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\DOI{10.5802/crmeca.270}
\datereceived{2024-10-07}
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\title{A survey on the positive mass theorem for asymptotically flat
initial data sets}

\alttitle{Enqu\^ete sur le th\'eor\`eme de la masse positive pour les
ensembles de donn\'ees initiales asymptotiquement plats}

\author{\firstname{Lan-Hsuan} \lastname{Huang}\CDRorcid{0000-0002-8918-3071}}
\address{Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA}
\email{lan-hsuan.huang@uconn.edu}

\thanks{NSF DMS-2304966, NSF DMS-1928930} 

\keywords{\kwd{ADM mass}
\kwd{ADM energy-momentum}
\kwd{Asymptotically flat initial data set}
\kwd{Dominant energy condition}
\kwd{pp-wave}}

\altkeywords{\kwd{Masse ADM}
\kwd{\'Energie-momentum ADM}
\kwd{Ensemble de donn\'ees initiales asymptotiquement plates}
\kwd{Condition d'\'energie dominante}
\kwd{Onde pp}}

\begin{abstract} 
In honor of Yvonne Choquet-Bruhat's 100th birthday, we present this
survey on the positive mass theorem. Originating from a conjecture in
general relativity regarding the ADM mass, the positive mass theorem
has significantly influenced geometry and analysis over the past four
decades and continued to inspire new connections. We review seminal
contributions as well as recent advances, and then we focus our
discussions on the equality case and the counter-examples arising from
pp-wave spacetimes. 
\end{abstract}

\begin{altabstract} 
En l'honneur du 100e anniversaire d'Yvonne Choquet-Bruhat, nous
pr\'esentons cette \'etude sur le th\'eor\`eme de la masse positive. Issu
d'une conjecture en relativit\'e g\'en\'erale concernant la masse de
l'ADM, le th\'eor\`eme de la masse positive a influenc\'e de mani\`ere
significative la g\'eom\'etrie et l'analyse au cours des quatre
derni\`eres d\'ecennies et continue d'inspirer de nouvelles connexions.
Nous passons en revue les contributions fondamentales ainsi que les
avanc\'ees r\'ecentes, puis nous concentrons nos discussions sur le cas
de l'\'egalit\'e et les contre-exemples provenant des espaces-temps
d'ondes pp. 
\end{altabstract}

\maketitle

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\def\xeqref#1{(\ref{#1})}

\section{Introduction}

I am honored to contribute this survey on the positive mass theorem in
celebration of Yvonne Choquet-Bruhat's 100th birthday. I had the chance
to meet her in 2008 at the Mittag-Leffler Institute while I was a PhD
student participating in the \emph{Geometry, Analysis, and General
Relativity} program.  She visited briefly to meet her collaborators,
including Piotr Chru\'sciel and Jim Isenberg who were key participants
of the program. During tea time, we gathered at a large oval table in
the sunroom. As I was just starting my research career, her warmth and
friendliness left a lasting impression.  We are truly fortunate to have
a pioneer like Choquet-Bruhat in the general relativity community. Her
groundbreaking work and kindness are an amazing example for us all and
continue to inspire us today.

Choquet-Bruhat made a fundamental contribution to the Cauchy problem,
ensuring that the initial value problem in general relativity is
well-posed. For more details, see her
survey~\cite{Choquet-Bruhat:2015}. Her research laid the groundwork for
the $3+1$ formulation of general relativity (ADM formalism), which is
crucial for understanding how spacetime evolves and is used in both
theoretical and numerical relativity. The ADM approach views spacetime
as a series of evolving three-dimensional slices from an initial data
set. It simplifies Einstein's field equations into constraint and
evolution equations. It also introduces ADM energy and linear momentum,
which measure the total mass of an asymptotically flat spacetime. These
quantities, defined on the initial data set, are essential for
understanding isolated gravitational systems and allow working with
just the initial data and constraint equations. We refer the reader to
her beautiful book~\cite{Choquet-Bruhat:2009}.

The positive mass conjecture asserts that the ADM mass is non-negative
(or, more precisely, that the ADM energy-momentum is future causal)
under suitable energy conditions. Choquet-Bruhat and Marsden proved a
very interesting case of the positive mass theorem, for metrics close
to the Euclidean metric~\cite{Choquet-Bruhat-Marsden:1976}, building on
an idea outlined by Brill and Deser~\cite{Brill-Deser:1968}. The
problem gained wider attention through questions posed by Geroch in his
1975 plenary address at the Joint Mathematical
Meeting~\cite{Geroch:1975}. This conjecture became one of the early
topics in mathematical general relativity and remains central to
developments in geometric analysis and general relativity. The search
for and affirmation of positivity conditions under various settings has
had a significant impact on mathematics, influencing areas like the
Yamabe problem in conformal geometry~\cite{Schoen:1984}.

While there were other partial results toward the positive mass
conjecture, the first groundbreaking result was obtained by Schoen and
Yau in 1979~\cite{Schoen-Yau:1979-pmt1}. For an earlier history of this
problem, we refer to the introduction of~\cite{Schoen-Yau:1979-pmt1}.
Not only did they completely settle the so-called Riemannian case, but
the minimal surface technique they introduced also revolutionized the
field. Since then, more general versions, higher-dimensional cases, and
new alternative proofs have emerged, making the topic even more dynamic
in recent years. Researchers have continued to develop innovative
techniques and approaches, expanding the scope and relevance of the
positive mass theorem. While the physically interesting case is three
spatial dimensions, there are various reasons and motivations for
seeking a general statement in higher dimensions. We will provide a
more comprehensive list of progress below. For now, we present the
statement.

\begin{theo}[Spacetime positive mass theorem] \label{thm:PMT}
Let  $n\ge 3$ and $(M, g, k)$ be an $n$-dimensional asymptotically flat
initial data set that satisfies the dominant energy condition. Suppose
either  $3 \le n \le 7$ or $M$ is spin. Then
{\[
E \ge |P|,
\]}\unskip
where $(E, P)$ is the ADM energy-momentum vector of $(M, g, k)$.
\end{theo}

The above theorem says that the ADM energy-momentum vector $(E, P)$ is
future-directed and causal. Thus, one can define its Lorentzian norm as
the \emph{ADM mass} $m=\sqrt{E^2 - |P|^2}$, which is a spacetime
invariant. 

The special case of Theorem \ref{thm:PMT} where $k \equiv 0$ is known
as the Riemannian case (or the time-symmetric case). In this case, $P =
0$, and the dominant energy condition implies nonnegative scalar
curvature. It is also conventional to denote the ADM energy $E$ by the
ADM mass $m$.  

\begin{theo}[Riemannian positive mass theorem] \label{thm:RPMT}
Let $n\ge 3$ and $(M, g)$ be an $n$-dimensional asymptotically flat
manifold with nonnegative scalar curvature. Suppose either  $3 \le n
\le 7$ or $M$ is spin. Then  $m\ge 0$. Furthermore,  $m=0$ if and only
if $(M, g)$ is isometric to Euclidean space.
\end{theo}

Schoen and Yau proved this case in dimension three in 1979 and
1981~\cite{Schoen-Yau:1979-pmt1, Schoen-Yau:1981-asymptotics} by
introducing minimal surface techniques and extended it to dimensions
less than eight ~\cite{Schoen-Yau:1979-pat, Schoen:1989}. They argued
that if the ADM mass $m<0$, then  there exists a codimension 1
area-minimizing minimal surface with the induced metric being
asymptotically flat with zero mass. The area-minimizing  property
allows for a conformal factor that further transforms the induced
metric to one with zero scalar curvature and negative mass,
contradicting the Riemannian positive mass theorem in one dimension
lower and ultimately conflicting with the Gauss--Bonnet Theorem when
$n=3$. Since the above argument does not address the case $m=0$, Schoen
and Yau gave a separate argument to show that if $m=0$, then the
manifold is Ricci flat and thus isometric to Euclidean space.

Lohkamp~\cite{Lohkamp:1999} observed a cut-off technique to simplify
the asymptotics of an asymptotically flat manifold with negative ADM
mass, showing that it can be compactified to a torus while maintaining
positive scalar curvature. Consequently, the Riemannian positive mass
theorem is equivalent to the torus rigidity theorem of Schoen and
Yau~\cite{Schoen-Yau:1979-minimal}. In more recent years, there have
been rapid advances and alternative proofs of the Riemannian positive
mass theorem for three dimensions. Huisken and
Ilmanen~\cite{Huisken-Ilmanen:2001} used the inverse mean curvature
flow, Li~\cite{Li:2018} employed the Ricci flow, Bray, Kazaras, Khuri,
and Stern~\cite{Bray-Kazaras-Khuri-Stern:2022} considered  level sets
of harmonic functions and Bochner identity, and Agostiniani, Mazzieri,
and Oronzio~\cite{Agostiniani-Mazzieri-Oronzio:2024} used Green's
function. 

For the general case $k \nequiv 0$ in dimension three, Schoen and
Yau proved $E \ge 0$ by introducing Jang's
equation~\cite{Schoen-Yau:1981-pmt2} and reduce the case to the
Riemannian positive mass theorem. Eichmair generalized the Jang
equation argument and proved the $E\ge 0$  theorem in dimensions less
than eight~\cite{Eichmair:2013}. These results also show that if $E=0$,
then $(M, g, k)$ can be isometrically embedded in Minkowski spacetime
with the second fundamental form $k$.  

For the  result $E \ge |P|$  as stated in Theorem~\ref{thm:PMT}, Witten
provided a novel proof in dimension three~\cite{Witten:1981,
Parker-Taubes:1982}, which directly generalizes to higher dimensions
for spin manifolds~\cite{Bartnik:1986, Ding:2008}. Eichmair, Huang,
Lee, and Schoen~\cite{Eichmair-Huang-Lee-Schoen:2016} extended the
minimal surface approach in the Riemannian case developed by Schoen and
Yau by introducing marginally outer trapped surfaces (MOTS) to prove
Theorem~\ref{thm:PMT}. They addressed the challenges posed by MOTS,
which, unlike  minimal surfaces, do not arise from a variation
principle of the initial data set. In the same paper, they also
provided an alternative approach, showing that the $E \ge 0$ theorem
implies $E \ge |P|$ by  a new density theorem. Recently, Hirsch,
Kazaras, and Khuri~\cite{Hirsch-Kazaras-Khuri:2022} provided an
alternative proof for the 3-dimensional case using level sets of
spacetime harmonic functions. We will briefly review those approaches
in Section~\ref{se:pos}. 

Challenges in higher dimensions $n\ge 8$ for the positive mass theorem
arise because minimal hypersurfaces and MOTS can have singularities in
these dimensions, causing the dimensional reduction argument to
potentially break down. Lohkamp proposed a program to address these
issues in a series of papers~~\cite{Lohkamp:2015, Lohkamp:2015b,
Lohkamp:2015c, Lohkamp:2016}. For the Riemannian case, Schoen and
Yau~\cite{Schoen-Yau:2022} introduced a different approach using
minimal slicings. Additionally, there are perturbation arguments to
avoid singularities of minimal hypersurfaces for dimensions $n=8$ by
Smale~\cite{Smale:1993} and $n=9,10$ by Chodosh, Mantoulidis, and
Schulze~\cite{Chodosh-Mantoulidis-Schulze:2023}. Separately, while we
shall only discuss $n\ge 3$ in this survey, there is also a Riemannian
positive mass theorem for $n=2$; see~\cite{Wong:2012}.

Neither the spinor proof in~\cite{Witten:1981, Parker-Taubes:1982} nor
the MOTS proof in~\cite{Eichmair-Huang-Lee-Schoen:2016} includes a
complete proof to characterize the equality case $E=|P|$ of
Theorem~\ref{thm:PMT}. The natural conjecture states that if $E = |P|$,
then $E = |P| = 0$, and thus by $E=0$ rigidity, $(M,g)$ can be
isometrically embedded into Minkowski space with induced second
fundamental form $k$. This would signify that the ADM mass $m
=\sqrt{E^2 - |P|^2} = 0$, uniquely characterizes Minkowski spacetime. 

The conjecture is shown to be true in dimensions $n=3, 4$, but
surprisingly, in dimensions $n\ge 5$ it has the subtlety depending on
asymptotic flat decay rates, to be specified  in
Section~\ref{se:equality} below. For $n \ge 5$,  an initial data set
with $E=|P|$ that does not exhibit the optimal asymptotic decay rate
can have  $E=|P|\neq 0$ and exists as a Cauchy hypersurface in a larger
class of spacetimes known as pp-waves (short for \emph{plane-fronted
waves with parallel rays}).  Minkowski spacetime is a special case of
pp-waves. Those pp-wave examples also give counter-examples to the
Bartnik stationary conjecture, see~\cite[Section 2]{Huang-Lee:2024}.

We summarize the current state of the art below.

\begin{theo}[Equality in the spacetime positive mass theorem]\label{theorem:main1}
Let $n\ge 3$ and $(M, g, k)$ be an $n$-dimensional asymptotically flat
initial data set that satisfies the dominant energy condition and has
$E=|P|$. 
\begin{itemize}
\item If either $n=3, 4$ or $(M, g, k)$ satisfies the optimal
asymptotic decay rates\footnote{We say  that $(M, g, k)$ satisfies the
\emph{optimal asymptotic decay rates} if  the asymptotics of $(g, k)$
defined by \xeqref{eq:decay}
below hold for some $q>0$ and 
$0 < \alpha < 1$  satisfying $q+\alpha > n-2$. This is an additional assumption
beyond the usual decay rate condition $q>\sfrac{n}{2}$ when $n\ge 5$.},
then $E=|P|=0$ and $(M, g)$ is isometrically embedded into Minkowski
spacetime with the induced second fundamental form $k$.  
\item If $n\ge 5$ and $M$ is spin, then $(M, g, k)$ is isometrically
embedded into a pp-wave spacetime with the induced second fundamental
form $k$.   
\end{itemize}
\end{theo} 

We can break down the equality theorem into two separate statements:
The first statement is to show that  $E=|P|$ implies $E=|P|=0$, which
is the part that requires the optimal asymptotics decay rate. The
second statement is to use the fact that $E=0$ to find an embedding of
the initial data set into Minkowski space. 

Witten sketched an idea for proving $E=|P|$ rigidity of spin manifolds
in his 1981 article~\cite{Witten:1981}. 
Ashtekar-Horowitz~\cite{Ashtekar-Horowitz:1982} and 
Yip~\cite{Yip:1987} gave arguments for $n=3$ under extra spacetime
assumptions. A complete, rigorous proof  was given in  the work of Beig
and Chru\'sciel for $n=3$ by  a spinor
argument~\cite{Beig-Chrusciel:1996} and  by Chru\'sciel and
Maerten~\cite{Chrusciel-Maerten:2006} to higher-dimensional spin
manifolds under the optimal asymptotic decay rates for dimensions $n
\geq 5$. For general initial data sets without the spin assumption,
Huang and Lee used the method of Lagrangian Multipliers introduced by
Bartnik~\cite{Bartnik:2005} to prove the first statement (under similar
optimal asymptotic decay rates if $n\ge 5$). They also gave an
alternative proof of $E=0$ rigidity~\cite{Huang-Lee:2023}. Unlike the
other proofs, their proof is self-contained in the sense that it does
not depend on how one proves that $E\ge |P|$ theorem or $E\ge 0$
theorem.  For $n=3$, Hirsch and Zhang proved the equality theorem for
$n=3$ from the level sets method~\cite{Hirsch-Zhang:2023}. 

On the other hand, Huang and Lee~\cite{Huang-Lee:2024}  discovered
those pp-wave counterexamples in dimensions $n >8$. Specifically, they
found asymptotically flat spacelike slices in a large class of
$(n+1)$-dimensional pp-wave spacetimes that are not isometric to
Minkowski spacetime, and those spacelike slices have  $E = |P| \neq 0$.
Hirsch and Zhang further improved these counterexamples to dimensions
$n \geq 5$ and showed that a spin asymptotically flat initial data set
which satisfies the dominant energy condition and $E=|P|$ must be
embeded into a pp-wave~\cite{Hirsch-Zhang:2024}. 

Given that multiple survey articles already exist on the Riemannian
positive mass theorem from different perspectives
(e.g.,~\cite{Schoen:1989, notices:2018, Huang-Lee:2022notices}), along
with a graduate textbook by Lee~\cite{Lee:book} on those topics, this
survey will primarily focus on the spacetime positive mass theorem,
with particular emphasis on the equality case. The outline of the
survey paper is as follows: In Section~\ref{se:ADM}, we set the stage
and include a variational aspect of the ADM energy-momentum.

In Section~\ref{se:pos}, we briefly review the different proofs of
positivity $E\ge |P|$. In Section~\ref{se:equality}, we discuss the
pp-waves and the equality case $E=|P|$.

\section{Definitions, notations, and basic facts}\label{se:ADM}

\subsection{Initial data sets, the dominant energy condition, and
asymptotic flatness}

Let $n\ge3$. An \emph{initial data set} is an $n$-dimensional manifold
$U$ equipped with a Riemannian metric $g$ and a  symmetric
$(0,2)$-tensor $k$. Loosely speaking, an initial data set consists the
information needed to determine the evolution of spacetime by viewing 
the triple $(U, g, k)$ as a hypersurface in spacetime $(\mathbf N,
\mathbf g)$ with the induced metric $g$ and second fundamental form
$k$. The \emph{Einstein equation} of spacetime is given by
{\[
\Ric_{\mathbf{g}} - \tfrac{1}{2} R_{\mathbf{g}} \mathbf{g} = \Lambda \mathbf{g} + T,
\]}\unskip
where $\Lambda\in \mathbb R$ is the cosmological constant and $T$
is the stress-energy tensor describing a matter field.  In this
article, we shall only discuss the case $\Lambda = 0$, but note that
there are also many progresses on the positive mass theorem for the
case $\Lambda < 0$; see, for example,~\cite{Wang.X:2001,
Chrusciel-Herzlich:2003, Andersson-Cai-Galloway:2008,
Chrusciel-Galloway-Nguyen-Paetz:2018, Huang-Jang-Martin:2020,
Sakovich:2021}. A spacetime is called \emph{vacuum} if $T$ is
identically zero. As an example of a non-vacuum matter field, the
stress-energy tensor of a perfect fluid is given by $T = p \mathbf{g}
+ (\rho + p) \mathbf{v} \otimes \mathbf{v}$, where $\rho$ is the
energy density, $p$ is the pressure, and $\mathbf{v}$ is the fluid
velocity. We do not assume any specific matter fields here; the results
apply to any matter field, as long as an energy condition is satisfied,
unless otherwise specified.

On an initial data set, one can define the \emph{mass density} $\mu$
and the \emph{current density} $J$ by
{\begin{eqnarray*} 
\mu&=&\tfrac{1}{2}\left(R_g  - | k |_g^2 + (\tr_g k )^2\right)\\
J&=&  \Div_g k - d(\tr_g k).
\end{eqnarray*}}\unskip
The first equation is referred to as the \emph{Hamiltonian constraint}
and the second is the \emph{momentum constraint}.  We denote the
\emph{constraint map} by $\Phi(g, k) : = (\mu, J)$. 

We say that $(U, g, k)$ satisfies the \emph{dominant energy condition}
if $\mu\ge |J|_g$.  If  $(U, g, k)$ is \emph{embedded in a spacetime}
$(\mathbf N, \mathbf g)$, meaning that $(U, g)$ is isometrically
embedded in to $\mathbf N$ with the induced metric $k$,  then the
dominant energy condition is equivalent to requiring $T(\mathbf X,
\mathbf n)\ge 0$ for all future causal vectors $\mathbf X$, where
$\mathbf n$ is the future-directed normal to the hypersurface. The
initial data set $(U, g, k)$ is said to be \emph{vacuum} if $\mu = 0,
J=0$.  See Figure~\ref{fig:DEC}.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig:DEC}For an initial data set $(U, g, k)$ embedded in
a spacetime, the dominant energy condition on $(U, g, k)$ states that
$\mu \ge |J|_g$, where $\mu = T(\mathbf{n}, \mathbf{n})$ and $J_i =
T(e_i, \mathbf{n})$, with $\mathbf{n}$ being the future-pointing unit
normal to $U$ and $e_i$ tangent to $U$. This condition is equivalent to
requiring that $T(\cdot, \mathbf{n})$ is future causal. Note that a
spacetime satisfying the dominant energy condition asserts that
$T(\mathbf{X}, \mathbf{Y}) \ge 0$ for all future causal vectors
$\mathbf{X}$ and $\mathbf{Y}$, which is a stronger condition.}
\end{figure}

When considering the perturbation of the dominant energy condition, it
turns out more effective to use the  \emph{modified constraint map
$\overline{\Phi}|_{(g,k)}$ at $(g,k)$}, introduced by Corvino and
Huang~\cite{Corvino-Huang:2020}.  The modified operator
$\overline{\Phi}|_{(g,k)}$ is defined on  initial data $(\gamma, \tau)$
as follows:
{\begin{equation} \label{eq:def-mco} 
 \overline{\Phi}_{(g, k)}(\gamma, \tau)= \Phi(\gamma, \tau)+  \left(0, \tfrac{1}{2} \gamma \cdot J\right),
\end{equation}}\unskip
where $J$ is the current density of $g$ and  $ (\gamma \cdot J)^i :=
g^{ij} \gamma_{jk} J^k$.  A fundamental property of the modified
operator is that the dominant energy condition is preserved under
perturbation. More precisely,  assume $(g,k)$ satisfies the dominant
energy condition $\mu\ge |J|_g$ in $M$. Suppose $(\gamma, \tau) $ is an
initial data set with $|\gamma - g|_g< 3$ in $M$ and  
{\[
\overline{\Phi}|_{(g,k)}(\gamma, \tau) = \overline{\Phi}|_{(g,k)}(g, k).
\]}\unskip
Then $(\gamma, \tau)$ also satisfies the dominant energy condition. 

The seminal work of Choquet-Bruhat says that if $(U, g, k)$ is a vacuum
initial data set, then   there exists a unique vacuum spacetime
development $(\mathbf N, \mathbf g)$ that evolves from $(U, g, k)$. 
However, note that in general an initial data set satisfying the
dominant energy condition may not have a spacetime development
satisfying an energy condition, or if it does, the spacetime
development may not be unique. 

For analysis on asymptotically flat manifolds, it is convenient to use
the weighted H\"older spaces, defined below.   Let $B$ be a closed ball
in $\mathbb R^n$ centered at the origin, and consider the exterior
region $\mathbb R^n\setminus B$.  For any $\ell= 0, 1, 2, \dots$,
$\alpha \in (0, 1)$, $q\in\mathbb R$, we define the \emph{weighted
H\"older space} $C^{\ell,\alpha}_{-q}(\mathbb R^n\setminus B)$ as the
space of functions $f$ on $\mathbb R^n\setminus B$ such that
{\[
\| f\|_{C^{\ell,\alpha}_{-q}(\mathbb R^n\setminus B)} := \sum_{|\alpha|
\le \ell} \sup_x \left| |x|^{ | \alpha| + q }\partial^{\alpha} f(x)
\right| +  \sum_{|\alpha| = \ell} \left[ |x|^{ | \alpha| + q
}\partial^{\alpha} f(x) \right]_\alpha<\infty. 
\]}\unskip
Now suppose that  $M$ is a manifold such that there is a compact set
$K\subset M$ and a diffeomorphism $M\setminus K \cong \mathbb
R^n\setminus B$.  Then one can define $C^{\ell,\alpha}_{-q}(M)$ by
choosing an atlas for $M$ that consists of the diffeomorphism
$M\setminus K \cong  \mathbb R^n\setminus B$ together with finitely
many precompact charts, and then using the
$C^{\ell,\alpha}_{-q}(\mathbb R^n\setminus B)$ norm on the noncompact
chart while using the $C^{\ell,\alpha}$ norm on the other ones.  The
definition of $C^{\ell,\alpha}_{-q}(M)$ also extends to tensors simply
by considering their components with respect to these coordinate
charts. 

 We say that an initial data set $(M,g, k)$ is \emph{asymptotically
flat with decay rate $q$}  if $M$ is a complete manifold and  there is
a compact set $K\subset M$ and a diffeomorphism $M\setminus K\cong
\mathbb R^n\setminus B$ for some closed ball $B\subset \mathbb R^n$
such that
{\begin{equation}\label{eq:decay}
(g-\delta,k)\in C^{2,\alpha}_{-q}(M)\times C^{1,\alpha}_{-1-q}(M)
\end{equation}}\unskip
where $\delta$ is a smooth symmetric $(0,2)$-tensor that coincides with
the Euclidean inner product on $M\setminus K\cong \mathbb R^n\setminus
B$. In addition, we assume $\mu, J\in L^1 (M)$, and sometimes it is
convenient to make a stronger assumption that $\mu, J \in
C^{0,\alpha}_{-n-q_0} (M)$ for some $q_0>0$. Throughout the article, we
assume $q>\sfrac{(n-2)}{2}$. 

\subsection{ADM energy and linear momentum, Regge--Teitelboim
Hamiltonian, and Killing initial data}

Assume $(M, g, k)$ is asymptotically flat with decay rate
$q>\sfrac{(n-2)}{2}$ in the above sense. We define the \emph{ADM energy}
$E$ and the \emph{ADM momentum} $P$ as
{\begin{eqnarray*}
E(g, k) &=& \frac{1}{2(n-1)\omega_{n-1}}  \lim_{r\to\infty}\int_{|x|=r}\sum_{i,j=1}^n \left(\frac{\partial g_{ij}}{\partial x^i} - \frac{\partial g_{ii}}{\partial x^j} \right)\nu_0^j\, \mathrm{d}\sigma_0\\
P_i (g, k)&=&\frac{1}{(n-1)\omega_{n-1}} \lim_{r\to\infty}  \int_{|x|=r}\sum_{j=1}^n ( k_{ij} - (\tr_g k)g_{ij}) \nu_0^j\, \mathrm{d}\sigma_0\qquad i= 1, 2, \ldots, n. 
\end{eqnarray*}}\unskip
Here, the integrals are computed in the coordinate chart $M \setminus K
\cong_x \mathbb R^n\setminus B$, $\nu_0^j= x^j / |x|$, $\sigma_0$ is
the $(n-1)$-dimensional Euclidean Hausdorff measure, and $\omega_{n-1}$
is the volume of the standard unit sphere in $\mathbb R^n$. 

Notice that if the decay rate $q>n-2$, then the integrand in the above
expressions decays too quickly, so $E=0$ and $P=0$ trivially. On the
other hand, a decay rate $q>\sfrac{(n-2)}{2}$ is sufficient to ensure that
the ADM energy-momentum is well-defined in the following sense. While
the ADM energy and linear momentum are defined using a specific
asymptotically flat coordinate chart, they can be shown to be
coordinate invariant, depending only on the structure at infinity,
see~\cite{Bartnik:1986}. Furthermore, although the ADM energy and
linear momentum are defined entirely on the initial data set, the
Lorentzian norm $-E^2 + |P|^2$ is a spacetime invariant. It means that,
while the ADM energy-momentum vectors of two asymptotically flat
initial data sets embedded in the same spacetime may differ, they have
the same Lorentzian norm, see~\cite{Chrusciel:1988}. 

To understand heuristically how the decay rate $q>\sfrac{(n-2)}{2}$ plays
a role, we expand the constraint equations in asymptotically flat
coordinates:
{\begin{eqnarray*}
\mu &=& \frac{1}{2} (R_g - |k|^2 + (\tr_g k)^2) =\frac{1}{2}  \sum_{i,j} \frac{\partial}{\partial x^j} \left( \frac{\partial g_{ij} }{\partial x^i} - \frac{\partial g_{ii}} {\partial x^j}\right)  + O(|x|^{-2-2q})\\
J_i &=& (\Div_g k)_j - (\tr_g k)_{,j} =  \sum_{i} \frac{\partial}{\partial x^i} ( k_{ij} - (\tr_g k)g_{ij}) + O(|x|^{-2-2q}).
\end{eqnarray*}}\unskip
Observe that the ADM energy-momentum arises from the flux integrals of
the divergence terms on the right-hand sides. The decay rate
$q>\sfrac{(n-2)}{2}$ ensures that the exponent of the error terms, $-2-2q<
-n$, decays rapidly enough to be integrable on the asymptotically
flat end. Together with the assumption that both $\mu$ and $J$ are
integrable, the flux integrals over the coordinate sphere have a limit
as the radius of the sphere going to infinity. 

The following density theorem states that initial data sets with the
decay rate $q= n-2$ are dense among those with the weaker decay rates
$q > \sfrac{(n-2)}{2}$, and that the ADM energy-momentum varies
continuously. This has several applications. For example, to prove the
positivity of the ADM mass as stated in Theorem~\ref{thm:PMT},   it
suffices to verify the inequality for an asymptotically flat initial
data set with a decay rate $q=n-2$.  For more details about the density
theorem, see, for example,~\cite[Section
6]{Eichmair-Huang-Lee-Schoen:2016}.

\begin{theo}[Density theorem (loosely stated)]\label{th:density}
Let $(M, g, k)$ be an $n$-dimensional asymptotically flat initial data
set at the decay rate $q>\sfrac{(n-2)}{2}$ satisfying the dominant energy
condition. For any $\epsilon>0$, there exists an asymptotically flat
initial data set $(M, \bar g, \bar k)$ at the decay rate $q=n-2$ such
that $(\bar g, \bar k)$ satisfies the dominant energy condition, is
$\epsilon$-close to $(g, k)$ in $C^{2,\alpha}_{-q}(M)\times
C^{1,\alpha}_{-1-q}(M)$, and $|(\bar E, \bar P) - (E, P) | < \epsilon$.

Furthermore, we can arrange $(M, \bar g, \bar k)$ to have either the
strict dominant energy  condition $\bar \mu > |\bar J|_{\bar g}$
everywhere  in $M$ or to be vacuum $\bar \mu = |\bar J|_{\bar g} = 0$
outside a large compact subset of $M$.
\end{theo}

The ADM energy-momentum  has a variational characterization.

 \begin{definition}[Regge--Teitelboim Hamiltonian]\label{definition:functional}
Let $M$ be an $n$-dimensional manifold that can carry an asymptotically
flat initial data set.  Let $(f_0, X_0)$ be a pair of a function and a
vector field on $M$ (which we will often call a \emph{lapse-shift
pair}) such that $(f_0, X_0)$ is smooth and is equal to a constant
$(a,b)\in \mathbb R\times \mathbb R^n$ in the exterior coordinate chart
for $M\setminus K$.  We define the \emph{Regge--Teitelboim Hamiltonian}
$\mathcal{H}_{\mathrm{RT}}$ corresponding to $(f_0, X_0)$ by, for
asymptotically flat initial data set $(g, k)$ 
{\begin{equation}\label{equation:functional}
\mathcal{H}_{\mathrm{RT}} (g, k) =(n-1)\omega_{n-1} \left[ a E(g,k) + b \cdot P(g,k)\right]- \int_M \langle \Phi(g, k) , (f_0, X_0)\rangle_g\, \mathrm{d}\mu_g.
\end{equation}}\unskip
\end{definition}

In the case of vacuum initial data sets, as originally considered by
Regge and Teitelboim~\cite{Regge-Teitelboim:1974}, the Hamiltonian can
recover the energy $ E $ and the linear momentum by choosing the
lapse-shift pair to be asymptotic to the translation vector fields of
Minkowski spacetime. Specifically, $(f, X) \to (1, 0)$ yields the
energy $ E $, while $(f, X) \to (0, \sfrac{\partial}{\partial x^i})$
yields $ P_i $, the $ i $-th component of the linear momentum. The
task of ``minimizing'' the ADM energy-momentum $(E, P)$ is
transformed into minimizing the functional \xeqref{equation:functional}
as follows. Suppose the positivity of mass holds for vacuum $(g, k)$,
i.e., $ E(g, k) \ge |P(g, k)| $. Then, for any future causal vector
$(a, b)$, we have
{\begin{equation} \label{eq:causal}
a E(g, k) + b \cdot P(g, k) \ge a E(g, k) - |b| |P(g, k)| \ge (a - |b|) |P(g, k)| \ge 0
\end{equation}}\unskip
with equality if and only if $E(g, k)= |P(g, k)|$ and $(a,b)$ is a
constant multiple of $(E(g, k), P(g, k))$. This is just a general fact
that for non-zero future causal vectors $\mathbf{X}$ and
$\mathbf{Y}$, their Lorentzian inner product $ -\langle \mathbf{X},
\mathbf{Y} \rangle \ge 0 $ with equality if and only if $\mathbf{X},
\mathbf{Y}$ are  both null and $\mathbf X$ is a constant multiple of
$\mathbf{Y}$. 

With the minimization task in mind, we compute the first variation of 
$\mathcal H_{\mathrm{RT}}$: for vacuum $(g, k)$, we have  
{\[
 D\mathcal{H}\big|_{(g,k)} (h, w) =- \int_M (h, w) \cdot D\Phi|_{(g,k)}^*(f_0, X_0) \, \mathrm{d}\mu_g
\]}\unskip
for all compactly supported $(h, w)$, where $D\Phi|_{(g,k)}^*$ denotes
the $L^2$-formal adjoint equation of the linearization $D\Phi|_{(g,k)}$
at $(g, k)$. The adjoint equation $D\Phi|_{(g,k)}^*$ has geometric
significance in the vacuum case, as discovered by
Moncrief~\cite{Moncrief:1975}. 

\begin{theo}\label{theorem:Moncrief}
Let $(U, g, k)$ be a vacuum initial data set and suppose that there
exists a nontrivial lapse-shift pair $(f, X)$ on $U$ solving 
{\[ 
D{\Phi}|_{(g, k)}^*(f, X) =0.
\]}\unskip
Then the vacuum spacetime development of $(U, g, k)$ admits a unique
global Killing vector field $\mathbf{Y}$ such that $\mathbf{Y}= f
\mathbf{n} + X$ along~$U$, where $\mathbf{n}$ is the future unit normal
to $U$.

Conversely, given a vacuum spacetime equipped with a global Killing
vector field $\mathbf{Y}$ and a spacelike hypersurface $U$ with induced
initial data $(g, k)$,  if we decompose $\mathbf{Y}=f \mathbf{n} + X$
along~$U$, then the lapse-shift pair $(f, X)$ must lie in the kernel of
$D{\Phi}|_{(g, k)}^*$.
\end{theo}
In light of the above theorem, a solution $(f, X)$ to $D{\Phi}|_{(g,
k)}^*(f, X) =0$ is referred to as the \emph{Killing initial data}.

An analogous statement for a non-vacuum case was established by Huang
and Lee~{\cite[Theorem 6]{Huang-Lee:2024}} using the modified
constraint operator. A spacetime $(\mathbf{N}, \mathbf{g})$ is said to
be a \emph{null perfect fluid} with velocity $\mathbf{v}$ and pressure
$p$ if $\mathbf{v}$ is either future null or zero at each point and the
Einstein tensor takes the form: 
{\begin{equation}\label{eq:nulldust}
\Ric_{\mathbf g} - \tfrac{1}{2} R_{\mathbf g} \mathbf g = p \mathbf{g} + \mathbf{v}\otimes \mathbf{v}.
\end{equation}}\unskip

\begin{theo}
Let $(U, g, k)$ be an initial data set satisfying the dominant energy
condition. Assume there exists a nontrivial lapse-shift pair $(f,X)$ on
$U$ solving the system
{ \begin{equation}\label{equation:pair}
D\overline{\Phi}|_{(g,k)}^*(f,X)=0 \quad \mbox{and} \quad fJ + |J|_g X=0.
\end{equation}}\unskip
 and assume that $f$ is  nonvanishing in $U$. Then the following holds:
\begin{enumerate}[(1)]
\item The dominant energy scalar $\sigma(g, k):= \mu - |J|_g$ is
constant on $U$.\label{item:constancy}
\item  $(U, g, k)$ is embedded inside a null perfect fluid spacetime
$(\mathbf{N}, \mathbf{g})$ that satisfies the spacetime  dominant
energy condition and admits a global Killing vector field  $\mathbf{Y}$
equal to $f\mathbf{n} +X$ along~$U$, where $\mathbf{n}$ is the future
unit normal to $U$. 
\label{item:Bartnik-spacetime}
\end{enumerate}
There is also a converse statement. 
 \end{theo} 
We remark that the system \xeqref{equation:pair} is closely related to
the improvability of the dominant energy condition.
See~\cite{Huang-Lee:2024}.  

We refer to a solution $(f, X)$ to $D\overline {\Phi}|_{(g, k)}^*(f, X)
=0$  as the \emph{modified} Killing initial data. These solutions play
an important role in the equality theorem of the positive mass theorem;
see Section~\ref{se:equality}. An overall strategy for proving the
equality theorem is to show that the initial data set achieving
equality has one or more modified Killing initial data.

\section{The proofs of positivity $E\ge |P|$} \label{se:pos}

We outline the currently available proofs of the positivity theorem,
Theorem~\ref{thm:PMT}.

\subsection{Marginally outer trapped hypersurfaces}

We briefly revisit the minimal hypersurface proof of the Riemannian
positive mass theorem, originally developed by Schoen and Yau. This
proof uses induction on the dimension $3 \leq n \le 7$ and proceeds
by contradiction. Assume there exists an asymptotically flat Riemannian
manifold $(M, g)$ with nonnegative scalar curvature and negative mass
$m < 0$. A density argument allows us to assume $(M, g)$ has
harmonic asymptotics and positive scalar curvature. These conditions
imply that the coordinate planes $x^n = \pm \Lambda$ act as barriers
for minimal hypersurfaces for sufficiently large $\Lambda$.

Consider an $(n-1)$-dimensional vertical cylinder $\partial C_\rho$
with a large radius $\rho$. For each $h \in [-\Lambda, \Lambda]$,
there exists an area-minimizing hypersurface $\Sigma_{\rho, h} \subset
C_\rho$ whose boundary is the height $h$ sphere on $\partial
C_\rho$. If $n \le 7$, this hypersurface is smooth. Each
$\Sigma_{\rho, h}$ lies between the hyperplanes $x^n = \pm
\Lambda$. The area of $\Sigma_{\rho, h}$ is minimized by some
$h_\rho \in (-\Lambda, \Lambda)$. This ``height-picking'' step is
crucial in dimensions $n \geq 4$ because the corresponding surface
$\Sigma_{\rho, h_\rho}$ satisfies
{\begin{equation*}
\left.\frac{\mathrm{d}^2}{\mathrm{d} t^2}\right|_{t=0} \mathrm{area}(\Phi_t (\Sigma_{\rho, h_\rho})) \geq 0
\end{equation*}}\unskip
for any variation $\Phi_t$ that is a vertical translation along
$\partial C_\rho$. Without the height-picking step, the above
inequality only holds for compactly supported variations $\Phi_t$,
which is not sufficient. (When $n = 3$, one can use the so-called
logarithmic cut-off trick to approximate variations that are vertical
translations along $\partial C_\rho$ by compactly supported
variations. However, this trick does not work in higher dimensions.) A
smooth subsequential limit $\Sigma_{\infty}$ of $\Sigma_{\rho,
h_{\rho}}$ as $\rho \to \infty$ is an $(n-1)$-dimensional
asymptotically flat manifold with zero energy and is a stable minimal
hypersurface of $M$.

For $n = 3$, the stability of $\Sigma_\infty$ and its asymptotics
are incompatible with the Gauss--Bonnet theorem. In higher dimensions,
one can construct a conformal factor that changes the metric on
$\Sigma_\infty$ to one that is asymptotically flat with zero scalar
curvature and negative mass, thus violating the Riemannian positive
mass theorem in dimension $n-1$.

Our approach to the spacetime positive mass theorem generalizes the
minimal hypersurface proof using marginally outer trapped hypersurfaces
(MOTS). A hypersurface $\Sigma$ is a MOTS if $\theta = 0$, where
$\theta = H_\Sigma + \mathrm{tr}_\Sigma k$ is the expansion
scalar. In the Riemannian case, $\theta$ reduces to the mean
curvature.

Let $(M, g, k)$ be an $n$-dimensional asymptotically flat initial
data set satisfying the dominant energy condition $\mu \geq |J|$ and
$E < |P|$. By our density theorem, Theorem~\ref{th:density}, we may
assume $(M, g, k)$ has harmonic asymptotics and satisfies the strict
dominant energy condition $\mu > |J|$. Furthermore, we assume $P$
points in the vertical direction $-\partial_n$. The harmonic
asymptotics and $E < |P|$ assumptions imply that the planes $x^n =
\pm \Lambda$ act as barriers for MOTS for sufficiently large
$\Lambda$. Consider an $(n-1)$-dimensional vertical cylinder
$\partial C_r$. For each $h \in [-\Lambda, \Lambda]$, the existence
theorem of MOTS~\cite{Eichmair:2009} guarantees the existence of a MOTS
$\Sigma_{r, h}$ with boundary equal to the height $h$ sphere on
$\partial C_r$. This MOTS is smooth if $n < 8$ and lies between the
planes $x^n = \pm \Lambda$, being stable in the sense of MOTS with
boundary. See Figure~\ref{fig:MOTS}.

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig:MOTS}Under the (contradictory) assumption $E<|P|$,
two coordinate hyperplanes $x^n=\Lambda$ and $x^n= -\Lambda$, together
with the lateral side of a cylinder of large radius, provide barriers
for the existence of a MOTS $\Sigma_r$ with prescribed boundary.}
\end{figure}

A major challenge, distinct from the minimal surface approach,  is that
MOTS do not arise from a variational principle. Thus, there is no
obvious alternative to  the height-picking step of minimizing the area
with respect to the height $h$. To overcome this, we introduce a new
functional $\mathcal{F}$ on hypersurfaces with boundary on $\partial
C_r$ such that for some $h_r \in (-\Lambda, \Lambda)$, we have
{\[
\left.\frac{\mathrm{d}}{\mathrm{d} h}\right|_{h = h_r} \mathcal{F}(\Sigma_{r, h}) \geq 0.
\]}\unskip
This inequality, along with the MOTS stability, parallels the area
inequality in the Riemannian case. Extracting a smooth subsequential
limit $\Sigma_{\infty}$ of $\Sigma_{r, h_r}$ as $r \to \infty$,
we find that $\Sigma_{\infty}$ is an $(n-1)$-dimensional
asymptotically flat manifold with zero energy and is a stable MOTS in
$M$. We can construct a conformal factor that changes the metric on
$\Sigma_\infty$ to one with zero scalar curvature and decreases the
energy of $\Sigma_\infty$, thus violating the Riemannian positive
mass theorem in dimension $n-1$.

\subsection{Boost argument}

The positive mass theorem $E\ge |P|$ can also be derived from the
positive energy theorem $E\ge 0$  using the following reduction
argument. Assume that $0< E < |P|$. Fix $\theta \in (0,1)$ such that
$E' := (E - \theta |P|)/(1 - \theta^2) < 0$. Using the  density
theorem, Theorem~\ref{th:density}, we may assume that $(M,g,k)$
satisfies $\mu= 0 $ and $J = 0$ outside a large compact set. According
to~\cite{Christodoulou-OMurchadha:1981}, there exists a vacuum
spacetime development of the asymptotically flat end of $(M, g, k)$
in which the end of $(M, g, k)$ can be deformed to a boosted slice
(of slope $\theta$) that has energy $E'$. The deformed initial data
$(M', g', k')$ satisfies the conditions of the $E \ge 0$ theorem,
leading to a contradiction. This reduction is well known in the
mathematical relativity community when the initial data set is assumed
to be a spacelike slice of an asymptotically flat spacetime; see,
e.g.,~\cite{Chrusciel:1989}.

\subsection{Spinor proof}

Witten's proof is notable for its use of spinors, offering a different
perspective on the positive mass theorem.  A spinor field $\psi$ on a
manifold is a section of a spinor bundle, which can be roughly thought
of as a field of ``spinor-valued'' functions. For a 3-dimensional
initial data set $(M, g, k)$, $M$ can be given a spin structure. In
higher dimensions, this method requires $M$ has a spin structure, which
allows us to use spinor fields $\psi$ on $M$. For convenience, we
consider $(M, g, k)$ embedded in a spacetime $(\mathbf N, \mathbf g)$
and choose an orthonormal frame $\{ e_0, e_1, \ldots, e_n\}$ where $e_0$
is the timelike vector. One can also use a framework that is a purely
initial data perspective. Define a new connection $\tilde \nabla_i =
\nabla_i + ({1}/{2}) k_{ij} e_j e_0$ and the Dirac operator 
${\slashed D} = e_i \cdot \tilde \nabla_i = e_i \cdot \nabla_i +
({1}/{2}) k_{ij} e_i e_j e_0$ where $\nabla_i$ is the induced
connection on $M$. The Witten equation states
{\[
\slashed D^2 \psi =\tilde \nabla^* \tilde \nabla \psi + \tfrac{1}{2} (\mu + Je_0)\cdot  \psi.
\]}\unskip
For any $\psi$ asymptotic to a constant spinor $\psi_0$ at infinity, 
integrating the above Witten's equation and analyzing the boundary term
{\begin{eqnarray}\label{eq:Witten}
&&
\int_M |\tilde \nabla \psi|^2 - |\slashed D \psi|^2 + \tfrac{1}{2}
\langle \psi, (\mu+Je_0) \cdot \psi \rangle \, \mathrm{d}\mu 
\nonumber\\&&\qquad
= \lim_{r \to \infty} \int_{|x|=r} \langle \psi, \tilde
\nabla_\nu \psi + \nu \cdot \slashed D \psi \rangle \,
\mathrm{d}\sigma= \tfrac{1}{2} (n-1) \omega_{n-1} \left(  |\psi_0|^2 E+
\langle \psi_0, P_j e_j e_0\cdot \psi_0\rangle \right)
\end{eqnarray}}\unskip
where ${\nu}$ is the outward unit normal. By choosing ${\psi}$ to solve
${\slashed{D}} \psi =0$ 
with $\psi$ asymptotic to a certain constant
spinor $\psi_0$ such that $\langle \psi_0, P_j e_j e_0\cdot
\psi_0\rangle = -|\psi_0|^2 |P|$, one gets
{\[
\tfrac{1}{2} (n-1) \omega_{n-1} |\psi_0|^2 (E-|P|) = \int_M |\tilde
\nabla \psi|^2 + \tfrac{1}{2} \langle \psi, (\mu+Je_0) \cdot \psi
\rangle \, \mathrm{d}\mu\ge 0
\]}\unskip
where we use the dominant energy condition in the last inequality. 

The argument also directly leads the $E=0$ rigidity. Since $E=0$
implies $|P|=0$, for any constant spinor $\psi_0$  at infinity, one can
let the harmonic spinor $\psi$ asymptotically approach $\psi_0$ in
\xeqref{eq:Witten} to obtain $\tilde \nabla \psi=0$, meaning that $\psi$
is, in fact, a parallel spinor. One then argue that the spacetime
metric is flat along $M$. See~\cite{Parker-Taubes:1982} for more
details. 

\subsection{Level set method using spacetime harmonic functions}

We provide an overview of the proof by  Hirsch, Kazaras, and
Khuri~\cite{Hirsch-Kazaras-Khuri:2022}. Additionally, a recent survey 
by Bray, Hirsch, Kazaras, Khuri, and
Zhang~\cite{Bray-Hirsch-Kazaras-Khuri-Zhang:2023} discusses the level
set method applied to positive mass theorems and other applications.

Let $(\Omega, g, k)$ be a   3-dimensional compact initial data set.
Consider a scalar-valued function $u$. For simplicity, we assume that
$u$ has no critical points. Let $\Sigma_t = u^{-1}(t)$ be a level set
of $u$, and denote the unit normal to $\Sigma_t$ by $\eta =
\sfrac{\nabla u}{|\nabla u|}$.  By applying the Gauss equation to the
level set $\Sigma_t$ and using $A(e_\alpha, e_\beta) = |\nabla u|^{-1}
\nabla^2 u(e_\alpha, e_\beta)$, we obtain
{\begin{equation} \label{eq:level-set}
|\nabla u| \Ric(\eta, \eta) = \tfrac{1}{2} |\nabla u| (R_g - R_\Sigma) +|\nabla u|^{-1} \left( |\nabla |\nabla u||^2 - \tfrac{1}{2} |\nabla^2 u|^2 + \tfrac{1}{2} (\Delta u)^2 -(\Delta u) (\nabla^2 u) (\eta, \eta) \right).
\end{equation}}\unskip
We have the modified Bochner identity:
{\begin{equation*}
    \Delta |\nabla u| = \frac{1}{|\nabla u|} \big(|\nabla^2 u|^2 + \langle \nabla u, \nabla \Delta u\rangle \big)+ |\nabla u| \Ric(\eta, \eta)   - |\nabla u|^{-1} |\nabla |\nabla u||^2.
\end{equation*}}\unskip
Equating the previous two identities gives
{\begin{equation} \label{eq:laplace}
 \Delta |\nabla u| =  \frac{1}{|\nabla u|} \langle \nabla u, \nabla \Delta u\rangle +\frac{1}{2} |\nabla u|  \left[ R_g - R_\Sigma +\frac{ |\nabla^2 u|^2 + (\Delta u)^2-2|\nabla u|(\Delta u) \nabla^2 u (\eta, \eta)  }{|\nabla u|^2} \right].
\end{equation}}\unskip

So far, the identity holds for any function $u$. It was observed
in~\cite{Hirsch-Kazaras-Khuri:2022} that this identity takes a more
convenient form when $u$ satisfies the \emph{spacetime harmonic
equation} $\Delta u + (\tr_g k) |\nabla u|=0$, as motivated below.
Suppose the initial data set $(\Omega, g, k)$ is embedded in a
spacetime $(\mathbf N, \mathbf g)$ with the unit normal $\mathbf n$.
Then the Hessian operator $\bnabla^2$ of $\mathbf g$ restricted on
$\Omega$ is given by 
{\[
\bnabla^2 u(e_i, e_i) = \nabla^2 u(e_i, e_j) + k_{ij} \mathbf n (u).
\]}\unskip
Since $\mathbf n (u)$ involves the spacetime vector $\mathbf n$ and is
not determined on the initial data set $(M, g, k)$, it turns out to be
effective to require $\mathbf n (u)=|\nabla u|$; namely, the spacetime
vector $\bnabla u$ is null.  By taking the trace, it gives the
spacetime harmonic equation. 

Substituting $u$, which solves the spacetime harmonic equation, into
\xeqref{eq:laplace} yields
{\[
\Delta |\nabla u| = \frac{1}{2} |\nabla u| \left[ 2\mu  - R_\Sigma + \frac{\big|\nabla^2 u + k |\nabla u|\big|^2 }{|\nabla u|^2} \right]  - \langle \nabla u, \nabla (\tr_g k) \rangle-\langle \nabla^2 u, k\rangle.
\]}\unskip
Consider the identity on an asymptotically flat initial data set $(M,
g, k)$ and let $\Omega$ be a large coordinate sphere. Consider $u$ with
the asymptotics $u(x) \to a\cdot x$ as $|x|\to \infty$ where $a\in
\mathbb R^3$ and $|a|=1$. Integrating the previous identity  over
$\Omega$ and applying the coarea formula gives the following integral
formula:
{\begin{equation*}
\int_{\partial \Omega} \left( \nu (|\nabla u|)   +  k(\nabla u,
\nu)\right) \, \mathrm{d}\sigma=\frac{1}{2} \int_{\underline
u}^{\overline u} \int_{\Sigma_t} \left(2\mu +2 J(\eta)- R_{\Sigma_t} +
\frac{\big|\nabla^2 u + k |\nabla u|\big|^2}{|\nabla u|^2 } \right)\,
\mathrm{d}\sigma \mathrm{d} t
\end{equation*}}\unskip
where $\nu$ is the outward unit normal to $\partial \Omega$. By
carefully analysis, one can show that the integral $\int_{\underline
u}^{\overline u} \int_{\Sigma_t}R_{\Sigma_t}$ is asymptotically
nonpositive, and the boundary term is asymptotic to $16\rmpi (E+ a \cdot
P)$. To summarize, one obtain 
{\[
E + a\cdot P \ge \frac{1}{16\rmpi } \int_M \left(\mu + J(\eta) +
\frac{\big|\nabla^2 u+ k |\nabla u|\big|^2}{|\nabla u|^2 } \right)\,
\mathrm{d}\sigma \mathrm{d} t \ge 0.
\]}\unskip
This completes the proof. 

\section{pp-waves and the equality case $E=|P|$} \label{se:equality}

\subsection{pp-waves}\label{se:pp}

A pp-wave spacetime (or just a pp-wave for short) is a Lorentzian
metric $\mathbf{g}$ defined on $\mathbb{R}^{n+1}$ equipped with a
global coordinate chart $\{ u, z, x^1, \ldots, x^{n-1}\}$  where
{\begin{equation} \label{eq:pp}
\mathbf{g} = 2 \mathrm{d} u \mathrm{d} z+ S \, \mathrm{d} z^2
+\sum_{a=1}^{n-1} (\mathrm{d} x^a)^2.
\end{equation}}\unskip
Here, the scalar-valued function $S$ depends on $z$ and $x^1, \ldots,
x^{n-1}$, but not on $u$. The vector $\mathbf{Y} :=
\sfrac{\partial}{\partial u}$ is a covariantly constant, nowhere
vanishing null vector field. Note that a pp-wave is a special case of a
null perfect fluid described by \xeqref{eq:nulldust} with pressure $ p =
0 $ and carries a global null Killing vector field $\mathbf{Y}$ with
$\mathbf{v} = \eta \mathbf{Y}$ for some scalar function $\eta$. The
term ``pp-waves'' stands for \emph{plane-fronted} waves (the wave
fronts are constant $z$ slices, whose induced metric is flat) with
\emph{parallel rays} ($\mathbf{Y}$ is a parallel null vector). These
waves can describe situations in which gravitational waves are
idealized as propagating along one direction, uniformly across flat
wave fronts perpendicular to this direction. 

\begin{example}
The Minkowski metric $\mathbf{g} = -\mathrm{d} t^2 + \sum_{a=1}^{n}
(\mathrm{d} y^a)^2$ is a special case of the pp-wave metric. To see
this, we can introduce the null coordinates $U = y^n - t$ and $Z =
\frac{1}{2} (y^n + t)$ and re-express the Minkowski metric as
$\mathbf{g} = 2 \mathrm{d} U \mathrm{d} Z+ \sum_{a=1}^{n-1} (\mathrm{d}
y^a)^2$. Thus, we recover the pp-wave metric by setting $S=0$ in
\xeqref{eq:pp}. Another choice of coordinates is to set
{\begin{equation*}
U =u - (x^1+ \dots + x^{n-1}), \quad Z = z, \quad y^a  = x^a +  z,
\end{equation*}}\unskip
and express the Minkowski metric as 
{\begin{equation} \label{eq:M}
\mathbf{g} = 2 \mathrm{d}u\mathrm{d}z +   \mathrm{d}z^2
+\sum_{a=1}^{n-1} (\mathrm{d}x^a)^2.
\end{equation}}\unskip
Then, we recover the pp-wave metric \xeqref{eq:pp} by setting $S=1$. 

The second expression \xeqref{eq:M} of the Minkowski metric is more
relevant to what we will discuss below. Notice that the induced metric
from \xeqref{eq:M} on the constant $u$ slices is the Euclidean metric
with the induced second fundamental form $k\equiv 0$. 
\end{example}

We consider the constant $u$-slices in \xeqref{eq:pp}, assuming that
$S>0$ everywhere. Additionally, we use $x^n$ to denote the $z$
coordinate in \xeqref{eq:pp}. 

\begin{lemm}\label{lemma:pp-initial}
We assume $S>0$. Let $g$ and $k$ be respectively the induced metric and
second fundamental form on the constant $u$-slices in \xeqref{eq:pp}.
Then $(\mathbb R^n, g, k)$ is an initial data set where  $g$ and $k$
are given by
{\begin{eqnarray*}
g&=& S\, (\mathrm{d}x^n)^2  + \sum_{a=1}^{n-1} (\mathrm{d}x^a)^2\\
k_{ab}&=& 0 \quad \mbox{for } a, b = 1, \ldots, n-1\\
k_{jn} &=& \tfrac{1}{2} S^{-\frac{1}{2}}S_{,j} \quad \mbox{for } j = 1,
\ldots, n.
\end{eqnarray*}}\unskip
We also have 
{\begin{eqnarray*}
\mu &=&  -\tfrac{1}{2} S^{-1} \Delta' S\\
J &=& \frac{1}{2} S^{-\frac{3}{2}} (\Delta' S)  \frac{\partial}{\partial x^n},
\end{eqnarray*}}\unskip
and in particular, $\mu=|J|_g$. 
\end{lemm}

\begin{proof}
Along a constant $u$-slice we have $\mathbf{Y} =
\sfrac{\partial}{\partial u} = S^{-\frac{1}{2}} \mathbf n + S^{-1}
\sfrac{\partial }{\partial x^n}$ where $\mathbf n = \sfrac{\bnabla
u}{|\bnabla u|}$.  To compute the second fundamental form, we use
that $\mathscr L_{\sfrac{\partial}{\partial u}} \mathbf g = 0$ because
$\sfrac{\partial}{\partial u}$ is Killing and thus 
{\begin{eqnarray*}
k_{ij} &=& \tfrac{1}{2} \mathscr L_{\mathbf n} \mathbf g (\partial_i, \partial_j) = - \tfrac{1}{2} f^{-1} \mathscr L_X \mathbf g  (\partial_i, \partial_j)= -\tfrac{1}{2} S^{\frac{1}{2}} \mathscr L_X g  (\partial_i, \partial_j)\\
&=&  -\tfrac{1}{2} S^{\frac{1}{2}} \big(X_{i,j} + X_{j,i} - 2\Gamma_{ij}^n X_n\big) = S^{\frac{1}{2}} \, \Gamma_{ij}^n.
\end{eqnarray*}}\unskip
Computing the Christoffel symbols gives the desired formula for $k$. 
The computations for $\mu, J$ are more involved, and we refer to
Lemma~C.2 of~\cite{Huang-Lee:2024}.
\end{proof}

It easily follows that the dominant energy condition holds if and only
if  $\Delta' S\le0$.  Note that such a function $S$ cannot exist for
$n=3$ because Liouville's theorem says that any superharmonic function
on $\mathbb{R}^2$ that is bounded below must be constant.  In the next
lemma, we summarize the properties that $S$ have such that $(\mathbb
R^n, g, k)$ is asymptotically flat with the decay rate $q$. 

\begin{lemm}\label{lemma:S}
Let $n > 3$. There exist nonconstant smooth functions $S$ on
$\mathbb{R}^n$ with the following properties:
\begin{enumerate}[(1)]
\item $\Delta' S \le 0$ everywhere, strictly negative somewhere, and
$\Delta' S$ is integrable on $\mathbb{R}^n$. \label{item:superharmonic}
\item $S \equiv 1$ in $\{ |x^n| \ge C \}$ for some constant $C > 0$.\vspace*{2pt} \label{item:slab}
\item $ {\lim}_{\rho \to \infty} \int_{|x'| = \rho} -
{\sum}_{a = 1}^{n-1} \sfrac{\partial S}{\partial x^a} \sfrac{x^a}{|x'|} \,
\mathrm{d}\sigma$ exists and is positive. \label{item:mass}
\item For each nonnegative integer $\ell$ and each $\alpha \in (0, 1)$,
we have $S - 1 \in C^{\ell, \alpha}_{-q} (\mathbb{R}^n)$ with $q = n -
3 - (\ell + \alpha)$. \label{item:decay}
\end{enumerate}
\end{lemm}

\begin{proof}
Let $F$ be a smooth nonnegative function on $\mathbb{R}^{n-1}$ with
coordinates $x' = (x^1, \ldots, x^{n-1})$, such that $F = O(|x'|^{-s})$
for some $s > n - 1$. We can solve $\Delta'\psi = -F$ on
$\mathbb{R}^{n-1}$ via convolution with the fundamental solution of the
Laplacian on Euclidean $\mathbb{R}^{n-1}$. As long as $ F $ is not
identically zero, $\psi(x')$ will be a positive, globally superharmonic
function on $\mathbb{R}^{n-1}$. For $ n > 3 $, it must have the
expansion 
{\[
\psi(x') = A |x'|^{3-n} + \text{(lower order terms)},
\]}\unskip
and since $\psi$ is positive, the constant $ A $ must also be positive.
Now define 
{\[
S(x', x^n) = 1 + \phi(x^n) \psi(x'),
\]}\unskip
where $\phi$ is chosen to be any nontrivial, compactly supported,
smooth, nonnegative function on $\mathbb{R}$. Note that $\Delta' S =
-\phi(x^n) F(x') \leq 0$ and is strictly negative somewhere. It is
straightforward to verify that $ S $ satisfies Items
(1), (2), and (3).
Since the derivatives of $ S $ in the $ x^n $ direction do not decay
any faster than $ |x'|^{3-n} $, we can only conclude that $ S - 1 $ and
its derivatives of any order are $ O(|x|^{3-n}) $. Thus, $ S - 1 \in
C^{\ell,\alpha}_{-q} (\mathbb{R}^{n}) $ with $ q = n - 3 - \ell -
\alpha $ by the definition of weighted H\"older spaces.
\end{proof}


\begin{theo}\label{example:pp}
For each $n\ge 5$, there exist  complete, asymptotically flat initial
data sets $(\mathbb{R}^n, g, k)$ that satisfy $\mu=|J|$ and  $E=|P|>0$.
These examples have asymptotic decay rate $q>\sfrac{(n-2)}{2}$.
\end{theo}

\begin{proof}
We first describe the case $n>8$ by Huang and
Lee~\cite{Huang-Lee:2024}. Choose any $ S $ as in Lemma~\ref{lemma:S}
and use this choice in Lemma~\ref{lemma:pp-initial} to construct
initial data $(\mathbb{R}^n, g, k)$. We claim that for $ n > 8 $, this
is the desired example. By construction $(g, k)$ is clearly complete.
The main task is to show that $(g, k)$ is asymptotically flat at the
decay rate $ q > \sfrac{(n-2)}{2} $.

Recall that our asymptotic flatness condition requires $ g_{ij} -
\delta_{ij} \in C^{2,\alpha}_{-q}(\mathbb{R}^n) $ and $ k_{ij} \in
C^{1,\alpha}_{-q}(\mathbb{R}^n) $ for some $ \sfrac{(n-2)}{2} < q < n-2 $,
and $(\mu, J) \in L^1(\mathbb{R}^n)$. For our $(g, k)$ from
Lemma~\ref{lemma:pp-initial}, this is equivalent to requiring that $ S
- 1 \in C^{2,\alpha}_{-q} $ for some $ \sfrac{(n-2)}{2} < q < n-2 $ and
that $\Delta' S$ is integrable. By Item~(4) of
Lemma~\ref{lemma:S}, this imposes the condition on $n$:
{\[
 (n-3)-(2+\alpha) > \frac{n-2}{2} \text{ for some } \alpha\in(0,1),\quad \mbox{or equivalently, } \quad n>8.
\]}\unskip
To see that $E=|P|>0$, we evaluate the ADM energy-momentum by
integrating over large capped cylinders. The caps do not contribute,
and we can see that
{\begin{equation*}
E = -P_n =  \frac{1}{2(n-1)\omega_{n-1}} \lim_{\rho \to \infty} \int_{|x'|=\rho} -\sum_{a=1}^{n-1}\frac{\partial S}{\partial x^a} \frac{x^a}{|x'|} \, \mathrm{d}\mu >0,
\end{equation*}}\unskip
and $P_1=\dots=P_{n-1}=0$. 

To extend the example down to dimensions $ n \ge 5 $, Hirsch and Zhang
observed that instead of choosing the constant $ u $-slices in
\xeqref{eq:pp} as above, one can choose other spacelike slices, such as
graphs over the $ u $-slices. By choosing appropriate graphing
functions, they can improve the derivatives in the $ x^n $ direction
and thus construct examples for $ n \ge 5 $.
See~\cite{Hirsch-Zhang:2024}.
\end{proof}

\subsection{Proof that $E=|P|$ implies $E=|P|=0$}

We review the proof  by Huang and Lee~\cite{Huang-Lee:2020} for the
first statement of the equality theorem.
\begin{theo}\label{thm:E=P}
Let $(M, g, k)$ be an $n$-dimensional asymptotically flat initial data
set with decay rate~$q$ and assume that the positive mass theorem is
true near $(g,k)$. Furthermore, we make the stronger asymptotic
assumption that with this decay rate $q$, 
{\[
q+\alpha>n-2,
\]}\unskip
where $\alpha$ is the H\"older exponent in the definition of asymptotic
flatness. If $(g,k)$ satisfies the dominant energy condition,  then for
each asymptotically flat end, $E=|P|$ implies that $E=|P|=0$. 
\end{theo}

We set up a variational setting to prove the above theorem. Let $(M, g,
k)$ be an asymptotically flat initial data set satisfying the dominant
energy condition,  as well as the assumption $E=|P|$.   Given a scalar
function $f_0$ and a vector field $X_0$ so that $(f_0, X_0)\to (a,b)$,
we introduce the \emph{modified Regge--Teitelboim Hamiltonian} $\mathcal
H$ corresponding to $(g,k)$ and $(f_0, X_0)$ by 
{\begin{equation*}
\mathcal H(\gamma, \tau) =(n-1)\omega_{n-1} \left[ a E(\gamma,\tau) + b
\cdot P(\gamma, \tau)\right] - \int_M \overline{\Phi}|_{(g,k)}(\gamma,
\tau) \cdot (f_0, X_0)\, \mathrm{d}\mu_g
\end{equation*}}\unskip
where the volume measure $\mathrm{d}\mu_g$ and the inner product in the integral
are both with respect to~$g$. The functional is obtained from the
classical Regge--Teitelboim Hamiltonian in 
Definition~\ref{definition:functional}   by replacing the usual
constraint operator with the modified constraint operator
$\overline{\Phi}|_{(g,k)}$. We also choose the volume form with respect
to $g$. 

Define the constraint set $\mathcal{C}_{(g,k)} $ to be the set of
asymptotically flat initial data sets $(\gamma,\tau)$ such that
$\overline{\Phi}_{(g,k)}(\gamma, k) = \overline{\Phi}|_{(g,k)}(g, k)$.
A key property of the modified constraint operator is that if  $(g, k)$
satisfies the dominant energy condition, then the dominant energy
condition holds for all $(\gamma, \tau)\in \mathcal{C}_{(g,k)}$ close
to $(g, k)$. Choosing the pair  $(f_0, X_0)$ asymptoting to $(a,b) :=
(E,-P)$, we apply the positive mass inequality to conclude that $(g,
k)$ locally minimizes the functional $\mathcal{H}$ on
$\mathcal{C}_{(g,k)} $ as  below: For $(\gamma, \tau) \in
\mathcal{C}_{(g,k)}$, by the same computation as in \xeqref{eq:causal},
we have
{\begin{eqnarray}
\mathcal{H} (\gamma, \tau) - \mathcal{H} (g, k) &=& (n-1) \omega_{n-1}
\left[ a E(\gamma, \tau) + b \cdot P(\gamma, \tau) \right] - (n-1)
\omega_{n-1} \left[ a E + b \cdot P \right] \label{eq:1}\Seqnsplit
&\ge& (n-1) \omega_{n-1} \left[ E E(\gamma, \tau) - P \cdot P(\gamma,
\tau) \right] \ge 0,\label{eq:2}
\end{eqnarray}}\unskip
where the inequality $ E(\gamma, \tau) \ge |P(\gamma, \tau)| $ is used.
The equality holds when $(\gamma, \tau) = (g, k)$.

Using the method of Lagrange multipliers, there exists a Lagrange
multiplier $(f_1, X_1)$ asymptotic to zero such that
{\[
\left. D\mathcal{H} \right|_{(g, k)} (h, w)  =\int_M (f_1, X_1) \cdot
D\overline{\Phi}|_{(g,k)}(h, w) \, \mathrm{d}\mu_g = \int_M D\overline
\Phi|_{(g, k)}^* (f_1, X_1) \cdot (h, w) \, \mathrm{d}\mu_g 
\]}\unskip
Since the left hand side equals $-\int_M (h,w) \cdot
D\overline{\Phi}|_{(g,k)}^* (f_0, X_0)  \, \mathrm{d}\mu_g$, just as
the first variation of the classical Regge--Teitelboim Hamiltonian
\xeqref{equation:functional}, we construct a pair $(f, X) = (f_0, X_0)
+ (f_1, X_1)$ that solves $D\overline{\Phi}|_{(g,k)}^*(f, X)=0$ and is
asymptotic to $(E, -P)$. To summarize, the above construction yields a
modified Killing initial data $(f, X)$  with the prescribed asymptotics
$(f, X)\to (E, -P)$. 

Note that in the non-vacuum case, there need not be a corresponding
spacetime Killing vector field. However, intriguingly, the spinor proof
by Beig and Chru\'sciel uses the harmonic spinor to construct $(f, X)$
that satisfies a similar over-determined equation with the asymptotic
to $(E, -P)$. Their key observation is that for asymptotically flat
initial data sets with the optimal decay rate, this implies $E=|P|=0$. 
It completes the proof. 

\subsection{Proof that $E=|P|=0$ implies embedding in Minkowski spacetime}

Suppose $ E = 0 $. We extend the above variational setting to show that
the initial data set can be embedded into Minkowski
space~\cite{Huang-Lee:2023}. Unlike other
proofs~\cite{Schoen-Yau:1981-pmt2, Witten:1981, Parker-Taubes:1982,
Eichmair:2013, Hirsch-Zhang:2023}, this proof does not rely on the
specific method used to establish the $ E \ge 0 $ theorem. Instead, it
characterizes the minimizer of the modified Regge--Teitelboim
Hamiltonian by producing additional modified Killing initial data,
under the assumption that the positivity of mass holds.

\begin{theo}\label{thm:main}
Let $(M, g, k)$ be a asymptotically flat initial data set with decay
rate~$q>\sfrac{(n-2)}{2}$ and assume that the positive mass theorem is
true near $(g,k)$. If $(g,k)$ satisfies the dominant energy condition
and  $E=0$, then $(M,g, k)$ is embedded into the Minkowski spacetime. 
\end{theo}

As in the proof of Theorem~\ref{thm:E=P}, we use the fact that $(g, k)$
minimizes a modified Regge--Teitelboim Hamiltonian $\mathcal H$ on the
constraint set $\mathcal{C}_{(g,k)}$. Then we invoke the method of
Lagrange multiplier to construct a lapse-shift pair $(f, X)$ that
solves $D\overline \Phi|_{(g, k)}^*(f, X)=0$ with the prescribed
asymptotics $(f, X)\to (a, b)$, where $(a,b) = (E, -P)$.  

When $E=|P|=0$, it turns out that $(g,k)$ minimizes the modified
Regge--Teitelboim Hamiltonian for many choices of $(a, b)$, because the
last term of \xeqref{eq:1} is alway zero. By similar computations as in
\xeqref{eq:2}, we  can choose any $a$ and  $b$ with $a= |b|$. This
allows us to construct an entire $(n+1)$-dimensional space of
lapse-shift pairs that can be thought of as being asymptotic to the
$(n+1)$-dimensional space of translational Killing fields on Minkowski
space as we approach spatial infinity.  Moreover, having so many
solutions implies that $(g,k)$ is vacuum, 
by~\cite[Corollary~6.6]{Huang-Lee:2024}.  

Once we know that $(g,k)$ is vacuum, it follows that each of these
lapse-shift pairs is actually vacuum Killing initial data for $(g,k)$
by Moncrief's theorem, and we can extend them to become actual Killing
fields on the vacuum development of $(g,k)$. The final step is to show
that having such a space of Killing fields on an asymptotically flat
Lorentzian manifold that are asymptotic to the $(n+1)$ translational
directions implies that the Lorentzian manifold must be Minkowski
space. 

\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}

{The author was partially supported by NSF DMS-2304966. Part of the
survey was completed while the author was in residence at the Simons
Laufer Mathematical Sciences Institute in Berkeley, California, during
the Fall 2024 semester, supported by the NSF DMS-1928930.} 
 
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