,
corresponding to $r=+\infty$.}
\end{figure}


Here are some of the most important  properties of $\KK(a,m)$:
\begin{itemize}
\item   $\KK(a, m)$  possesses  a canonical  family of  null  pairs,
called \textit{principal null pairs},   of the  form $(\la e_4,
\la^{-1} e_3) $, with $\la&gt;0$ an arbitrary scalar function, and 
{\begin{equation}\label{eq:PrincipalNull}
e_4= \frac{r^2+a^2}{|q|^2} \partial_{t} +\frac{\De}{|q|^2}
\partial_{r} + \frac{a}{|q|^2} \partial_{\phi}, \quad
e_3  = \frac{r^2+a^2}{\De} 
\partial_{t} -\partial_{r} +\frac{a}{\De} \partial_{\phi}.
\end{equation}}\unskip
\item  The horizontal structure, perpendicular to $e_3, e_4$, denoted
$\HH$,  is spanned by the  vectors 
{\begin{equation}
\label{eq:canonicalHorizBasisKerr-intro}
e_1=\frac{1}{|q|}\partial_{\th},\quad e_2=\frac{a\sin\th}{|q|}
\partial_{t}+
\frac{1}{|q|\sin\th}
\partial_{\phi}.
\end{equation}}\unskip
The distribution generated by $\HH$ is non-integrable  for $a\neq 0$.
\item The  horizontal  structure  $( e_3,  e_4, \HH)$  has   the
remarkable property  that all components of the Riemann curvature 
tensor  $\R$, decomposed  relative  to them,    vanish  with the
exception of those which  can be deduced  from $\R(e_a, e_3, e_b, e_4),
a, b=1,2.$
 \item $\KK(a, m)$  possesses  the Killing vectorfields $\T, \Z$ which,
in BL coordinates, are given by   $\T=\partial_t, \Z=\partial_\phi$. 

  \item In addition to the symmetries  generated  by $\T, \Z$,  $\KK(a,
m)$  possesses   also  a  non-trivial  Killing tensor\footnote{Given by
the expression $ \C=-a^2 \cos^2\th \g+O$,  $O=|q|^2 ( e_1\otimes
e_1+ e_2\otimes e_2)$.}, i.e.\ a symmetric   2-tensor $\C_{\a\b}$
verifying the property $\D_{(\ga} \C_{\a\b)}=0$. The tensor carries the
name  of its discoverer B. Carter, see~\cite{Carter}, who  made use of
it   to show that the   geodesic flow   in Kerr is integrable.  Its
presence, in addition to $\T$ and $\Z$,  as a higher order symmetry,   
is at the heart of what   Chandrasekhar, see~\cite{Chand3},  called the
most striking feature  of Kerr,   ``the separability of all the
standard equations of mathematical physics in Kerr geometry''. 

\item  The Carter  tensor can be used to define the Carter   operator
{\begin{equation}
\CC=\D_{\a}( \C^{\a\b}\D_{\b}),
\end{equation}}\unskip
a second order operator  which commutes  with $\Box_{a,m}$. This
property  plays a  crucial  role in  the proof of    the   stability 
result,   Theorem~\ref{MainThm-firstversion}, more precisely in  Part
II of~\cite{GKS-2022}.
\end{itemize}
 


\subsection{Main conjecture}


\begin{proof}[Conjecture][Stability of Kerr conjecture]
\textit{Vacuum, asymptotically flat,  initial data sets,  sufficiently
close to  $\KK(a,m)$, $|a|/m &lt;1$,  initial data,  have  maximal
developments with complete future null infinity  and with   domain of
outer communication\footnote{This presupposes the existence of an event
horizon. Note that the  existence of such an event horizon   can only
be established  a posteriori,  upon the completion of the proof of the
conjecture.}   which approaches  (globally)  a nearby Kerr solution.}  
\let\qed\relax
\end{proof}

  
\subsection{Notions of   stability} 
\lab{section:NotionsofLNL-stability} 


It is instructive  to   consider the problem of the stability of  the
Kerr family in  the  broader context of  nonlinear PDEs of the form
{\begin{equation}
\label{eq:Nonl-NN}
\NN[\phi]=0.
\end{equation}}\unskip
Consider  a stationary solution $\phi_0$ of \eqref{eq:Nonl-NN}. There
are two  distinct notions of stability, \textit{orbital stability},
according to which  small perturbations    of $\phi_0$    lead to
solutions $\phi$ which remain close to $\phi_0$  for all time, and 
\textit{asymptotic stability (AS)} according to which the perturbed
solutions converge as $t\to \infty$ to $\phi_0$. In the case where
$\phi_0$ is non trivial, i.e.\ it  depends on the spatial variables, 
there is a third notion,  called \textit{asymptotic orbital stability
(AOS)}, to describe the fact that  the perturbed solutions may converge
to a different stationary solution. This happens if $\phi_0$ belongs to
a multi-parameter smooth family of stationary solutions, or by applying
a gauge transform to $\phi_0$ which keeps the equation
invariant\footnote{In  Kerr, both cases are present.}. 

  For quasilinear equations\footnote{Orbital stability can be 
established directly (i.e.\ without   establishing the stronger version)
only  in rare occasions,   such as for  hamiltonian equations with weak
nonlinearities.}, such as  EVE,   a proof of stability means
necessarily  AS  or AOS   stability.      Both    require    a   
detailed  understanding of the \textit{decay properties}    of the  
linearized equation, i.e. 
{\begin{equation}
   \label{eq:Linearized-LL}
   \LL[\phi_0]\psi =0,
   \end{equation}}\unskip
with $\LL[\phi_0]$ the Fr\'echet  derivative   $\NN'[\phi_0]$. This  
is,  essentially,  a linear hyperbolic  system  with variable
coefficients  which,  typically,  presents   instabilities\footnote{In 
unstable situations~\eqref{eq:Linearized-LL} may have  
exponentially growing solutions, see for example~\cite{DKSW}.}.
  
  In the exceptional  situation,  when  stability can ultimately  be
established,    one  can  tie  all   the  instability   modes to  the  
following   properties of the nonlinear equation:  
\begin{enumerate}[(M1)]
\item  If  $\phi_\la $ is a family of stationary  solutions,  near
$\phi_0$,     verifying    $\NN[\phi_\la ]=0$.  Then  $\psi_0
=((\mathrm{d}/\mathrm{d}\la) \phi_\la)_{\la=0} $ verifies
$\NN'[\phi_0] \psi_0=0$,
i.e.\ $\psi_0$ is a nontrivial, stationary,  bound state  of the
linearized  equations~\eqref{eq:Linearized-LL}.  
\item If   $\Phi_\la $ is a smooth  family of  diffeomorphisms  
of the background manifold,   $\Phi_0=I$,   such that $
\NN[\Phi^*_\la(\phi_0)]=0$. Then $\Psi_0=((\mathrm{d}/\mathrm{d}\la) 
\Phi^*_\la(\phi_0))_{\la=0}$ verifies $\NN'[\phi_0]\Psi_0=0$, i.e.\ 
$\Psi_0$ is also a stationary bound state  of the linearized
equation~\eqref{eq:Linearized-LL}.  
\end{enumerate}
These  linear  instabilities  are responsible for the fact   that a
small perturbation of  the   fixed stationary solution $\phi_0$  may
not converge to  $\phi_0$ but to  another nearby  stationary
solution\footnote{The methodology of tracking this asymptotic final
state, in general different from $\phi_0$, is usually referred to as
modulation.  See   for example~\cite{Ma-Me,Me-R}  for how
modulation theory  can be used   to deal with some examples of scalar
nonlinear dispersive equations.}.

  To prove the asymptotic  convergence   of  $\phi$   to  a  final
state $\phi_f$  we  need  to establish  sufficiently strong rates  of
decay\footnote{To control the nonlinear terms of the equation.} for
$\phi-\phi_f$.          Rates of decay however are strongly 
\textit{coordinate dependent},   i.e.\  dependent on   the  choice of 
the   diffeomorphism (or gauge) $\Phi$ in which decay is measured.  
Thus, to prove  a  nonlinear stability   result we need to know both 
the final   state $\phi_f$ and the coordinate system $\Phi_f$   in
which sufficient decay,  and thus   convergence  to $\phi_f$, can be
established. The difficulty  here is that neither $\phi_f$ nor $\Phi_f$
can be determined a-priori  (from  the  initial perturbation),  they
have  to emerge dynamically  in the process of convergence. Moreover, 
in all   examples  involving  nonlinear wave equations in $1+3$
dimensions, the nonlinear terms  have to also cooperate, that is  an
appropriate  version  of the so called  null condition  has to be
verified.


To  summarize,  given a nonlinear system  $\NN[\phi]=0$ which possesses
both a smooth family of stationary solutions $\phi_\la$   and   a
smooth family of invariant diffeomorphisms a proof of the nonlinear
stability of $\phi_0$  requires the following ingredients:
\begin{itemize}
\item  The only non-decaying modes of the linearized  equation
$\LL(\phi_0)\psi=0$ are those due to  the items M1--M2 above. In
particular there are no exponentially growing  modes.
\item A  dynamical construction of both the final state $\phi_f$ and
the final gauge $\Phi_f$ in which convergence to the final state takes
place.
\item The nonlinear  terms in   the equation
{\begin{eqnarray*}
\LL[\phi_f] \psi = N(\psi) 
\end{eqnarray*}}\unskip
obtained by expanding  the equation $\NN[\phi]$  near $\phi_f$, in the
gauge  given by  the diffeomorphism $\Phi_f$,  has  to verify  an
appropriate version of the null condition.
\end{itemize}
   
      
\subsection{Stability of Minkowski space} 
\lab{section:stabilityMink}

In the  case of the Kerr family  one has  to deal  not only with  a
2-parameter  family of  solutions,  corresponding to   the parameters
$ (a, m)$, but also  with the entire group of
diffeomorphisms\footnote{Indeed, according to the covariant properties
of the Einstein vacuum equations  we  cannot distinguish between $\g$
and $\Phi^*\g$,   for any diffeomorphism $\Phi$ of $\MM$.} of  $\MM$.  
To understand  these difficulties  it helps to   them   with   those 
arising    in the simplest  case  when $a=m=0$, i.e.\  stability of
Minkowski space.



   Until very recently  the only   space-time for which  full nonlinear
stability had been established was the Minkowski space.  The  first
proof in~\cite{Ch-Kl}  fits in the category  of  asymptotic stability,
i.e.\ convergence back  to  Minkowski space,  and            is based 
on  some     important  PDE  advances    of late last century:
\begin{enumerate}[(i)]
\item  Robust    approach, based on the  vectorfield method,  to  
derive quantitative   decay  based on generalized energy estimates and
commutation with (approximate)  Killing and conformal Killing 
vectorfields. 

\item   The  \textit{null condition} identifying the deep 
mechanism for     nonlinear stability,  i.e.\  the specific structure of
the nonlinear terms   which   enables stability  despite the low decay
of the  perturbations. 

  \item    Elaborate bootstrap argument according to which  one
makes  educated assumptions  about the behavior of  solutions  to 
nonlinear wave equations  and then proceeds, by a long sequence of
a-priori estimates,  to       show that   they are in fact  satisfied.
This  amounts to a \textit{conceptual linearization}, i.e.\ a method  by
which the equations become, essentially, linear\footnote{Note that in
the context of  EVE, and other quasilinear hyperbolic systems, this   
differs substantially from the usual notion of linearization around a
fixed background.}  without   actually linearizing them.
 \end{enumerate}
The main innovation in the  proof in~\cite{Ch-Kl}  is   the choice  of 
an  appropriate   gauge   condition,  readjusted dynamically  through 
the  convergence process,   by  a  continuity argument, which  allows
one to separate   the curvature  estimates,   treated by hyperbolic
methods,   from the    estimates for the connection coefficients. The
key point is that   these   latter  verify  transport or elliptic
equations in which the curvature terms appear as sources.  Thus both 
the curvature  components and  connection coefficients  can be 
controlled by a bootstrap argument. The  gauge condition is based on 
the  constructive choice  of a maximal  time function $t$ and two
outgoing optical functions  $\uint$\footnote{The interior optical
function is initialized  on  a  timelike  geodesic  from the  initial
hypersurface.} and $\uext$\footnote{The exterior  optical  function 
$\uext$ is initialized  on   the last slice $t=t_*$, by  the 
construction of a    foliation (inverse lapse foliation)  initialized
at space-like infinity.  It   is thus readjusted  dynamically  as
$t_*\to \infty$.}  covering  the interior and exterior  parts of  the
spacetime.

  
Another novelty of~\cite{Ch-Kl}  was the reliance on    null frames 
adapted to   the  $S$-foliations   induced    by the level surfaces of
$t$ and $u$. These  define   integrable horizontal structures (in the
language of   part I of~\cite{GKS-2022}),   by contrast with the
non-integrable ones  used in the proof of
Theorem~\ref{MainThm-firstversion}  and discussed in
Section~\ref{sec:nonintegrabilityhorizontalstructure}. The functions $
t, u$ and  this  integrable horizontal structure can be used to define
approximate Killing vectorfields used in estimating the curvature. 


   
\subsection{Main difficulties in the Kerr stability problem}
    \lab{subs:MainDiffic}
    
    
  There are a few   major obstacles in passing from the stability of
Minkowski  to that of  Kerr:
   \begin{enumerate}[(1)]
\item The first one was     already discussed  in  Section~\ref{section:NotionsofLNL-stability} in the general context of the
stability of a stationary solution $\phi_0$.  In the case  when
$\phi_0$ is trivial    there are no nontrivial bound states    for the
linearized problem and thus   we expect  that the final state   does
actually coincide  with  $\phi_0$.  This  is precisely  the case  for  
the special member  of the Kerr family  $a=m=0$, i.e.\ the Minkowski 
space\footnote{Note however  that  even though the linearized system
around Minkowski  does not contain instabilities,   the proof of the
nonlinear stability of the Minkowski space  in~\cite{Ch-Kl}    takes
into account (in a fundamental way!)   general covariance. Indeed  the
presence of  the   ADM mass affects the causal structure  of  the far,
asymptotic,  region of the perturbed space-time.} $(\RRR^{1+3}, m)$.   
On the other hand, in perturbations of Kerr,  general covariance
affects the entire construction of the spacetime. In the proof of
Theorem~\ref{MainThm-firstversion}  the crucial  concept  of a GCM
admissible  spacetime  is meant to  deal  with  both  finding the final
parameters and the  gauge in which convergence to the final state takes{\unskip\break}
place.

  
\item A  fundamental insight   in the stability of the Minkowski space 
was that the Bianchi identities  decouple at first order from the null
structure equations which allows one to control curvature first, as a
Maxwell type system (see~\cite{Ch-Kl0}), and then proceed with the rest
of the solution. This  cannot work for  perturbations of Kerr due to
the fact that some of the null components\footnote{With respect to  the
\textit{principal null directions of Kerr,} i.e.\ a distinguished  null
pair which   diagonalizes  the  full curvature tensor, the middle
component $P=\rho+ i\rhod$ is nontrivial.} of the  curvature tensor 
are non-trivial in Kerr.

 \item    Even if  one succeeds in tackling the above mentioned issues,
there are still major obstacles  in understanding the  decay properties
of the solution.  Indeed,  when one  considers  the simplest,   
relevant, linear equation  on a fixed Kerr background,  i.e.\ the scalar
wave equation ${\Box}_{a,m} \psi=0$,  one  encounters serious
difficulties  to  prove decay.   Below is a  very short description of
these:
\begin{itemize}
\item \textit{The problem of   trapped null geodesics.}   This 
concerns the existence of null geodesics\footnote{In the Schwarzschild
case, these geodesics are associated with the celebrated photon sphere
$r=3m$.} neither crossing the event horizon nor escaping to  null
infinity, along which solutions   can concentrate for arbitrary long
times.  This   leads  to degenerate  energy--Morawetz estimates  which
require a very delicate analysis.

\item \textit{The   trapping   properties of the     horizon.}       
The horizon  itself  is ruled  by null  geodesics, which do not    
communicate with  null infinity and can thus concentrate energy.       
This problem was solved   by understanding      the  so called    
red-shift effect associated to  the event horizon,  which    
counteracts this type of  trapping.

\item \textit{The problem of superradiance.} This  is  the failure of
the stationary Killing  field $\T=\partial_t$ to be everywhere timelike
in the domain of outer communication\footnote{The stationary  Killing 
vectorfield $\T$ is timelike only outside of the so-called
ergoregion.}, and thus, of the associated  conserved energy to be
positive. Note that this problem is absent in Schwarzschild and, in
general,   for axially symmetric solutions of EVE. In both cases
however there still is a degeneracy along the horizon.
\item   \textit{Superposition problem.}       This is  the  problem  of
combining the estimates  in the near  region, close to the horizon,
(including the  ergoregion  and trapping) with estimates in the
asymptotic region, where the spacetime is close to   Minkowski. 
\end{itemize}
\item Though, as seen above,  the analysis of  the scalar wave equation
in Kerr  presents formidable difficulties,  it is itself just  a vastly
simplified model problem.  A more realistic  equation is the so called
spin 2 wave equation, or Teukolsky equation,  which presents many new
challenges\footnote{Unlike  the  scalar wave equation $\Box_{a,m}\psi=0$, which is conservative, the Teukolsky equation is not,  and  we
thus  lack  the most basic ingredient  in  controlling     the  
solutions of the equation, i.e.\ energy estimates.}.

\item The full linearized system, whatever  its  formulation, 
presents   many additional difficulties   due to its  complex 
tensorial  structure and  the  huge gauge covariance of the 
equations\footnote{As mentioned  earlier, rates of decay are heavily
dependent on a proper choice of gauge, thus affecting the issue of
convergence.}.   The crucial  breakthrough in this regard is the 
observation, due to Teukolsky~\cite{Teuk},  that the  extreme
components of the  linearized curvature tensor are both gauge invariant
(see below in  Section~\ref{sect:LinearStab}) and  verify decoupled
spin 2  equations, that is  the Teukolsky equations mentioned above.   
\item   A crucial simplification of the linear theory, by comparison to
the  full nonlinear case, is that  one   can  separate the treatment of
the gauge  invariant   extreme  curvature components form all the other
gauge invariant quantities. In  the nonlinear case this separation is
no longer true, all quantities need to be treated simultaneously. 
Moreover,  methods based on  separation of variables, developed  to
treat scalar and and spin 2 wave  equations in Kerr,    are  
incompatible  with the nonlinear setting which requires, instead, 
robust methods   to derive decay.  
\end{enumerate}
   
   
  
\subsection{Linear stability}
\lab{sect:LinearStab}


Linear stability  for the vacuum  equations is formulated in the
following way.  Given the Einstein tensor $\G_{\a\b}=\R_{\a\b}- 1/2 \R
\g_{\a\b}$  and a stationary solution  $\g_0$, i.e.\ a fixed Kerr
metric, one   has to solve the  system of equations
{\begin{equation}
\label{eq:Lin.Grav}
\G'(\g_0) \, \de \g =0.
\end{equation}}\unskip
The  covariant properties of the  Einstein equations, i.e.\ the
equivalence between a solution $\g$  and $\Phi^*(\g)$, leads  us to
identify $\de\g$  with   $\de\g+\Lie_X (\g_0)$ for arbitrary 
vectorfields    $ X$    in $\MM$, i.e.      
{\begin{equation}
\label{eq:linearcovariance}
\de\g\equiv  \de\g+\Lie_X (\g_0).
\end{equation}}\unskip
One  can now attempt to  formulate a version of linear stability 
for~\eqref{eq:Lin.Grav}, loosely related to   the   nonlinear stability of
Kerr    conjecture,  as follows.

\begin{definition}[By   linear stability of the Kerr  metric $\g_0$   we understand a
result  which achieves the following]
{\upshape Given  an  appropriate initial data  for  a perturbation 
$\de\g$,   find a  vectorfield $X$ such that, after projecting  away
the   bound states   generated by the parameters $a, m$, according to 
M1--M2  in Section~\ref{section:NotionsofLNL-stability},    a solution 
of the form  $\de\g+\Lie_X\g_0$      to~\eqref{eq:Lin.Grav},   decays,
relative  to an  appropriate  null  frame of $\KK(a_0, m_0)$,    
sufficiently fast  in time.}
\end{definition} 

   \begin{remark}
{\upshape The definition above is necessarily  vague.       What    is the
meaning of sufficiently fast?       In fact    various  components of
the metric $\de\g$, relative   to  the canonical null frame  of $\KK(a,
m)$,   are expected to decay at different  polynomial rates, some of
which  are  not even integrable.      Unlike in the nonlinear context,
where one needs precise   rates of    decay  for components   of the
curvature tensor  and Ricci coefficients (as well as    their
derivatives)  to control the nonlinear terms, in linear  theory     any
type of  nontrivial   control of  solutions  may be  regarded as
satisfactory\footnote{Thus, for example,  in their well known  linear
stability result around Schwarzschild~\cite{D-H-R}, the authors 
derive   satisfactory results (compatible    with  what is needed in
nonlinear theory)  for  components of the  curvature  tensor, and some 
Ricci coefficients, but not all.  Similar   comments apply 
to~\cite{HKW,Johnson}.}. Thus linear stability, as formulated
above,   can only be regarded as  a vastly simplified  model problem.
Nevertheless  the study of linear stability of the Kerr family  has
turned    out to be    useful in various ways, as we shall see below.} 
   \end{remark}
   
   

   Historically,     the following  versions of linear stability  have
been   considered.
   \begin{enumerate}[(a)]
\item \textit{Metric perturbations.} At the level of the metric 
itself, i.e.\  as above in~\eqref{eq:Lin.Grav}.  
\item  \textit{Curvature perturbations.}  Via  the Newman--Penrose
(NP) formalism,  based  on null frames. 
   \end{enumerate}
The strategy  followed  in  both  cases\footnote{In the article we 
refer mainly to  the curvature perturbation approach.}     is:
\begin{itemize}
\item  Find  components  of the metric (in  case (a)) or curvature
tensor  (in case (b)), invariant  with respect to linearized gauge
transformations (\ref{eq:linearcovariance}),   which verify   decoupled
wave equations.    The main insight of this type was  the discovery, by
Teukolsky~\cite{Teuk},  in the context of  (b) above,  that the extreme
components of the  linearized curvature tensor  verify   both  these
properties. 
\item Analyze  these components   by showing one of the following:\Tg&lt;/p&gt;
\begin{itemize}
\item[--]   There  are no exponentially  growing modes. This  is known as
mode stability.
\item[--] Boundedness  for all time.
\item[--] Decay (sufficiently fast)    in time.
\end{itemize} 
\item Find a linearized  gauge condition, i.e.\ a vectorfield $X$, such
that  all remaining (gauge dependent) components (at the metric or
curvature level)   inherit  the property mentioned above: no 
exponentially growing modes,  boundedness, or decay in time.  In the
physics literature  this  is  known as the problem of  reconstruction. 
   \end{itemize}
   
   
   
\subsubsection{Mode stability}  
\lab{section:ModeStab}

   All results on the  linear  stability of Kerr   in the   physics
literature  during the  10--15 years after   Roy Kerr's  1963 
discovery,   often called  the ``Golden Age of Black Hole Physics'',  
are based  on mode decompositions. One makes use of the
separability\footnote{See  discussion in  Section~\ref{section:Kerr}.}
of the  linearized  equations,  more precisely the Teukolsky equations,
on a fixed Kerr background, to derive   simple  ODEs  for the
corresponding modes.     One can then show, by ingenious methods,  that
these modes  cannot exhibit exponential growth.       The  most
complete  result  of    this type  is due to Bernard Whiting~\cite{Whit}.

   
   The obvious limitation of these results    are as follows:
   \begin{itemize}
\item  They  are far from   even  establishing    the  boundedness of 
general solutions  to  the Teukolsky equations, let alone  to establish
quantitative decay  for the   general solutions.
\item  Results based on mode decompositions   depend strongly on the
specific symmetries   of   Kerr which   cannot be adapted to 
perturbations  of Kerr.
   \end{itemize}
   
     Robust  methods  to deal  with both issues have been developed  in
the  mathematical community, based on the vectorfield method which I 
discuss below. 

 
\subsubsection{Classical vectorfield method} 

The  vectorfield method, as an analytic tool to derive   decay,   was
first   developed in connection   with the wave  equation  in Minkowski
space.     As well known,  solutions of the wave equation  $ \Box \psi=0$ 
in the Minkowski space  $\RRR^{n+1}$   both conserve energy 
and decay uniformly in time   like $t^{-({n-1}/{2})}$. While
conservation of energy  can be established  by a simple integration by
parts,  and is thus robust to perturbations of the Minkowski metric, 
decay   was first derived   either using  the   Kirchhoff formula  or
by Fourier methods, which are manifestly not robust.  An integrated
version of local energy  decay, based on an  inspired  integration by
parts argument,   was first derived by   Morawetz~\cite{Mor1,Mor2}.  The first    derivation of decay  based  on   the 
commutations properties of  $\Box$ 
with     Killing and  conformal
Killing  vectorfields    of Minkowski space    together with    energy
conservation  appear in~\cite{Kl-vectorfield,Kl-vect}.   The
same  method also  provides  precise  information about  the    decay
properties  of  derivatives of solutions  with respect to   the
standard null frame  of Minkowski space,   an important motivating
factor in the discovery of the  null condition~\cite{Kl-ICM,Chr,Kl-null}.    

The crucial feature  of the  methodology  initiated by these
papers,    to which we  refer      as the classical vectorfield method,
is that  it can be easily   adapted to perturbations of  the Minkowski
space.   As such the method     has had numerous applications to
nonlinear wave equations and played an important  role in  the proof of
the nonlinear stability of Minkowski space, as discussed in 
Section~\ref{section:stabilityMink}. It  has also been applied    to   later
versions  of the stability  of Minkowski   
in~\cite{KlNi1,KlNi2,Lind-Rodn,Bi,Lind,huneau,HVas2,graf,Shen:Mink}  and
extensions of it   to  Einstein equation coupled with  various matter 
fields in~\cite{BiZi,FJS,Lind-T,BFJT,Lf-Ma,I-Pau}.  

 
\subsubsection{New vectorfield method}
\lab{section:NewVf}
  
  
  To  derive    decay estimates    for solutions of wave equations  on
a Kerr background one has to  substantially refine the classical
vectorfield  method. The  new vectorfield method  is an extension of
the classical  method  which  compensates  for   the lack of  enough
Killing and conformal Killing vectorfields   in Kerr   by  
introducing,   new,  cleverly  designed,   vectorfields  whose
deformation tensors have    coercive properties    in different regions
of spacetime, not necessarily causal.    The method  has    emerged  in
the last 20  years   in connection to the study of boundedness and
decay  for the    scalar wave equation in Schwarzschild and Kerr, see
Section~\ref{section:QuantitativeDecay}  for more details.

\subsection{Model problems}  
\lab{sect:ModelProblems}


To  solve  the stability of Kerr conjecture  one has to   deal
simultaneously  with all the difficulties  mentioned above.  This is,
of course, beyond the abilities  of  mere  humans.  Instead  the
problem was tackled in  a  sequence  of  steps based on  a variety  of
simplified  model problems,  in  increasing  order of   difficulty.  
To start with  we can classify  model problems based on the following
criteria:
\begin{enumerate}[(1)]
\item Whether  the result refers to Schwarzschild i.e.\ $a=0$,  slowly
rotating Kerr i.e.\ $|a|\ll m$   or   full non-extremal  Kerr $|a|&lt;m$.
\item Whether the result  refers to  linear or nonlinear stability.
\item Whether the result, in linear theory,   refers  to scalar wave
equation,  i.e.\ spin $0$,  Teukolsky equation,  i.e.\  spin  $2$,  or
the full linearized system.
\item Whether  the  stability   result, in linear theory,  is a mode
stability result, a boundedness result  or one that establishes 
quantitative decay. 
\end{enumerate}

   
\section{Short survey  of  model problems}
\lab{sect:ShortSurveyModel}

  
I  give  below a short  outline of the main developments  concerning
linear and nonlinear model problems for  Kerr stability.  

      
\subsection{Mode stability results} 
\lab{section:Modestability}
       
       
These are  mode stability  results, using the method of separation of
variables,   obtained in the physics community  roughly during the
period 1963--1990.  They  rely on what   Chandrasekhar  called     the
most striking feature of Kerr  i.e.\  ``\textit{the separability  of all
the standard equations of mathematical physics in Kerr geometry}''. 
     \begin{enumerate}[(1)]
\item \textit{Regge--Wheeler}~\cite{Re-W}.   Even before the discovery of the
Kerr solution physicists were interested in the mode  stability of 
Schwarzschild space, i.e.\ $\KK(0, m)$. The first important    result 
goes back to  Regge and  Wheeler~\cite{Re-W},  in  which they
analyzed   linear, metric perturbations, of the Schwarzschild metric.
They   showed that in a suitable gauge, Equation~\eqref{eq:Lin.Grav} 
decouples into even-parity and odd-parity perturbations, corresponding
to axial and polar perturbations.    The  most important  discovery in
that paper  is that of  the master Regge--Wheeler equation, a wave 
equation  with a  favorable  potential, verified by an  invariant 
scalar component $\phi$ of  the metric, i.e.
{\begin{equation}
\label{eq:Regge-W}
\Box_{m}
\phi= V\phi, \quad  V=\frac{4}{r^2}\left(1-\frac{2m}{r}\right).
\end{equation}}\unskip
where $\Box_m$ 
denotes the  wave operator of the Schwarzschild  
metric of mass $m$.   The R--W   study was completed by 
Vishveshwara~\cite{Vishev} and Zerilli~\cite{Ze}.  A gauge-invariant 
formulation of
\textit{metric perturbations}  was then given by Moncrief~\cite{Moncr}. 
\item  \textit{Teukolsky}~\cite{Teuk}. The curvature perturbation approach,
near Schwarzschild,    based on the  Newman--Penrose  (NP) formalism    
was  first  undertaken  by Bardeen--Press~\cite{Bar-Press}. This
approach  was later extended to the Kerr family by Teukolsky 
\cite{Teuk}, see also~\cite{P-T},   who made the  important  discovery
that  the extreme  curvature components, relative   to  a principal
null frame,  are  gauge invariant  and  satisfy  decoupled, separable, 
wave equations. The equations, bearing the name of   Teukolsky,  are
roughly   of the  form
{\begin{equation}
     \label{Teuk}
     \Box_{a,m} 
     \psi=\LL[\psi]
 \end{equation}}\unskip
     where $\LL[\psi]$ is a first order linear operator   in $\psi$.
\item \textit{Chandrasekhar}~\cite{Chand2}.  In~\cite{Chand2}   Chandrasekhar 
initiated  a  transformation theory relating  the  two   approaches. 
He exhibited a transformation which connects  the Teukolsky  equations 
to a       Regge--Wheeler  type equation.  In the particular case of
Schwarzschild  the transformation  takes the Teukolsky equation      
to the Regge--Wheeler  equation in~\eqref{eq:Regge-W}. The 
\textit{Chandrasekhar  transformation}  was further  elucidated and
extended  by  Wald~\cite{Wald} and recently by  Aksteiner and al~\cite{Akst}. 
Though originally  it was    meant only   to    unify the 
Regge--Wheeler approach with that of Teukolsky, the  Chandrasekhar
transformation, and various   extensions  of it,  turn out  to  play a 
fundamental   role in   the stability  problem. 

\item \textit{Whiting}~\cite{Whit}.   As mentioned before, the
\textit{full  mode stability}, i.e.\ lack of exponentially growing
modes,  for the Teukolsky equation on Kerr is  due  to 
Whiting\footnote{For the analogous result in the case of the  scalar
wave equation, see~\cite{Fins1}. See also~\cite{Yacov}  for a stronger
quantitative  version which was used in~\cite{mDiRySR2014}.}, 
see~\cite{Whit}.   Stronger quantitative  versions  were   proved 
in~\cite{AMPW,Fins2,Teix}. 

\item \textit{Reconstruction}. Once  we know  that the Teukolsky
variables, i.e.\ the extreme components of the curvature tensor   verify
mode stability, i.e.\ there are no exponentially growing modes, it still
remains  to  deal with the problem of reconstruction, i.e.\ to  find a
gauge relative to which all other components of the curvature  and
Ricci coefficients enjoy the same property.  We refer the  reader  to
Wald~\cite{Wald}  and the   references within  for a treatment of this 
issue in the physics literature.
 \end{enumerate} 


     
\subsection{Quantitative decay for the scalar wave equation}
\lab{section:QuantitativeDecay}

     
As mentioned in Section~\ref{section:ModeStab},  \textit{mode
stability}     is far from establishing  even the  boundedness of
solutions. To achieve that\footnote{The first   realistic boundedness
result for solutions of the scalar wave equation in Schwarzschild
appears in~\cite{K-Wald} based on a clever use of the energy method 
which takes into account  the degeneracy of $\T$ at the horizon.}  and,
more  importantly, to   derive realistic decay estimates,  one needs 
an entirely different approach based  on what one may   call the  ``new
vectorfield  method'',  in  section~\ref{section:NewVf}.    The method 
has    emerged    in connection to the study of boundedness and decay 
for the    scalar wave equation in $\KK(a,m)$,
{\begin{equation}
 \label{eq:WaveKerr}
  \Box_{a,m}
  \phi=0. 
 \end{equation}}\unskip
  

  The starting and most  demanding   part of the new method,  which (in
the case $a=0$)        appeared first in the  pioneering work of Blue
and Soffer~\cite{B-S1}, is  the derivation of  a global,  combined 
\textit{Energy--Morawetz}   estimate  which degenerates   in the
trapping region.  Once an Energy--Morawetz estimate is established  one 
can  commute  with the   Killing    vectorfields   of the background
manifold, and  the so called red shift  vectorfield  introduced 
in~\cite{DaRo1},   to derive uniform bounds  for  solutions.  The most
efficient  way  to also  get decay, and  solve the \textit{superposition  problem} (see Section~\ref{subs:MainDiffic}),
originating in~\cite{Da-Ro3},  is  based on   the presence of   family
of \textit{$r^p$-weighted},  quasi-conformal  vectorfields  defined in
the non-causal,   far $r$ region of spacetime\footnote{These replace
the scaling  and inverted time translation  vectorfields used 
in~\cite{Kl-vectorfield}  or their  corresponding  deformations used 
in~\cite{Ch-Kl}.  A recent improvement of the method  allowing one to
derive higher order decay  can be found in~\cite{AnArGa}.}. 

 
   The first Energy--Morawetz  type  results  for scalar wave  
Equation~\eqref{eq:WaveKerr}   in Schwarzschild, i.e.\ $a=0$,   are due
to  Blue--Soffer~\cite{B-S1,B-S2} and  Blue--Sterbenz~\cite{B-St},  
based on a modified version of the classical  Morawetz  integral energy
decay  estimate.     Further developments appear in the works of
Dafermos--Rodnianski, see~\cite{DaRo1,Da-Ro3},  and  
Marzuola--Metcalfe--Tataru--Tohaneanu~\cite{MaMeTaTo}.  The vectorfield
method    can also be extended to derive decay   for axially
symmetric solutions   in Kerr, see~\cite{I-Kl1} and\footnote{In his
Princeton PhD thesis Stogin   establishes a Morawetz estimate  even for
the full subextremal   case $|a|&lt;m$.}~\cite{St}, but it is  known to
fail   for  general  solutions  in Kerr, see  Alinhac~\cite{Al}.

       
In the absence of axial symmetry   the derivation of    an   
Energy--Morawetz estimate   in    $\KK(a,m)$ for  $|a/m|\ll 1 $ requires
a more refined  analysis involving     both the vectorfield method and
either  micro-local methods or mode decompositions.   The first   full
quantitative  decay\footnote{See also~\cite{DaRo2}  for the first proof
of  boundedness of solutions, based on mode decompositions.}   result,
based on micro-local analysis techniques, is due to
Tataru--Tohaneanu~\cite{Ta-Toh}.  The   derivation of    such an
estimate  in the full sub-extremal case $|a|&lt;m$  is even more subtle 
and   was  achieved  by
Dafermos--Rodnianski--Shlapentokh--Rothman~\cite{mDiRySR2014}  by 
combining the  vectorfield method with a  full  separation of variables
approach.    A  purely physical   space proof  of  the  Energy--Morawetz
estimate   for small  $|a/m|$,  which avoids both  micro-local analysis
and mode  decompositions, was pioneered   by  Andersson--Blue
in~\cite{A-B}.     Their method, which  extends the classical
vectorfield method to  include    second order  operators (in this case
the Carter operator, see Section~\ref{section:Kerr}),    has the usual
advantages of  the classical  vectorfield method, i.e.\ it is robust with
respect to perturbations.    It is for this reasons 
that~\cite{GKS-2022}   relies  on it  in the proof of
Theorem~\ref{MainThm-firstversion}.  

     
       
\subsection{Linear  stability  of Schwarzschild}  
\lab{section:LinSchw}
  
  
 A  first  quantitative  proof   of the  linear stability of
Schwarzschild  spacetime  was established\footnote{A  somewhat weaker
version of  linear stability of Schwarzschild    was   subsequently  
proved   in~\cite{HKW} by using the original, direct,   Regge--Wheeler,
Zerilli  approach combined with the vectorfield method and    adapted
gauge  choices. See also~\cite{Johnson} for an alternate  approach  of 
linear stability of Schwarzschild using wave coordinates.}  by 
Dafermos--Holzegel--Rodnianski  (DHR) in~\cite{D-H-R}.   Notable in their
analysis  is   the treatment of  the Teukolsky equation  in  a fixed
Schwarzschild background.   While  the Teukolsky equation    is
separable, and  amenable to mode analysis, it  is not  variational and 
thus cannot  be treated    directly  by      energy  type estimates. As
mentioned earlier  in Section~\ref{section:Modestability},
Chandrasekhar   was able to relate  the Teukolsky equation to the   
Regge--Wheeler (RW)  equation, which is both  variational and coercive
(the potential  $V$  has a  favorable sign). In~\cite{D-H-R}  the
authors  rely   on a physical space  version of   the  Chandrasekhar
transformation.   Once  decay  estimates  for   the RW  equation   
have   been established, based on the technology developed earlier for
the scalar wave equation in Schwarzschild,    the authors  recover  
the  expected boundedness and  decay for  solutions to  the original  
Teukolsky equation.  

The remaining work in~\cite{D-H-R}   is to       derive  similar
control   for the other curvature  components and  the   linearized 
Ricci coefficients associated to the double null foliation.  This  
last step  requires carefully chosen gauge conditions, which the
authors   make  within the framework of  a double null   foliation, 
initialized  both  on the  initial hypersurface and the  background
Schwarzschild horizon\footnote{The authors use a a scalar condition for
the linearized lapse along the event horizon (part of what the authors
call future normalized gauge), itself initialized from initial  data,
see (212) and (214) in~\cite{D-H-R}.}. This gauge fixing  from initial
data leads to sub-optimal decay estimates  for some  of the metric
coefficients\footnote{See (250)--(252)  and  (254) in~\cite{D-H-R}.}  
and is  thus inapplicable to the nonlinear case. This deficiency  was 
fixed in  the PhD thesis of E. Giorgi,  in the context of the  linear
stability of  Reissner--Nordstr\"om, see~\cite{Giorgi-RN1,Giorgi-RN2},  
by relying on  a linearized version of the GCM
construction  in~\cite{KS}.


\subsection{Linear stability of  Kerr for small angular momentum}
\lab{section:LinstabKerr}

 
The first  breakthrough result on the linear stability of Kerr,  
for $|a|/m\ll 1$,  is due  to  Ma~\cite{Ma},  followed by~\cite{D-H-R-Kerr}. 
Both results  are  based on a generalization of the
Chandrasekhar transformation to Kerr which   takes the Teukolsky
equations, verified by the extreme  curvature components,    to  
generalized versions of the  Regge--Wheeler (gRW)  equation.    Relying
on separation of variables  and vectorfield techniques,  similar to
those developed   for  the scalar wave equation in slowly rotating 
Kerr, the authors  derive Energy--Morawetz and $r^p$ estimates    for
the  solutions  of the gRW  equations. Note that these  results   were 
recently extended to the full subextremal range, $|a|&lt;m$, 
in~\cite{Y-R1,Y-R2} and\footnote{This latter   uses   the
micro-local framework of H\"afner, Hintz and Vasy, see~\cite{HHV}.}~\cite{Millet}. 

 The first   stability results for the full linearized   Einstein
vacuum equations   near   $\KK(a, m)$,   for $|a|/m\ll 1$,  appeared 
in~\cite{ABBMa2019,HHV}.  The first paper,  based on
the   GHP formalism\footnote{An adapted  spinorial version  of the   
NP formalism.} see~\cite{GHP},    builds  on the results of~\cite{Ma}  
while the second paper is based on an adapted  version of 
the   metric formalism and builds on the seminal work of the authors on
Kerr--de Sitter~\cite{HVas}. Though the ultimate  relevance  of these 
papers to  nonlinear stability  remains open,   they are  both 
remarkable results  in so far as they deal with difficulties that
looked insurmountable even ten years ago.    

\subsection{Nonlinear model problems}  

\subsubsection{Nonlinear stability of Kerr--de Sitter}  
\lab{section:Kerr-DS}

There is another important,  simplified,   nonlinear model problem
which has drawn  attention in recent years, due mainly to the  striking
achievement  of Hintz and Vasy~\cite{HVas}. This is the problem of
stability  of Kerr--de Sitter  concerning    the Einstein vacuum
equation  with  a strictly positive cosmological  constant
{\begin{equation}
\label{eq:Einstein-cosmological}
\R_{\a\b}  +\Lambda \g_{\a\b}=0, \quad \La&gt;0.
\end{equation}}\unskip

In their work, which   relies  in part   on the important  mode
stability result     of   Kodama and  Ishibashi~\cite{K-I},      Hintz
and Vasy were able    prove the nonlinear   stability   of the
stationary part  of  Kerr--de Sitter with small angular momentum, the
first nonlinear stability  result of any nontrivial  stationary
solutions   for the  Einstein equations\footnote{This is also   the
first  general nonlinear  stability  result  in GR  establishing  
asymptotic stability towards a family of solutions, i.e.\ full
quantitative   convergence  to a final state  close,  but  different
from the  initial one.}. It is important to note that,  despite  the  
fact that,  formally,  the Einstein vacuum
Equation~\eqref{eq:EVE-intro} is the limit\footnote{To pass  to the
limit  requires one  to  understand  all global in time solutions
of~\eqref{eq:Einstein-cosmological} with $\La=1$, not only  those which
are small perturbations of   Kerr--de Sitter,  treated by~\cite{HVas}.}
of~\eqref{eq:Einstein-cosmological} as $\La\to 0$, the  global 
behavior  of  the corresponding solutions   is   radically
different\footnote{Major   differences between   formally close
equations occur in many other   contexts. For example, the 
incompressible Euler equations   are formally the limit of the
Navier--Stokes equations as the viscosity tends to zero. Yet, at fixed 
viscosity, the global properties of the Navier--Stokes equations are
radically different   from that  of the  Euler equations.}.   

 
The main  simplification in the case of stationary solutions 
of~\eqref{eq:Einstein-cosmological} is that  the  expected decay   rates 
of perturbations  near   Kerr--de Sitter     is exponential,   while in
the case $\La=0$ the decay is  lower degree polynomial\footnote{While
there is exponential decay in the stationary part treated
in~\cite{HVas}, note that lower degree polynomial decay is expected in
connection to the  stability of the complementary causal region 
(called   cosmological   or expanding)  of the full  Kerr--de Sitter 
space, see e.g.~\cite{Vo}.},  with various components of tensorial
quantities  decaying at different  rates,  and the slowest decaying
rate\footnote{Responsible  for carrying  gravitational waves at large
distances so that they are  detectable.} being no better than $t^{-1}$.
The  Hintz--Vasy result was  recently revisited  in the work of
Fang~\cite{Fang2,Fang1}  using   composite  methods  which 
bridge the gap between the spectral methods of~\cite{HVas} and the 
more  conventional  vectorfield methods used in the field.


\subsubsection{Nonlinear stability of Schwarzschild}  

 
The first  nonlinear stability result of the Schwarzschild space  was
established  in~\cite{KS}.    In its simplest version, the  result 
states the following.


\begin{THEOREM}[Klainerman--Szeftel~\cite{KS}]\label{MainThm-firstversion-KS}
The future globally hyperbolic   development  of  an   \emph{axially
symmetric, polarized},  asymptotically  flat   initial data set,
sufficiently close  (in a specified  topology)  to a Schwarzschild 
initial data set   of  mass $m_0&gt;0$,  has a complete    future null
infinity  $\II^+$ and converges  in  its causal past 
$\JJ^{-1}(\II^{+})$  to another  nearby  Schwarzschild solution of mass
$m_f$ close to $m_0$, see Figure~\ref{fig:schwarzchildbootreg} below. 
 \end{THEOREM}

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig:schwarzchildbootreg}
The GCM admissible space-time $\mathcal{M}$. By comparison to
Figure~\ref{fig1-introd},  $\MM$  does not have $\Mtop$,  the past  
boundaries  $\CC_0\cup\CCb_0$ and future boundary   $\CCb\cup\CCb_*$ 
are null and   the horizontal structures (induced by  geodesic
foliations)  are integrable.   As  in Theorem~\ref{MainThm-firstversion}, 
the crucial GCM  sphere $S_*$ is  defined
and constructed with no reference to the initial
data.}
\end{figure}
                                             
 
The restriction to  axial polarized perturbations  is the  simplest 
assumption   which insures  that the final state is itself
Schwarzschild  and thus avoids the additional complications  of  the
Kerr stability problem (see   the introduction in~\cite{KS} for a full
discussion of the result).  

  The proof is based  on   an adaptation of  GCM admissible spacetimes 
similar to  that   discussed below in Section~\ref{section:Proof}.
There are however several  important   simplifications to be noted:
\begin{itemize}
\item    The assumption of  polarization makes the  constructions of 
the GCM  spheres $S_*$ and   spacelike hypersurface $\Si_*$
significantly simpler, see Chapter 9 in~\cite{KS}, by comparison to 
the general  case treated in~\cite{KS-GCM1,KS-GCM2,Shen}. 
\item The  corresponding  GCM admissible spacetime  has only two
components   $\MM=\Mext\cup\Mint $ and the  null horizontal 
structures, defined  on  each component,    are   integrable. 
\item   As in the case of the scalar wave equation on Schwarzschild
space the main    spin-2 Teukolsky   wave equations   can be treated
(via   the passage to the Regge--Wheeler equation)  by  a  vectorfield
approach. This is no longer true  in   Kerr and even less so in 
perturbations of Kerr.  
\end{itemize}



Recently Dafermos--Holzegel--Rodnianski--Taylor~\cite{DHRT} have  
extended\footnote{The novelty of~\cite{DHRT},  compared to~\cite{KS}, 
is the well preparation of  the initial data, based on an additional  
three  dimensional modulation. Note however  that~\cite{DHRT}  requires
substantially stronger   asymptotic conditions for the initial data
compared to~\cite{KS}.}  the result of~\cite{KS}      by properly 
preparing a co-dimension 3 subset of the initial data such that the
final state is  still Schwarzschild. Like in~\cite{KS},  the starting
point of~\cite{DHRT} is to anchor  the entire construction   on a  far
away\footnote{That is  $r\gg u$, similar to the dominant in $r$
condition (3.3.4) of~\cite{KS}.} GCM type  sphere $S_*$, in the sense
of~\cite{KS-GCM1,KS-GCM2}, with no  direct reference to the
initial data. It also  uses  the same definition of the  angular
momentum as in (7.19) of~\cite{KS-GCM2}. Finally, the spacetime
in~\cite{DHRT} is separated in an exterior region $\Mext$ and an
interior region $\Mint$,   with the ingoing  foliation of $\Mint$, 
initialized based on the information induced by $\Mext$,  as
in~\cite{KS}. Note, however, that~\cite{DHRT} does not use the geodesic
foliation of~\cite{KS}, but instead both  $\Mint$ and $\Mext$ are
foliated  by double null    foliations, and thus, the process of
estimating the gauge dependent variables is somewhat different.  The
proof also makes use  of the same definition  of   the  angular
momentum   as  introduced   in~\cite{KS-GCM2} and  used in~\cite{KS}.


\section{Kerr stability for small angular momentum}
\lab{section:Proof}

   
\begin{THEOREM}\lab{MainThm-firstversion}
The future globally hyperbolic   development  of  a general,  
asymptotically  flat,    initial data set, sufficiently close  (in a
suitable  topology)  to a   $\KK(a_0, m_0) $   initial data set,   for
sufficiently  small $|a_0|/m_0$,  has a complete    future null
infinity  $\II^+$ and converges  in  its causal past 
$\JJ^{-1}(\II^{+})$  to another  nearby Kerr spacetime $\KK(a_f, m_f)$
with parameters    $(a_f, m_f)$ close to the initial ones $(a_0, m_0)$,
see Figure~\ref{fig3} below.
 \end{THEOREM}  
 
 
 \begin{figure}
 \includegraphics{fig03}
 \caption{\label{fig3}The Penrose diagram of the  final   space-time in  
 Theorem~\ref{MainThm-firstversion} with initial hypersurface $\Si_0$,  future
space-like boundary $\AA$, and $\II^+$ the complete future null
infinity.  The null  hypersurface $\HH_+$ is the future event horizon
of  the final Kerr.    This was recently shown to  be  a smooth
hypersurface in~\cite{Chen-K}.}
 \end{figure}
 


 The precise version of   the result,    all the main  features of the 
architecture of its proof,  as well    as detailed proofs  for  most of
the main steps  are to be found in~\cite{KS:Kerr}.   The full proof
relies  also on~\cite{GKS-2022}  as well as~\cite{KS-GCM1,KS-GCM2} on
GCM spheres, and the extension~\cite{Shen} to GCM hypersurfaces.  In
what follows I    describe  some of the main features of the proof.



\subsection{The bootstrap  region}


 The proof   of Theorem~\ref{MainThm-firstversion}  is  centered  
around    a  limiting  argument for    a family of  carefully
constructed finite  \textit{generally covariant modulated   (GCM)
admissible  spacetimes} $\MM=\Mext\cup\Mtop\cup\Mint$.    As can be
seen in Figure~\ref{fig1-introd}  below, the  future boundary of the 
spacetime is given by $\AA\cup\Sitop\cup \Si_*$ where  $\Si_*$  is a
spacelike, generally covariant modulated (GCM) hypersurface, that is a
hypersurface verifying  a  set of  crucial, well-specified,  geometric
conditions,  essential to  the proof of   convergence  to a final
state. 


\begin{figure}
\includegraphics{fig04}
\caption{\label{fig1-introd}
The Penrose diagram of  a finite GCM admissible    space-time
$\MM=\Mext\cup\Mtop\cup\Mint$. The spacetime  is prescribed in the
initial layer  $\LL_0$ and has $\AA\cup\Sitop\cup \Si_*$ as future
boundary, with $\Si_*$  a spacelike ``generally covariant modulated
(GCM)'' hypersurface. Its past boundary,  $\BB_{1}\cup\BBb_{1}$, is  itself
part of the construction.  $\Mext$ is   initialized  by   the GCM
hypersurface $\Si_*$ while  $\Mint$ is initialized  on  $\TT $ by the
foliation induced by $\Mext$. The main inovation is the  GCM sphere
$S_*$, defined  and constructed  with no reference  to the initial data
prescribed   in   the  initial data layer $\LL_0$.}
\end{figure}


The capstone as well as  the most original part   of the entire
construction is the  sphere $S_*$, the  future boundary  of $\Si_*$, 
which verifies a  set of rigid, extrinsic and intrinsic,  conditions.  
Once $\Si_*$ is specified the whole  GCM admissible spacetime $\MM$  
is  determined by  a more  conventional  construction, based on 
geometric transport type equations. More precisely $\Mext$  can be
determined  from $\Si_*$ by  a specified  outgoing  foliation
terminating in the timelike boundary $\TT$,  $\Mint$ is determined from
$\TT$ by a  specified  incoming  one, and $\Mtop$  is a complement of
$\Mext\cup\Mint$   which makes $\MM$ a   causal domain\footnote{This is
required because of the fact that, in  the~\cite{KS} construction,  the
future  boundary of $\Mext\cup\Mint$ is not causal. By contrast,   
in~\cite{KS}, $\MM=\Mext\cup \Mint$.}.
The past boundary
$\BB_1\cup\BBb_1$ of $\MM$, which is itself  to be constructed,  is
included in the initial layer $\LL_0$ in which the  spacetime is
assumed  to be known, i.e.\ a small perturbation of a Kerr solution.  
The passage from  the initial data specified  on $\Si_0$ to  the
initial layer  spacetime $\LL_0$ is  justified by Shen
in~\cite{Shen:Kerrext} by arguments similar to those of~\cite{KlNi1,KlNi2},   
based  on the mathematical methods  and
techniques introduced in~\cite{Ch-Kl}.

  
Each of the spacetime regions $\Mext, \Mint,\Mtop $ come equipped  with
specific geometric  structure including specific   choices of  null   
frames and functions such as $r,  u, \ub$. These are first defined on
$\Si_*$ and then transported   to  $\Mext, \Mint,\Mtop $.



\subsection{Main intermediary results}


The proof of Theorem~\ref{MainThm-firstversion} is divided in nine
separate steps, Theorems M0--M8. These steps are  briefly described 
below, see Section 3.7 in~\cite{KS:Kerr} for the precise statements:
\begin{enumerate}[(1)]
\item \textit{Theorem M0 (Control of the initial data in the bootstrap
gauge)}.  The smallness of the initial perturbation  is   given  in the
frame of the initial data layer $\LL_0$.  Theorem M0 transfers  this
control to the bootstrap gauge in the initial data layer. 
 \item \textit{Theorems M1--M2 (Decay estimates for $\a$ (Theorem M1) and $\aa$ (Theorem M2))}.  This is achieved   using Teukolsky equations and a Chandrasekhar type transform in perturbations of Kerr. 
\item \textit{Theorems M3--M5 (Decay estimates for all curvature,
connection and metric components)}. This is done making use of the GCM
conditions on $\Si_*$ as well as    the control of $\a$ and $\aa$
established in Theorems M1 and M2. The proof proceeds in the following
order:
\begin{itemize}
\item Theorem M3 provides the  crucial  decay  estimates on   $\Si_*$, 
\item Theorem M4 provides  the  decay estimates on  $\mbox{}^{(\mathrm{ext})}\MM$,
\item Theorem M5 provides  the  decay estimates on  $\mbox{}^{(\mathrm{int})}\MM$ and
$\mbox{}^{(\mathrm{top})}\MM$.
\end{itemize} 
\item \textit{Theorems M6 (Existence of a bootstrap spacetime)}. This
theorem shows that there exists a GCM admissible spacetime satisfying
the bootstrap assumptions, hence initializing the bootstrap procedure.
\item \textit{Theorems M7 (Extension of the bootstrap region)}. This
theorem shows the existence of a slightly larger GCM admissible
spacetime satisfying estimates improving the bootstrap assumptions on
decay.  
\item \textit{Theorem M8 (Control of the  top derivatives estimates)}. 
This is based on an induction argument relative to the number of
derivatives,  energy--Morawetz estimates and the Maxwell like character
of the Bianchi identities. 
 \end{enumerate}\Tg&lt;/p&gt;
The paper~\cite{KS:Kerr} provides the proof of Theorem M0, Theorems M3
to M7, and half of Theorem M8 (on the control of Ricci coefficients and
metric components). The proof of Theorems M1 and M2, and of the other
half of Theorem M8 (on the control of curvature components), based  on 
nonlinear wave equations techniques, are provided in~\cite{GKS-2022}.
The construction of GCM spheres in~\cite{KS-GCM1,KS-GCM2}, and
of GCM hypersurfaces in~\cite{Shen} are used in the proof of Theorems
M6 and M7 to construct respectively the terminal GCM sphere  $S_*$ and
the last slice hypersurface $\Si_*$ from $S_*$. 

 \subsection{Main new   ideas of the proof}
 \label{section:Mainideas}
 
 Here is  a short description of the main new ideas in the proof  of  
Theorem~\ref{MainThm-firstversion}  and how they compare  with ideas 
used in   other  nonlinear  results.

\subsubsection{GCM admissible spacetimes} \label{sec:GCMadmissiblespacetimes}
\mbox{} \vspace*{-1pc}
 \begin{itemize}
\item As mentioned  already     the crucial concept  in the proof of
Theorem~\ref{MainThm-firstversion}  is that of  a  GCM admissible
spacetime,  whose  construction is  anchored  by the GCM sphere $S_*$
in Figure~\ref{fig1-introd}.    GCM spheres\footnote{See the  
discussion in the introductions to~\cite{KS-GCM1,KS-GCM2}.},
are codimension 2 compact surfaces, unrelated   to the initial
conditions,   on which  specific geometric quantities take
Schwarzschildian values  (made possible by taking into account the full
general covariance of the Einstein vacuum equations). In addition to
these  extrinsic conditions the sphere $S_*$ is endowed with a choice
of ``effective\footnote{This  is meant to insure  the rigidity of the
uniformization map, see~\cite{KS-GCM2}.}  isothermal coordinates'',  
$(\th, \vphi)$ verifying the following properties:\Tg&lt;/p&gt;
 \begin{itemize}
\item[--]  The metric on $S_*$  takes the form $ g= \mathrm{e}^{2\phi} r^2(
(\mathrm{d}\th)^2+ \sin^2 \th (\mathrm{d}\vphi)^2)$.
 \item[--]   The integrals on $S_*$ of  the  $\ell=1$ modes\footnote{This
is a natural generalization of $\ell=1$ spherical harmonics.}   $
J^{(0)} :=\cos\th$,   $ J^{(-)} :=\sin\th\sin\vphi$ and $ J^{(+)}
:=\sin\th\cos\vphi $   vanish identically.
 \end{itemize}
\item Given the GCM sphere $S_*$ and the effective isothermal
coordinates $(\th, \vphi)$ on it,  the GCM procedure  allows us, in
particular,  to define the   mass $m$, the angular momentum $a$ and a
virtual axis of rotation which converge, in the limit,  to the final
parameters $a_f, m_f$ and the  axis of rotation of the final
Kerr\footnote{Previous  definitions of the angular momentum  in General
Relativity  were given  in~\cite{Rizzi,Chen,Chen2}, see
also~\cite{Sz} for a comprehensive discussion of the subject.}.   I
refer the reader to  Section 7.2 in~\cite{KS-GCM2} for  the  intrinsic
definition of $a$ and of the virtual axis of symmetry on a GCM sphere.

 \item  The boundary $\Si_*$, called a GCM  hypersurface,  is
initialized at $S_*$  and verifies additional conditions.  In the 
polarized setting the  first such construction appears~\cite{KS}.  The
general case is treated in~\cite{Shen}.   

\item   The concepts of         GCM   spheres   has appeared first
in~\cite{KS} in the context of polarized  symmetry.   The  construction
of GCM spheres,  without any  symmetries, in realistic perturbations of
Kerr, is treated in~\cite{KS-GCM1,KS-GCM2}\footnote{See also
chapter 16 of~\cite{DHRT} in the particular case of perturbations of
Schwarzschild, where the same concept  appears instead under the name 
``teleological''.}.   

 \item  The main novelty of  the GCM approach   is that it    relies on
gauge conditions initialized at a far away co-dimension $2$ sphere 
$S_*$,  with no direct reference to the initial conditions. Previously
known geometric constructions, such as  in~\cite{Ch-Kl,KlNi1,KLR},    
were based on codimension-1 foliations constructed on spacelike or null
hypersurfaces and initialized    on the initial
hypersurface\footnote{The first such  construction  appears in the
proof of  the nonlinear stability of the Minkowski  space~\cite{Ch-Kl}
where the ``inverse lapse foliation'' was  constructed on the ``last
slice'',    initialized  at  spacelike  infinity $i^0$. Similar
constructions, where the last slice is null rather than spacelike, 
appear in~\cite{KlNi1,KLR}.}. Gauge conditions initialized from the
future with no direct reference to the initial conditions, which was
initiated in~\cite{KS}, have since been used in other works,
see~\cite{Giorgi-RN1,Giorgi-RN2,graf,DHRT}.

 \item  The GCM  construction introduces the following new  important 
conceptual   difficulty.  The  foliation on $\Si_*$, induced from the
far away  sphere $S_*$, needs to be  connected, somehow,    to the
initial  conditions (i.e.\ the initial layer $\LL_0$ in 
Figure~\ref{fig1-introd}). This is achieved in  both~\cite{KS} and~\cite{KS:Kerr} 
by transporting\footnote{That is, one transport sthe
$\ell=1$ modes of some quantities from $S_*$ to $S_1$, see Section
8.3.1 in~\cite{KS:Kerr}.} the   sphere   $S_*$  to  a  sphere $S_1$ in
the   the initial  layer        and  compare  it, using the rigidity
properties  of the GCM conditions, to a sphere of  the initial data
layer. This  induces  a new foliation  of  the initial  layer  which 
differs substantially from  the  original one, due to a shift  of the
center of mass frame of the  final black hole, known  in the physics
literature as a gravitational wave recoil\footnote{ See  section 8.3 
in~\cite{KS:Kerr} for the details.}.
 \end{itemize}
 

 \subsubsection{Non integrability of the horizontal structure}
 \label{sec:nonintegrabilityhorizontalstructure}
 
 
 As mentioned in Section~\ref{section:Kerr},  the canonical  horizontal
structure    induced by the principal null directions $(e_3, e_4)$ 
in~\eqref{eq:PrincipalNull}   of Kerr are non integrable.  The lack of
integrability is    dealt with by    the  Newman--Penrose (NP) 
formalism by  general null frames $(e_3, e_4, e_1, e_2)$, with  $e_1,
e_2$ a specified  basis\footnote{Or rather  the complexified vectors
$m=e_1+ \mathrm{i}e_2$ and $\ov{m} = e_1-\mathrm{i} e_2$.}   of the horizontal structure
induced by the null pair   $(e_3, e_4)$.  It thus reduces all
calculations to  equations  involving    the  Christoffel  symbols  of
the frame, as scalar quantities. This  un-geometric feature of the
formalism makes it difficult to use it in  the nonlinear setting of 
the Kerr  stability  problem.  Indeed  complex calculations   depend   
on higher  derivatives  of  all   connection coefficients of the NP 
frame rather than only  those which are geometrically significant. 
This  seriously affects  and complicates  the structure of non-linear
corrections and makes  it difficult to avoid  artificial gauge type
singularities\footnote{There are   no    smooth,  global   choices  of 
a basis $(e_1, e_2)$.  The choice~\eqref{eq:canonicalHorizBasisKerr-intro} in Kerr, for example, is
singular  at $\th=0, \uppi$.}. This difficulty is avoided 
in~\cite{Ch-Kl} by working with  a  tensorial approach  adapted to  
$S$-foliations, i.e.\ $\{e_3, e_4\}^\perp$ coincides, at every point,
with the tangent space to $S$.  

The paper~\cite{GKS-2022}    extends,  with minimal  changes,   the
tensorial approach  introduced in~\cite{Ch-Kl},  to general 
non-integrable  foliations.    The idea is  simply to    define  Ricci
coefficients $\chi, \chib,\eta, \etab, \ze, \xi, \xib,  \om, \omb$   
exactly  as in~\cite{Ch-Kl},  relative to an arbitrary   basis of 
vectors $(e_a)_{a=1,2} $ of 
$\HH :={\{e_3, e_4\}}^\perp$. In particular, 
the null fundamental  forms  $\chi$ and $\chib$, are given by
{\begin{eqnarray*}
\chib_{ab}=\g(\D_ae_3, e_b),\quad \chi_{ab}=\g(\D_ae_4, e_b).
\end{eqnarray*}}\unskip
Due to the  lack of integrability of  $\HH$,  the null fundamental 
forms  $\chi$ and $\chib$ are no longer symmetric.  They can be both
decomposed as follows
{\begin{eqnarray*}
\chi_{ab}=\tfrac{1}{2} \trch\de_{ab}+\tfrac{1}{2} \in_{ab}\atrch +\chih_{ab},
\quad  \chib_{ab}=\tfrac{1}{2} \trchb\de_{ab}+\tfrac{1}{2} \in_{ab} \atrchb
+\chibh_{ab},
\end{eqnarray*}}\unskip
where  the new scalars $\atrch$, $\atrchb$ measure the lack of
integrability  of  the horizontal structure. The null curvature
components are also defined  as in~\cite{Ch-Kl}, 
{\begin{eqnarray*}
\a_{ab}=\R_{a4b4},\quad  \b_a=\tfrac{1}{2} \R_{a434},\quad
\bb_a=\tfrac{1}{2} \R_{a334}, \quad  \aa_{ab}=\R_{a3b3},\quad 
\rho=\tfrac{1}{4} \R_{3434}, \quad  \rhod=\tfrac{1}{4}\dual \R_{3434}.
\end{eqnarray*}}\unskip
The  null structure and null Bianchi equations can then be derived  as 
in the integrable case, see chapter 7 in~\cite{Ch-Kl}. The only new
features are the presence of the scalars $\atrch, \atrchb $ in the 
equations. Finally  note that the equations acquire additional
simplicity if we pass to complex notations\footnote{The dual here is
taken with respect to the antisymmetric  horizontal 2-tensor
$\in_{ab}$.}, 
{\begin{equation}\label{eq:defcomplexquantitieshorizontalstruct}
\begin{array}{c}
 \displaystyle A := \a+\mathrm{i}\dual\a, \quad B:=\b+\mathrm{i}\dual\b, \quad
P :=\rho+\mathrm{i}\dual\rho,\quad 
\Bb :=\bb+\mathrm{i}\dual\bb, \quad 
\Ab :=\aa+\mathrm{i}\dual\aa,\\
 \displaystyle X := \chi+\mathrm{i}\dual\chi, \quad \Xb :=\chib+\mathrm{i}
 \dual\chib, \quad H :=\eta+\mathrm{i}\dual
\eta, \quad \Hb :=\etab+\mathrm{i}\dual \etab, \quad  Z :=\ze+\mathrm{i}\dual\ze.
\end{array}
\end{equation}}\unskip


 
\subsubsection{Frame transformations and choice of frames}


Given an arbitrary perturbation of Kerr,  there is no a-priori  reason
to prefer an horizontal structure  to any other one obtained from the
first by another perturbation of the same size. It is thus   essential 
that one   considers  all possible frame transformations from  one  
horizontal structure  $(e_4, e_3, \HH)$ to another one  $(e_4', e_3', 
\HH')$  together with the transformation  formulas $\Ga\to \Ga'$, $R\to
R'$  they generate. The most general  transformation formulas  between
two null frames is given in Lemma  3.1  of~\cite{KS-GCM1}. It depends
on two  horizontal  $1$-forms  $f, \fb$ and a  real scalar function
$\la$ and is given by
{\begin{eqnarray}
 \label{General-frametransformation-intro}
 \begin{array}{l}
\displaystyle  e_4'=\la\left(e_4 + f^b  e_b +\tfrac{1}{4} |f|^2  e_3\right),\\[0.5pc]
\displaystyle  e_a'= \left(\de_a^b +\tfrac{1}{2}\fb_af^b\right) e_b +\tfrac{1}{2}  
\fb_a  e_4 +\left(\tfrac{1}{2} f_a +\tfrac{1}{8}|f|^2\fb_a\right)   e_3,\\[0.5pc]
\displaystyle e_3'=\la^{-1}
\left( \left(1+\tfrac{1}{2}f\c\fb +\tfrac{1}{16} |f|^2  |\fb|^2\right) 
e_3 + \left(\fb^b+\tfrac{1}{4}
|\fb|^2f^b\right) e_b  + \tfrac{1}{4} |\fb|^2 e_4 \right).
 \end{array}
 \end{eqnarray}}\unskip
The  very important   transformation formulas  $\Ga\to \Ga'$, $R\to R'$
are given in Proposition 3.3 of~\cite{KS-GCM1}. 



The case of $ \KK(a, m)$, $a\neq 0$   presents  an interesting  new 
feature which can be described as follows:
\begin{itemize}
\item To capture  the  simplicity induced by the principle null
directions in Kerr it is natural  to work with  non-integrable frames
and   define all  main quantities  relative to adapted   frames for
which all quantities which vanish in Kerr are of  the size of the
perturbation.

\item A   crucial  aspect of all    important  results in GR, based on 
integrable  $S$- foliations,  is  that one can rely on elliptic Hodge
theory  on each $2$-surface  $S$. This is  no longer possible in 
context where   the  main quantities  and the basic  equations  they
verify are  defined  relative to non integrable frames.  
In~\cite{KS:Kerr,GKS-2022}   
this problem is dealt with  by
passing back and forth, whenever needed,   from the main non-integrable
frame  to a  well chosen adapted  integrable frame,   according to the
transformation formulas mentioned above.
\end{itemize}

\subsubsection{Renormalization procedure and the canonical complex 1-form $\Jk$}
\lab{sec:renormalizationprocedure}


The main complex quantities introduced in~\eqref{eq:defcomplexquantitieshorizontalstruct} take a particularly
simple form in the principal null frame~\eqref{eq:PrincipalNull} of
Kerr:
{\begin{eqnarray}\label{eq:particularsimpleforminprincipalnullKerr}
\begin{array}{c}
\displaystyle A =\Ab=B=\Bb=0, \quad P=-\frac{2m}{q^3},\\
\displaystyle \Xh =\Xbh=0,\quad \tr X=\frac{2}{q}
\frac{\De}{|q|^2}, \quad \tr\Xb=-\frac{2}{\ov{q}},\\
\displaystyle Z =\frac{aq}{|q|^2}\Jk, \quad H=\frac{aq}{|q|^2}\Jk, 
\quad \Hb=-\frac{a\ov{q}}{|q|^2}\Jk,
\end{array}
\end{eqnarray}}\unskip
where $q=r+\mathrm{i}a\cos\th$, and where the regular\footnote{Note that $\Jk$
is regular including at $\th=0, \uppi$.} complex 1-form $\Jk$ is given by
{\begin{equation}
 \label{eq:DefineJk}
\Jk_1=\frac{\mathrm{i}\sin\th }{|q|}, \quad \Jk_2=\frac{\sin\th }{|q|},
 \end{equation}}\unskip
see Sections 2.4.2 and 2.4.3 in~\cite{KS:Kerr}. 
In particular, the following holds for the complexified horizontal
tensors of~\eqref{eq:defcomplexquantitieshorizontalstruct} in the
principal null frame~\eqref{eq:PrincipalNull} of Kerr:
\begin{itemize}
\item the complex scalars $P$, $\tr X$ and $\tr\Xb$ are functions of
$r$ and $\cos\th$,

\item the non vanishing complex 1-forms $H$, $\Hb$ and $Z$ consist of
functions of $r$ and $\cos\th$ multiplied by $\Jk$,

\item the traceless symmetric complex 2-tensors $A$, $\Ab$, $\Xh$ and
$\Xbh$ vanish identically. 
\end{itemize}

Based on that observation, for a given horizontal structure perturbing
the one of Kerr,  one  can define a renormalization  procedure    by
which, once we have\footnote{The constants $m$ and $a$ are computed on 
the  GCM sphere $S_*$, see Section~\ref{sec:GCMadmissiblespacetimes}.
$r$, $\th$ and $\Jk$ are chosen on $S_*$, transported to $\Si_*$ and
then to $\MM$. The horizontal structure is also defined first on
$\Si_*$ and then transported  to $\MM$.} suitable  constants $(a, m)$,
suitable scalar functions $(r, \th)$,  and a  suitable   complex 1-form
$\Jk$,   and after  subtracting  the corresponding  values  in Kerr
computed from $(a, m, r, \th, \Jk)$ for   all the  Ricci and  curvature
coefficients,  we  obtain   quantities  which are  first order in the
perturbation. 


  More precisely, once $(a,m)$,  $(r, \th)$ and $\Jk$   have been
chosen, one renormalizes the quantities 
in~\eqref{eq:defcomplexquantitieshorizontalstruct} that do not vanish in
Kerr as follows\footnote{The renormalization is written here in the
case of a null pair $(e_3, e_4)$ with an ingoing normalization.}:
{\begin{eqnarray}\label{eq:renormalizedquantities}
\begin{array}{c}
\displaystyle \widecheck{P} := P+\frac{2m}{q^3},\quad 
\widecheck{\tr X}:=\tr X-\frac{2}{q}\frac{\De}{|q|^2}, 
\quad \widecheck{\tr \Xb}:=\tr\Xb+\frac{2}{\ov{q}},\\
\displaystyle \widecheck{Z} :=Z-\frac{aq}{{|q|}^2}\Jk, \quad 
\widecheck{H}:=H-\frac{aq}{|q|^2}\Jk, 
\quad \widecheck{\Hb}:=\Hb+\frac{a\ov{q}}{|q|^2}\Jk.
\end{array}
\end{eqnarray}}\unskip
  

\subsubsection{Principal geodesic and principal temporal structures}


In addition to the GCM gauge conditions on $\Si_*$, one  needs to
construct a gauge on $\MM$ which relates the non integrable horizontal
structure to the scalars $(r, \th)$ and the complex 1-form $\Jk$.   Two
such gauges were introduced in~\cite{KS:Kerr}:
\begin{itemize}
\item  Principal Geodesic (PG) structure, which is a generalization of
the geodesic foliation to non-integrable horizontal structures,
\item  Principal Temporal (PT) structure, which favors transport
equations along a null direction.
\end{itemize}

The PG structure\footnote{Note that,  in the integrable context
of~\cite{KS},  the PG structure coincides  with the standard
(integrable) geodesic foliation used there. Thus the PG structure,   
defined in~\cite{KS:Kerr}, is a suitable   generalization    to the
non-integrable case  of perturbations of Kerr.}   is well suited for
decay estimates, but  fails to be well posed.  Indeed,    due to the
lack of  integrability  of the horizontal structure,  we cannot 
control  the null structure equations\footnote{In integrable situation,
like in the case of $S$-foliations, the  Hodge systems on   the leaves
of the $S$-foliation allows us to avoid the loss.} without a loss of
derivative. The  PT structure, on the other hand,  is designed  so that
the  loss  of derivatives    in the null structure equations, in the
incoming or outgoing direction, is completely avoided. Note however
that the PT structure  is not well suited to the derivation of decay
estimates on $\Mext$ where $r$ can take arbitrary large values. 
In~\cite{KS:Kerr}  the authors   work with both  gauge conditions,
depending  on   the goal  they  want to achieve, and rely on the
transformation formulas~\eqref{General-frametransformation-intro} to
pass from  one to the other.

 
In the outgoing normalization both  the outgoing  PG and PT structures
consist  of  a choice   $(e_3, e_4, \HH)$, with $e_4$ null geodesic,
together with a scalar functions $r, \th$ and a complex $1$-form $\Jk$ 
such that $e_4(r)=1$, $e_4(\th)=0 $, $\nab_4(q\Jk)=0$ where
$q=r+\mathrm{i}a\cos\th$. In addition: 
  \begin{enumerate}[(1)]
\item  In a  PG structure   the gradient  of $r$,  given by 
$N=\g^{\a\b} \partial_\b r \partial_\a$,  is perpendicular  to  $\HH$,
\item   In a    PT  structure  $\Hb=-{a\ov{q}}/{|q|^2}\Jk$, i.e.\ 
$\check{\underline{H}}=0$ in view of~\eqref{eq:renormalizedquantities}.
\end{enumerate}
A similar definition  of incoming  PG and PT  structures  is obtained 
by interchanging the roles  of $e_3, e_4$. Note that both structures
still  need to be initialized. The  outgoing  PG and PT structures of
$\Mext$ are both initialized  on $\Si_*$ from the GCM frame of $\Si_*$,
while  the ingoing  PT structures of $\Mint$ and $\Mtop$ are
initialized on the the timelike hypersurface $\TT$, see 
Figure~\ref{fig1-introd}, using  the  data induced by the outgoing structures.


\subsubsection{Control of the extreme curvature components $A, \Ab$} 


It was already observed by Teukolsky that, in linear theory,   the
extreme components of the curvature  are both  gauge invariant and  
verify  decoupled  wave equations\footnote{See discussion in 
Section~\ref{sect:LinearStab}.}.   In  the  nonlinear context   this translates
to the statement that   the horizontal 2-tensors  $A, \Ab$, defined 
relative to an $O(\ep)$ perturbation   of the principal  frame of 
Kerr,  are $O(\ep^2)$-invariant, relative to  $O(\ep)$ frame
transformations\footnote{This means that $f, \fb$, $\la-1$ are $O(\ep)$
in the transformation formulas~\eqref{General-frametransformation-intro}.},
and  verify  tensorial
wave equations of the form 
{\begin{equation}
\squared_2A  
+L[A]= \err(\Gac, \Rc),\quad   \squared_2\Ab  
+\underline{L}\, [\Ab ]= \und{\err}(\Gac, \Rc).
\end{equation}}\unskip
Here $\squared_2$ 
denotes the wave operator on horizontal symmetric
traceless 2-tensors,   $L$ and $\underline{L}$ are linear first order
operators  and    $\check{\Gamma}, \check{R}$ denote the linearized Ricci  and
curvature coefficients.  The error terms  $ \err(\check{\Gamma}, \check{R}), \,
\underline{\err}(\check{\Gamma}, \check{R})$ are nonlinear  expressions  in  $\check{\Gamma}, \check{R}$.      

     In linear theory, i.e.\ when $\g$ is the Kerr metric and  the error
terms  are not present,    these equations have been treated 
by~\cite{D-H-R} in Schwarzschild\footnote{See discussion in 
Section~\ref{section:LinSchw}.}  and by~\cite{Ma,D-H-R-Kerr} in
slowly rotating\footnote{See discussion in Section~\ref{section:LinstabKerr}.} 
Kerr, i.e.\ $|a|/m\ll 1$.    More precisely
both results   derive realistic decay estimates for  $A, \Ab$.  The
methods  are however  not    robust. Indeed,  a crucial ingredient in
the proof, the   Energy--Morawetz estimates,  is based on  separation of
variables. The control of $A$ and $\Ab$ in perturbations of Kerr
in~\cite{GKS-2022} contains the following new features:
\begin{itemize}
\item \textit{Derivation of the gRW equation}. The derivation of the   
generalized Regge--Wheeler equations in Kerr, in~\cite{Ma,D-H-R-Kerr},  
is done starting with  the  complex, scalar, 
Teukolsky    equations, derived via  the  NP, or GHP   formalism,   by
applying  a Chandrasekhar type transformation.  Part I
of~\cite{GKS-2022}   extends their derivation, using  the
non-integrable horizontal formalism,  to perturbations of Kerr.  By
contrast with~\cite{Ma,D-H-R-Kerr,GKS-2022}   
derives     gRW equations for  the horizontal symmetric traceless
2-tensors\footnote{Derived from $A, \Ab$, see Definition 5.2.2 and
5.3.3 in~\cite{GKS-2022}.}   $\qf, \qfb$,  rather than for complex
scalars.   The main difficulty here is to make sure that the  
non-linear  error terms  verify  a favorable structure.       
\item \textit{Nonlinear  error terms}.   The control of the nonlinear
terms and their associated null structure was already understood in
perturbations of Schwarzschild in~\cite{KS} and is extended to
perturbations of Kerr in~\cite{GKS-2022}. 
\item \textit{Energy--Morawetz}. To derive energy--morawetz   estimates
for $A, \Ab$,~\cite{GKS-2022} (Part II) vastly extends the  
pioneering   idea   of Andersson  and Blue~\cite{A-B}, based on
commutations with  $\T, \Z$ and the second order Carter operator $\CC$,
developed in the context of the scalar wave equation in slowly rotating
Kerr,  to treat  the    tensorial  Teukolsky and gRW equations  in 
perturbations of Kerr.
\end{itemize} 

\subsubsection{Comment on the full  sub-extremal  range}

      

Though the full sub-extremal range  $|a|&lt;m$  remains open,     a large
part of  the  proof of stability does not require  the smallness of
$|a|/m$.  This is the case  for~\cite{KS-GCM1,KS-GCM2,Shen,KS:Kerr}. In
fact   the   smallness  assumption is only needed in~\cite{GKS-2022},
mostly  in the derivation of the main  energy--morawetz estimates in
parts II and III.

\section{Stability of the Kerr--Newman family}
\lab{section:KN}

As stated    in the introduction I make only passing remarks  about 
the stability  the Kerr--Neman family $\KN\!(a,m,e)$. To
start with, as  discussed in~\cite{Chand5}, unlike in the case of  
Kerr,     mode stability fails for   Kerr--Neman  spacetimes due to lack
of separability  and the coupling of the   gravitational   radiation
with the   electromagnetic   one.  So far

 As mentioned  earlier (see Section~\ref{section:LinSchw}   the first
results  on the linear stability of   the Reissner--Nordstr\"om  family,
i.e.\  special  case   of Kerr--Newman   with $a=0$, are due to  Giorgi
in~\cite{Giorgi-RN1,Giorgi-RN2}, based on   an approach similar
to that used  in~\cite{KS}.  The main novelty of those  works  were  to
identify  (1))  the  correct gauge invariant quantities   (in fact    a
couple of tensors), (2) the correct     Teukolsky-type  coupled  systems
they verify, (3) the  correct version of the Chandrasekhar 
transformation and   the correct  generalization of the Regge--Wheeler 
type system into  which  the  Teukolsky type system is transformed (4) 
Derivation of Morawetz-Energy type estimates   for this   system. The
same steps were than carried out   in the  more difficult   case of 
Kerr--Newman~\cite{Giorgi-KN1,Giorgi-KN2,Giorgi-KN3}  in
which linear stability is established   for $|a|, |Q|\ll M$. Recently
in~\cite{Giorgi-KN4}  the authors   derive  boundedness and decay for 
the Giorgi--Teukolsky system, in axial symmetry,   in the range $|a| \ll
M, |Q|&lt;M$.  The  linear   stability   of the  Kerr--Newman    family for
$|a|, |Q|\ll M$. was also established   by  He~\cite{He}  in the metric
framework  using the   powerful micro-local techniques   of Hintz and
Vasy.


\section*{Declaration of interests} 
The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Funding}

Supported by NSF grant 220103 the authors also likes to thank IHES for
hosting him during his sabbatical for Princeton, February 1--May 31.

\CDRGrant[NSF]{220103}

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