# The case of equality for the spacetime positive mass theorem

Sven Hirsch  Department of Mathematics, Duke University, Durham, NC, 27708,
USA<EMAIL_ADDRESS>and  Yiyue Zhang  Department of Mathematics,
University of California, Irvine, CA, 92697, USA<EMAIL_ADDRESS>
###### Abstract.

The rigidity of the spacetime positive mass theorem states that an initial
data set satisfying the dominant energy condition with vanishing mass can be
isometrically embedded into Minkowski space. This has been established by
Beig-Chrusciel and Huang-Lee under additional decay assumptions for the energy
and momentum densities $\mu$ and $J$. In this note we give a new and
elementary proof in dimension 3 which removes these additional decay
assumptions. Our argument is based on spacetime harmonic functions and
Liouville’s theorem.

##  1\. Introduction

One of the central objects studied in general relativity are isolated
gravitational systems such as stars, black holes and galaxies. Mathematically,
they are modeled by asymptotically flat initial data sets (IDS) which are
triples $(M,g,k)$ consisting of an asymptotically flat, complete, smooth
Riemannian 3-manifold $(M,g)$ together with a smooth, symmetric two-tensor
$k$.

More precisely, $(M,g)$ contains a compact set $\mathcal{C}\subset M$ such
that we can write
$M\setminus\mathcal{C}=\cup_{\ell=1}^{\ell_{0}}M_{end}^{\ell}$ where the ends
$M_{end}^{\ell}$ are pairwise disjoint and diffeomorphic to the complement of
a ball $\mathbb{R}^{3}\setminus B_{1}$. Furthermore, there exists a coordinate
system in each end satisfying

(1) $|\partial^{l}(g_{ij}-\delta_{ij})(x)|=O(|x|^{-\tau-l}),\quad
l=0,1,2,\quad\quad|\partial^{l}k_{ij}(x)|=O(|x|^{-\tau-1-l}),\quad l=0,1,$

for some $\tau>\tfrac{1}{2}$. To each initial data set $(M,g,k)$ we associate
the energy density $\mu$ and the momentum density $J$ defined by

(2) $\displaystyle\mu=\frac{1}{2}(R+(\text{Tr}_{g}k)^{2}-|k|^{2}),\quad\quad
J=\operatorname{div}_{g}(k-\text{Tr}_{g}kg)$

where $R$ is the scalar curvature of $g$. Moreover, we define the ADM energy
$E$ and linear momentum $P$ by

(3)
$E=\lim_{r\rightarrow\infty}\frac{1}{16\pi}\int_{S_{r}}\sum_{i}\left(g_{ij,i}-g_{ii,j}\right)\upsilon^{j}dA,\quad\quad
P_{i}=\lim_{r\rightarrow\infty}\frac{1}{8\pi}\int_{S_{r}}\left(k_{ij}-(\mathrm{Tr}_{g}k)g_{ij}\right)\upsilon^{j}dA$

where $\upsilon$ is the outer unit normal to the sphere $S_{r}$ and $dA$ is
its area element. In order to ensure that $E$ and $P$ are well-defined in
equation (3), we impose additionally $\mu,J\in L^{1}(M)$, and throughout this
paper we assume that $g\in C^{2,\alpha}(M)$ and $k\in C^{1,\alpha}(M)$.

A fundamental results about initial data sets is the positive mass theorem
(PMT):

######  Theorem 1.1.

Suppose $(M,g,k)$ is a complete asymptotically flat initial data set
satisfying the dominant energy condition (DEC) $\mu\geq|J|$. Then $E\geq|P|$.

This result has been first established by Schoen-Yau in [19] using the Jang
equation and by Witten in [21] using spinors. Further proofs have been given
by [8, 9, 10], and the important special case $k=0$ has been treated in [18,
14, 16, 6, 1]. We refer to [10] for a more detailed historical overview.

It has been conjectured that if $E=|P|$, the IDS embeds isometrically in
Minkowski spacetime with second fundamental form $k$. This has been already
confirmed under additional decay assumptions on $g$ and $k$ by Beig-Chrusciel
and Huang-Lee in [2, 12]. More precisely, Beig-Chrusciel assume additionally
$g_{ij}-\delta_{ij}\in C^{3,\alpha}_{-\tau}(M)$, $k_{ij}\in
C^{2,\alpha}_{-\tau-1}(M)$ and $\mu,J\in C^{1,\alpha}_{-3-\epsilon}(M)$ for
some constants $\tau>\frac{1}{2}$, $\epsilon>0$ and $0<\alpha<1$. Huang-Lee
assume additionally $\mu,J\in C^{0,\alpha}_{-3-\epsilon}(M)$ and
$\text{Tr}_{g}k\in C^{0}_{-2-\epsilon}(M)$. As observed in [10], the decay
condition $\text{Tr}_{g}k\in C^{0}_{-2-\epsilon}(M)$ can be omitted by
combining [12] with [10]. However, the general case is still an open question
and is for instance listed as conjecture in [15], page 226.

Furthermore, we would like to point on that [12] and [7] addressed the
rigidity conjecture in higher dimension under certain additional assumptions.
However, the situation becomes more subtle, see for instance the counter
example constructed in [12]. Finally, we would also like to point out the
paper [11] on the rigidity of asymptotically hyperbolic manifolds.

In this manuscript we establish the following result which removes the
additional decay assumptions required in previous papers:

######  Theorem 1.2.

Let $(M,g,k)$ be a complete asymptotically flat initial data set satisfying
the dominant energy condition $\mu\geq|J|$. Moreover, suppose that $E=|P|$.
Then $E=|P|=0$ and $(M,g,k)$ arises as spacelike slice of Minkowski spacetime
$\mathbb{R}^{3,1}$.

Our theorem is optimal in the sense that we merely need to assume $\mu,J\in
L^{1}(M)$ which is required to ensure that $E$ and $P$ are finite and
independent of the coordinate system used.

Our proof is short, elementary and relies on two ingredients: First, we use
the integral formula for spacetime harmonic functions $u$ established in [10].
Using the integral formula, we deduce that $E=|P|$ implies that all level-sets
of $u$ have vanishing Gaussian curvature. Second, we employ the fundamental
theorem of surfaces which states that if $(M,g,k)$ satisfies the Gauss and
Codazzi equations, $(M,g,k)$ embeds isometrically into Minkowski spacetime.
Combining the flatness of the level-sets with Liouville’s theorem, we verify
that the Gauss and Codazzi equations are indeed satisfied. We expect that this
method can also be applied in other settings such as asymptotically hyperbolic
manifolds.

Acknowledgements. The authors would like to thank Hubert Bray, Demetre
Kazaras, Marcus Khuri and Dan Lee for stimulating discussions.

##  2\. Preliminaries

There are several tools available to study IDS such as the Jang equation [19],
spinors [21] and marginally outer trapped surfaces [9]. In [10] a new method
to study IDS has been introduced: _spacetime harmonic functions_. The main
result of [10] states the following:

######  Theorem 2.1.

There exists an asymptotically linear, spacetime harmonic function $u\in
C^{2,\alpha}(M_{ext})$, i.e., a function $u$ solving the differential equation
$\Delta u=-\text{Tr}_{g}k|\nabla u|$, such that

(4) $\displaystyle
E-|P|\geq\frac{1}{16\pi}\int_{M_{ext}}\left(\frac{|\nabla^{2}u+k|\nabla
u||^{2}}{|\nabla u|}+2\mu|\nabla u|+2\langle J,\nabla u\rangle\right).$

We refer to [10] for a discussion of the exterior region $M_{ext}$ and to [4]
for a detailed motivation of spacetime harmonic functions. The above theorem
yields directly:

######  Corollary 2.2.

Let $(M,g,k)$ be an asymptotically flat initial data set satisfying the
dominant energy condition $\mu\geq|J|$ and suppose $E=|P|$. Then
$M=M_{ext}=\mathbb{R}^{3}$ and there exists an asymptotically linear spacetime
harmonic function $u\in C^{3,\alpha}(M)$ satisfying

(5) $\displaystyle\nabla^{2}u=$ $\displaystyle-k|\nabla u|,$ (6)
$\displaystyle\mu|\nabla u|=$ $\displaystyle-\langle J,\nabla u\rangle.$

Moreover, $|\nabla u|\neq 0$ and the level sets $\Sigma_{t}=\\{u=t\\}$ are
flat with second fundamental form $h=-k|_{T\Sigma_{t}}$.

###### Proof.

The identities for $\nabla^{2}u=-k|\nabla u|$ and $\mu|\nabla u|=-\langle
J,\nabla u\rangle$ follow immediately from the integral formula (4). This also
implies $h=-k|_{T\Sigma_{t}}$. Lemma 7.1 and Proposition 7.2 in [10]
established that $|\nabla u|\neq 0$ and $M=M_{ext}=\mathbb{R}^{3}$. The claim
$u\in C^{3,\alpha}(M)$ follows immediately from Schauder estimates in
combination with the non-vanishing of $|\nabla u|$. Finally, the claim that
the level sets have vanishing Gaussian curvature is implied from the following
computation, also see [5]. Since $\mu|\nabla u|=-\langle J,\nabla u\rangle$,
the Gaussian equations yield

(7) $\displaystyle\Delta|\nabla u|=\frac{1}{|\nabla u|}(-K|\nabla
u|+|k|^{2}|\nabla u|^{2}-\langle\operatorname{div}k,\nabla u\rangle|\nabla
u|)$

where $K$ is the Gaussian curvature of $\Sigma_{t}$. On the other side, we
have by the equation $\nabla^{2}u=-k|\nabla u|$

(8) $\displaystyle\Delta|\nabla u|=|k|^{2}|\nabla
u|-\langle\operatorname{div}k,\nabla u\rangle$

which finishes the proof. ∎

To prove rigidity of the spacetime PMT we will need to use every piece of
information given by this corollary.

##  3\. Proof of Theorem 1.2

Throughout this section we assume $E=|P|$, and let $u$ be the asymptotically
linear spacetime harmonic function from Corollary 2.2. Let $e_{3}=\frac{\nabla
u}{|\nabla u|}$. For a fixed level set $\Sigma$, we can express the level set
metric by $dx^{2}_{1}+dx^{2}_{2}$ which is possible since $\Sigma$ is flat.
Let $e_{1}=\partial_{x_{1}}$, $e_{2}=\partial_{x_{2}}$, then we extend
$e_{1}$, $e_{2}$ to the entire manifold such that $\\{e_{1},e_{2},e_{3}\\}$
forms an orthonormal frame. We use Greek letter $\alpha$, $\beta$ to denote
$e_{1},e_{2}$, and Roman letters $i,j,k,l$ to denote $e_{1},e_{2},e_{3}$.

We define $\bar{R}_{ijkl}=R_{ijkl}+k_{il}k_{jk}-k_{ik}k_{jl}$ and say that
$(M,g,k)$ satisfies the Gauss- and Codazzi equations if $\bar{R}_{ijkl}=0$ and
$\nabla_{i}k_{jk}-\nabla_{j}k_{ik}=0$ for all $i,j,k,l$.

######  Proposition 3.1.

Suppose $(M,g,k)$ is asymptotically flat, and satisfies the Gauss and Codazzi
equations. Then $(M,g,k)$ arises as a subset of Minkowski spacetime.

This is the Lorentzian version of the well-known fundamental theorem for
hypersurfaces. For the convenience of the reader we provide a proof in
Appendix A. In the next two lemma we demonstrate that the majority of the
Gauss- and Codazzi equations are already satisfied.

######  Lemma 3.2.

We have

(9) $\displaystyle 0=$ $\displaystyle\nabla_{1}k_{23}-\nabla_{2}k_{13},$ (10)
$\displaystyle 0=$
$\displaystyle\nabla_{\alpha}k_{\beta\beta}-\nabla_{\beta}k_{\alpha\beta},$
(11) $\displaystyle 0=$
$\displaystyle\nabla_{\alpha}k_{33}-\nabla_{3}k_{\alpha 3}.$

###### Proof.

The first identity follows from

(12) $\displaystyle\nabla_{1}k_{23}-\nabla_{2}k_{13}=$
$\displaystyle-\nabla_{1}\frac{\nabla^{2}_{23}u}{|\nabla
u|}+\nabla_{2}\frac{\nabla^{2}_{13}u}{|\nabla u|}=R_{2133}=0.$

Observe that $J_{\alpha}=0$ implies

(13)
$\displaystyle\nabla_{\beta}k_{\alpha\beta}-\nabla_{\alpha}k_{\beta\beta}+\nabla_{3}k_{\alpha
3}-\nabla_{\alpha}k_{33}=0.$

Thus, the last two identities follow from

(14) $\displaystyle\nabla_{3}k_{\alpha
3}-\nabla_{\alpha}k_{33}=-\nabla_{3}\frac{\nabla^{2}_{\alpha 3}u}{|\nabla
u|}+\nabla_{\alpha}\frac{\nabla^{2}_{33}u}{|\nabla u|}=R_{\alpha 333}=0.$

∎

######  Lemma 3.3.

We have

(15) $\displaystyle\bar{R}_{1212}=$ $\displaystyle 0,$ (16)
$\displaystyle\bar{R}_{\alpha\beta 3\alpha}=$ $\displaystyle 0,$ (17)
$\displaystyle\bar{R}_{\alpha 33\beta}=$ $\displaystyle A_{\alpha\beta}.$

where $A_{\alpha\beta}:=\nabla_{3}k_{\alpha\beta}-\nabla_{\alpha}k_{\beta 3}$.

###### Proof.

Using the Gauss equations we obtain

(18) $\displaystyle R_{1212}=2K-h_{11}h_{22}+h_{12}^{2}.$

Thus, the first identity follows from $K=0$ and $h=-k|_{\Sigma}$. Next, we
compute

(19) $\displaystyle R_{\alpha\beta
3\alpha}=(\nabla_{\alpha}\nabla_{\beta}-\nabla_{\beta}\nabla_{\alpha})\frac{\nabla_{\alpha}u}{|\nabla
u|}.$

Using the spacetime Hessian equation $\nabla^{2}u=-k|\nabla u|$, we obtain

(20) $\displaystyle\bar{R}_{\alpha\beta\alpha
3}=\nabla_{\alpha}k_{\alpha\beta}-\nabla_{\beta}k_{\alpha\alpha}=0,$

where the last equality follows from the previous lemma. Finally, the third
identity follows in the same spirit as the second one. ∎

Next, we show that $A_{\alpha\beta}$ is vanishing. This will be achieved by
PDE methods in combination with the asymptotics of $g,k$.

######  Lemma 3.4.

On each level set, there exists a twice differentiable function $F$ such that

(21) $\displaystyle\nabla^{\Sigma}_{\alpha\beta}F=|\nabla
u|^{-2}A_{\alpha\beta}.$

For the proof of this lemma we need to additionally assume that $g\in
C^{3}(M)$ and $k\in C^{2}(M)$. However, we provide an alternative approach to
the spacetime PMT rigidity in Appendix B. This approach does not require such
additional regularity of $g$ and $k$ and therefore establishes Theorem 1.2 in
full generality.

###### Proof.

We first show that $\partial_{2}(|\nabla u|^{-2}A_{11})=\partial_{1}(|\nabla
u|^{-2}A_{12})$ and $\partial_{1}(|\nabla u|^{-2}A_{22})=\partial_{2}(|\nabla
u|^{-2}A_{12})$. Since
$\langle\nabla_{e_{\alpha}}e_{3},e_{\beta}\rangle=-k_{\alpha\beta}$ and
applying Lemma 3.2, we obtain

(22) $\displaystyle\partial_{2}A_{11}=$
$\displaystyle\partial_{2}(\nabla_{3}k_{11}-\nabla_{1}k_{13})$ (23)
$\displaystyle=$
$\displaystyle\nabla_{2}(\nabla_{3}k_{11}-\nabla_{1}k_{13})-k_{2}^{\alpha}\nabla_{\alpha}k_{11}+2k_{21}\nabla_{3}k_{31}$
(24) $\displaystyle-
k_{21}\nabla_{3}k_{13}-k_{21}\nabla_{1}k_{33}+k_{2}^{\alpha}\nabla_{1}k_{1\alpha}$
(25) $\displaystyle=$
$\displaystyle\nabla_{2}(\nabla_{3}k_{11}-\nabla_{1}k_{13}).$

Therefore, we have

(26) $\displaystyle\partial_{2}A_{11}-\partial_{1}A_{12}$ (27)
$\displaystyle=$
$\displaystyle\nabla_{2}(\nabla_{3}k_{11}-\nabla_{1}k_{13})-\nabla_{1}(\nabla_{3}k_{12}-\nabla_{2}k_{13})$
(28) $\displaystyle=$
$\displaystyle\nabla_{3}\nabla_{2}k_{11}-2R_{231}^{i}k_{1i}-\nabla_{2}\nabla_{1}k_{13}-(\nabla_{3}\nabla_{1}k_{12}-R_{131}^{i}k_{i2}-R_{132}^{i}k_{1i})$
(29)
$\displaystyle+(\nabla_{2}\nabla_{1}k_{13}-R_{121}^{i}k_{i3}-R_{123}^{i}k_{1i})$
(30) $\displaystyle=$
$\displaystyle\nabla_{3}\nabla_{2}k_{11}-\nabla_{3}\nabla_{1}k_{12}-R_{2312}k_{12}-R_{2313}k_{13}+R_{1312}k_{22}$
(31) $\displaystyle+R_{1313}k_{23}-R_{1212}k_{23}-R_{1213}k_{33}$ (32)
$\displaystyle=$
$\displaystyle\nabla_{3}\nabla_{2}k_{11}-\nabla_{3}\nabla_{1}k_{12}-(k_{12}k_{23}-k_{22}k_{13})k_{12}-(k_{12}k_{33}-k_{23}k_{13}-\nabla_{3}k_{12}+\nabla_{1}k_{23})k_{13}$
(33)
$\displaystyle+(k_{11}k_{23}-k_{12}k_{13})k_{22}+(k_{11}k_{33}-k_{13}^{2}-\nabla_{3}k_{11}+\nabla_{1}k_{13})k_{23}$
(34)
$\displaystyle-(k_{11}k_{22}-k_{12}^{2})k_{23}-(k_{11}k_{23}-k_{13}k_{12})k_{33}$
(35) $\displaystyle=$
$\displaystyle\nabla_{3}\nabla_{2}k_{11}-\nabla_{3}\nabla_{1}k_{12}-(-\nabla_{3}k_{12}+\nabla_{1}k_{23})k_{13}+(-\nabla_{3}k_{11}+\nabla_{1}k_{13})k_{23}.$

Because
$\langle\nabla_{3}e_{\alpha},e_{3}\rangle=-\langle\nabla_{3}e_{3},e_{\alpha}\rangle=k_{\alpha
3}$, we deduce

(36) $\displaystyle\nabla_{3}\nabla_{2}k_{11}-\nabla_{3}\nabla_{1}k_{12}$ (37)
$\displaystyle=$
$\displaystyle\partial_{3}(\nabla_{2}k_{11})-\nabla_{\nabla_{3}e_{2}}k_{11}-2\nabla_{2}k(\nabla_{3}e_{1},e_{1})$
(38)
$\displaystyle-\partial_{3}(\nabla_{1}k_{12})+\nabla_{\nabla_{3}e_{1}}k_{12}+\nabla_{1}k(\nabla_{3}e_{1},e_{2})+\nabla_{1}k(e_{1},\nabla_{3}e_{2})$
(39) $\displaystyle=$ $\displaystyle-\langle
e_{1},\nabla_{3}e_{2}\rangle\nabla_{1}k_{11}-k_{23}\nabla_{3}k_{11}-2\langle\nabla_{3}e_{1},e_{2}\rangle\nabla_{2}k_{21}-2k_{13}\nabla_{2}k_{31}+\langle
e_{2},\nabla_{3}e_{1}\rangle\nabla_{2}k_{12}$ (40)
$\displaystyle+k_{13}\nabla_{3}k_{12}+\langle
e_{2},\nabla_{3}e_{1}\rangle\nabla_{1}k_{22}+k_{13}\nabla_{1}k_{32}+\langle
e_{1},\nabla_{3}e_{2}\rangle\nabla_{1}k_{11}+k_{23}\nabla_{1}k_{13}$ (41)
$\displaystyle=$ $\displaystyle
k_{23}(\nabla_{1}k_{13}-\nabla_{3}k_{11})-k_{13}(\nabla_{1}k_{32}-\nabla_{3}k_{12}).$

Combing Equation (35) and (41) yields

(42) $\partial_{2}A_{11}-\partial_{1}A_{12}=2A_{12}k_{13}-2A_{11}k_{23}.$

Moreover, we have $\partial_{\alpha}|\nabla u|=-k_{\alpha 3}$ which implies

(43) $\displaystyle\partial_{2}(|\nabla u|^{-2}A_{11})-\partial_{1}(|\nabla
u|^{-2}A_{12})$ (44) $\displaystyle=$ $\displaystyle|\nabla
u|^{-2}(\partial_{2}A_{11}-\partial_{1}A_{12})+A_{11}\partial_{2}|\nabla
u|^{-2}-A_{12}\partial_{1}|\nabla u|^{-2}$ (45) $\displaystyle=$
$\displaystyle|\nabla u|^{-2}(2A_{12}k_{13}-2A_{11}k_{23})+2A_{11}|\nabla
u|^{-2}k_{23}-2A_{12}|\nabla u|^{-2}k_{13}$ (46) $\displaystyle=$
$\displaystyle 0.$

Therefore, $|\nabla u|^{-2}A_{11}dx_{1}+|\nabla u|^{-2}A_{12}dx_{2}$ is
closed, where $dx_{1}$ and $dx_{2}$ are the dual 1-forms of $e_{1}$ and
$e_{2}$. Since the topology of a level set is trivial, there exists on each
level set a function which we suggestively denote by $F_{1}$ such that
$dF_{1}=|\nabla u|^{-2}A_{11}dx_{1}+|\nabla u|^{-2}A_{12}dx_{2}$. Replacing
the roles of $e_{1}$ and $e_{2}$, there exists another function $F_{2}$ such
that $dF_{2}=|\nabla u|^{-2}A_{12}dx_{1}+|\nabla u|^{-2}A_{22}dx_{2}$. Next,
we compute

(47) $\displaystyle d(F_{1}dx_{1}+F_{2}dx_{2})=$ $\displaystyle\frac{\partial
F_{1}}{\partial x_{2}}dx_{2}\wedge dx_{1}+\frac{\partial F_{2}}{\partial
x_{1}}dx_{1}\wedge dx_{2}$ (48) $\displaystyle=$ $\displaystyle(|\nabla
u|^{-2}A_{12}-|\nabla u|^{-2}A_{12})dx_{2}\wedge dx_{1}=0.$

Thus there exists an $F$ with $dF=F_{1}dx_{1}+F_{2}dx_{2}$. ∎

######  Lemma 3.5.

On each level set, $F$ is a linear function with respect to $x_{1}$ and
$x_{2}$, i.e. $\nabla^{2}_{\Sigma}F=0$.

###### Proof.

First observe that $F$ is superharmonic on each level set, i.e.

(49) $\displaystyle\Delta^{\Sigma}F\geq 0$

which follows immediately from

(50) $\Delta^{\Sigma}F=|\nabla u|^{-2}(A_{11}+A_{22})=-|\nabla
u|^{-2}J_{3}=|\nabla u|^{-2}\mu\geq 0.$

Since $\partial^{l}k_{ij}=O(|x|^{-\tau-l-1})$, $\tau>\frac{1}{2}$, $l=0,1$, we
obtain

(51)
$F_{11}=\nabla^{\Sigma}_{11}F=\nabla_{3}k_{11}-\nabla_{1}k_{13}=O(|x|^{-\tau-2}).$

Therefore, integrating $F_{11}$ twice on the level set, we see that $F=L+B$,
where $L$ is a linear function with respect to $\\{x_{1},x_{2}\\}$, and $B$ is
a bounded function. Combining this with our previous observation yields
$\Delta^{\Sigma}B=\Delta^{\Sigma}F\geq 0$. Thus, $B$ is constant in view of
Liouville’s theorem. ∎

###### Proof of Theorem 1.2.

Since $\nabla^{2}_{\Sigma}F=0$, $(M,g,k)$ satisfies the Gauss and Codazzi
equations which completes the proof in view of the Proposition 3.1. ∎

##  Appendix A The fundamental theorem of hypersurfaces

###### Proof of Proposition 3.1.

We follow the proof of [17], page 100. Let $U$ be a compact subset of $M$. We
construct the metric $\bar{g}=-dt^{2}+g_{t}$ on
$(-\varepsilon,\varepsilon)\times U$ by prescribing

(52) $\displaystyle\partial_{t}g_{t}(\partial_{i},\partial_{j})=$
$\displaystyle 2\bar{\nabla}^{2}_{ij}t,$ (53) $\displaystyle\bar{g}|_{t=0}=$
$\displaystyle g,$ (54)
$\displaystyle\partial_{t}(\bar{\nabla}^{2}_{ij}t)-(\bar{\nabla}^{2}t)^{2}_{ij}=$
$\displaystyle 0,$ (55) $\displaystyle\bar{\nabla}^{2}_{ij}t|_{t=0}=$
$\displaystyle k_{ij}$

where
$(\bar{\nabla}^{2}t)^{2}_{ij}=(\bar{\nabla}^{2}t)_{i}^{l}(\bar{\nabla}^{2}t)_{jl}$.
We will use Roman letters $\\{i,j,k,l\\}$ to denote indices tangential to $M$.
By standard ODE existence theory there exists a small $\varepsilon>0$ such
that we can solve the above equation for $t\in(-\varepsilon,\varepsilon)$.
Next, we take a cover $\\{U_{i}\\}$ of $M$. According to the asymptotics of
$(M,g,k)$, there exists a uniform $\varepsilon>0$ for each $U_{i}$. Therefore,
we can patch together above’s construction and $(M,g)$ can be embedded in
$((-\varepsilon,\varepsilon)\times M,\bar{g})$ with the second fundamental
form $k$.

To verify the flatness of $\bar{g}$ we proceed exactly as in [17]. It suffices
to verify that the curvatures $\bar{R}_{tijt}$, $\bar{R}_{ijkl}$ and
$\bar{R}_{tijk}$ are vanishing. Observe that
$\langle\bar{\nabla}t,\bar{\nabla}t\rangle=-1$ implies
$\bar{\nabla}_{i}\bar{\nabla}_{t}t=0$. Combining this with equation (54)
yields

(56) $\displaystyle 0=$
$\displaystyle\partial_{t}(\bar{\nabla}^{2}_{ij}t)-(\bar{\nabla}^{2}t)^{2}_{ij}$
(57) $\displaystyle=$
$\displaystyle\bar{\nabla}_{t}\bar{\nabla}_{i}\bar{\nabla}_{j}t+(\bar{\nabla}^{2}t)^{2}_{ij}$
(58) $\displaystyle=$
$\displaystyle\bar{\nabla}_{i}\bar{\nabla}_{t}\bar{\nabla}_{j}t-\bar{R}_{tijt}+(\bar{\nabla}^{2}t)^{2}_{ij}$
(59) $\displaystyle=$
$\displaystyle\partial_{i}(\bar{\nabla}_{t}\bar{\nabla}_{j}t)-\bar{\nabla}^{2}t(\partial_{t},\bar{\nabla}_{i}\partial_{j})-\bar{\nabla}^{2}t(\partial_{j},\bar{\nabla}_{i}\partial_{t})-\bar{R}_{tijt}+(\bar{\nabla}^{2}t)^{2}_{ij}$
(60) $\displaystyle=$ $\displaystyle-\bar{R}_{tijt}.$

Since $\bar{R}_{tijt}=0$, $\bar{\nabla}_{t}\partial_{t}=0$ and
$\bar{\Gamma}_{ti}^{t}=0$, we obtain

(61) $\displaystyle\partial_{t}(\bar{R}_{tijk})=$
$\displaystyle(\bar{\nabla}_{t}\bar{R})_{tijk}+\bar{R}_{tljk}\bar{\Gamma}_{ti}^{l}+\bar{R}_{tilk}\bar{\Gamma}_{tj}^{l}+\bar{R}_{tljk}\bar{\Gamma}_{tk}^{l}$
(62) $\displaystyle=$
$\displaystyle(\bar{\nabla}_{j}\bar{R})_{titk}+(\bar{\nabla}_{k}\bar{R})_{tijt}+\bar{R}_{tljk}\bar{\Gamma}_{ti}^{l}+\bar{R}_{tilk}\bar{\Gamma}_{tj}^{l}+\bar{R}_{tljk}\bar{\Gamma}_{tk}^{l}$
(63) $\displaystyle=$
$\displaystyle\partial_{j}(\bar{R}_{titk})-\bar{R}_{litk}\bar{\Gamma}_{jt}^{l}-\bar{R}_{tilk}\bar{\Gamma}_{jt}^{l}+\partial_{k}(\bar{R}_{tijt})-\bar{R}_{lijt}\bar{\Gamma}_{kt}^{l}-\bar{R}_{tijl}\bar{\Gamma}_{kt}^{l}$
(64)
$\displaystyle+\bar{R}_{tljk}\bar{\Gamma}_{ti}^{l}+\bar{R}_{tilk}\bar{\Gamma}_{tj}^{l}+\bar{R}_{tljk}\bar{\Gamma}_{tk}^{l}$
(65) $\displaystyle=$
$\displaystyle-\bar{R}_{litk}\bar{\Gamma}_{jt}^{l}-\bar{R}_{tilk}\bar{\Gamma}_{jt}^{l}-\bar{R}_{lijt}\bar{\Gamma}_{kt}^{l}-\bar{R}_{tijl}\bar{\Gamma}_{kt}^{l}+\bar{R}_{tljk}\bar{\Gamma}_{ti}^{l}+\bar{R}_{tilk}\bar{\Gamma}_{tj}^{l}+\bar{R}_{tljk}\bar{\Gamma}_{tk}^{l}.$

According to the Codazzi equation, $\bar{R}_{tijk}|_{t=0}=0$, and thus
$\bar{R}_{tijk}=0$. Next, we compute

(66) $\displaystyle\partial_{t}(\bar{R}_{ijkl})=$
$\displaystyle(\bar{\nabla}_{t}\bar{R})_{ijkl}+\bar{R}_{sjkl}\bar{\Gamma}_{ti}^{s}+\bar{R}_{iskl}\bar{\Gamma}_{tj}^{s}+\bar{R}_{ijsl}\bar{\Gamma}_{tk}^{s}+\bar{R}_{ijks}\bar{\Gamma}_{tl}^{s}$
(67) $\displaystyle=$
$\displaystyle(\nabla_{k}\bar{R})_{ijtl}+(\nabla_{l}\bar{R})_{ijkt}+\bar{R}_{sjkl}\bar{\Gamma}_{ti}^{s}+\bar{R}_{iskl}\bar{\Gamma}_{tj}^{s}+\bar{R}_{ijsl}\bar{\Gamma}_{tk}^{s}+\bar{R}_{ijks}\bar{\Gamma}_{tl}^{s}$
(68) $\displaystyle=$
$\displaystyle-\bar{R}_{ijsl}\bar{\Gamma}_{kt}^{s}-\bar{R}_{ijks}\bar{\Gamma}_{lt}^{s}+\bar{R}_{sjkl}\bar{\Gamma}_{ti}^{s}+\bar{R}_{iskl}\bar{\Gamma}_{tj}^{s}+\bar{R}_{ijsl}\bar{\Gamma}_{tk}^{s}+\bar{R}_{ijks}\bar{\Gamma}_{tl}^{s}$
(69) $\displaystyle=$
$\displaystyle\bar{R}_{sjkl}\bar{\Gamma}_{ti}^{s}+\bar{R}_{iskl}\bar{\Gamma}_{tj}^{s}.$

According to the Gauss equations, $\bar{R}_{ijkl}|_{t=0}=0$, and thus
$\bar{R}_{ijkl}=0$. Therefore, $\bar{M}$ is flat which implies together with
the asymptotical flatness of $M$ that $\bar{M}$ is a subset of Minkowski
spacetime. ∎

##  Appendix B Killing development

Another way to proof the rigidity for the spacetime PMT, is to construct a
spacetime using our spacetime harmonic function, and demonstrating that this
spacetime is Minkowski space. For this purpose, we define on
$\tilde{M}^{4}=\mathbb{R}\times M^{3}$ the Lorentzian metric

(70) $\tilde{g}=2d\tau du+g.$

This so-called _Killing Development_ is motivated by [2, 10], though we note
that the Killing Development in [2, 10] was obtained from three, rather than a
single vector field. Since $M^{3}=\mathbb{R}^{3}$, we have
$\tilde{M}^{4}=\mathbb{R}^{4}$, and thus it suffices to show that $\tilde{g}$
is flat. The flatness of $\tilde{g}$ follows essentially from the Gauss and
Codazzi equations computed in Section 3. We present here another approach
which has the advantage that it does not require the additional regularity
assumptions $g\in C^{3}(M)$ and $k\in C^{2}(M)$ used in Lemma 3.4, and
therefore establishing Theorem 1.2 in full generality.

We first claim that we can write

(71) $g=(|\nabla
u|^{-2}+a^{2}+b^{2})du^{2}+2adudx_{1}+2bdudx_{2}+dx_{1}^{2}+dx_{2}^{2},$

for some functions $a,b\in C^{2}(M^{3})$. This essentially follows from the
flatness of the level-sets of $u$, but let us elaborate more on this
construction:

To write $g$ in the above form, we need to define globally defined coordinates
$x_{1},x_{2}$. To do so, we begin with introducing global polar coordinates.
Given some point $p_{0}\in M^{3}$, let $\Gamma:(-\infty,+\infty)\to M^{3}$ be
the integral curve through $p_{0}$ with respect to the vector field $\nabla
u$. We define the function $\rho(p)=d(p,\Gamma\cap\Sigma_{u(p)})$ where $d$
denotes the distance within the level set $\Sigma_{u(p)}$. Since $u\in
C^{3}(M)$ and $|\nabla u|\neq 0$, the second fundamental form of
$\Sigma_{u(p)}$ is $C^{1}$. On each level set $\Sigma_{t}$ of $u$, we can
write the metric $g_{\Sigma_{t}}$ as $d\rho^{2}+\rho^{2}d\theta^{2}$. We would
like $g$ to have globally such a form, i.e., we need to define an angle
function $\theta(p)\in[0,2\pi)$, for any $p\in M^{3}\backslash\Gamma$. To
uniquely determine $\theta(p)$, we fix another point $p_{1}\in M^{3}$ not
contained in the image $\text{im}(\Gamma)$. Let
$\Gamma_{1}:(-\infty,\infty)\to M^{3}$ be the integral curve through $p_{1}$
with respect to the vector field $\nabla u$. Since $|\nabla u|\neq 0$, we have
$\text{im}(\Gamma)\cap\text{im}(\Gamma_{1})=\varnothing$. We set
$\theta(\Gamma_{1})=0$. Thus, the Lorentzian metric $\tilde{g}$ can be written
in the form

(72) $\tilde{g}=2d\tau du+(|\nabla
u|^{-2}+a_{0}^{2}+\rho^{-2}b_{0}^{2})du^{2}+2a_{0}dud\rho+2b_{0}dud\theta+d\rho^{2}+\rho^{2}d\theta^{2}$

for some functions $a_{0},b_{0}\in C^{2}(M^{3}\backslash\Gamma)$, where the
$C^{2}$ regularity follows from the second fundamental form being $C^{1}$.
Finally, we make the change of coordinates $x_{1}=\rho\cos\theta$,
$x_{2}=\rho\sin\theta$ and set

(73) $\displaystyle a=a_{0}\cos\theta-
b_{0}\rho^{-1}\sin\theta,\;\;b=a_{0}\sin\theta+b_{0}\rho^{-1}\cos\theta$

to obtain

(74) $\tilde{g}=2d\tau du+(|\nabla
u|^{2}+a^{2}+b^{2})du^{2}+2adudx_{1}+2bdudx_{2}+dx_{1}^{2}+dx_{2}^{2}$

as desired.

The null vector $\partial_{\tau}=\tilde{\nabla}u$ is covariantly constant,
i.e. $\tilde{\nabla}^{2}u=0$, and the second fundamental form of
$(M^{3},g)\subset(\tilde{M}^{4},\tilde{g})$ is given by $k$. Thus,
$(\tilde{M}^{4},\tilde{g})$ is a pp-wave; see [3] for a more detailed
discussion of such spacetimes.

We denote with $\partial_{i}$ the vectors $\frac{\partial}{\partial x_{i}}$.
Then $\\{\partial_{1},\partial_{2},\partial_{u},\partial_{\tau}\\}$ form a
frame of $T\tilde{M}^{4}$ and $\\{\nabla u,\partial_{1},\partial_{2}\\}$ form
an orthogonal frame of $TM^{3}$. Using Mathematica, we obtain that the only
non-vanishing Ricci curvature terms are given by

(75) $\displaystyle\widetilde{\operatorname{Ric}}(\partial_{u},\partial_{1})=$
$\displaystyle\frac{1}{2}(-a_{x_{2}x_{2}}+b_{x_{1}x_{2}}),$ (76)
$\displaystyle\widetilde{\operatorname{Ric}}(\partial_{u},\partial_{2})=$
$\displaystyle\frac{1}{2}(a_{x_{1}x_{2}}-b_{x_{1}x_{1}}),$ (77)
$\displaystyle\widetilde{\operatorname{Ric}}(\partial_{u},\partial_{u})=$
$\displaystyle\frac{1}{2}(a_{x_{2}}-b_{x_{1}})^{2}-\frac{1}{2}\Delta_{\mathbb{R}^{2}}(|\nabla
u|^{-2}+a^{2}+b^{2})+a_{ux_{1}}+b_{ux_{2}}.$

A timelike, unit normal vector to the initial data set
$(M^{3},g,k)\subset(\tilde{M}^{4},\tilde{g})$ is given by

(78) $\displaystyle\hat{N}=|\nabla u|(-|\nabla
u|^{-2}\partial_{\tau}+\partial_{u}-a\partial_{1}-b\partial_{2}).$

Since $\tilde{R}=0$, we have
$\mu=\widetilde{\operatorname{Ric}}(\hat{N},\hat{N})$ and
$J=\widetilde{\operatorname{Ric}}(\hat{N},\cdot)$. The identity $\langle
J,\partial_{1}\rangle=\langle J,\partial_{2}\rangle=0$ yields
$\widetilde{\operatorname{Ric}}(\hat{N},\partial_{1})=\widetilde{\operatorname{Ric}}(\hat{N},\partial_{2})=0$.
Combining this with $\mu\geq 0$, we obtain
$\widetilde{\operatorname{Ric}}(\partial_{u},\partial_{u})\geq 0$. The
equation
$\widetilde{\operatorname{Ric}}(\hat{N},\partial_{1})=\widetilde{\operatorname{Ric}}(\hat{N},\partial_{2})=0$
also implies

(79) $a_{x_{2}x_{2}}=b_{x_{1}x_{2}}\quad\text{and}\quad
a_{x_{1}x_{2}}=b_{x_{1}x_{1}}.$

Thus, $\psi:=a_{x_{2}}-b_{x_{1}}$ only depends on $u$. Hence, there exists a
function $l$ such that $a=x_{2}\psi(u)+l_{x_{1}}$ and
$b=-x_{1}\psi(u)+l_{x_{2}}$. Inserting this into equation (77), we obtain

(80) $\Delta_{\mathbb{R}^{2}}\left(\frac{1}{2}|\nabla
u|^{-2}+\frac{1}{2}l_{x_{1}}^{2}+\frac{1}{2}l_{x_{2}}^{2}+l_{x_{1}}x_{2}\psi(u)-l_{x_{2}}x_{1}\psi(u)-l_{u}\right)\leq
0.$

Next, we define

(81) $\displaystyle F(u,x_{1},x_{2}):=\frac{1}{2}|\nabla
u|^{-2}+\frac{1}{2}l_{x_{1}}^{2}+\frac{1}{2}l_{x_{2}}^{2}+l_{x_{1}}x_{2}\psi(u)-l_{x_{2}}x_{1}\psi(u)-l_{u}.$

Another computation and the fact that $\tilde{\nabla}u$ is covariantly
constant, yield that the only non-vanishing Riemann curvature terms in the
frame $\\{\partial_{1},\partial_{2},\partial_{\tau},\nabla u\\}$ are given by

(82) $\displaystyle\tilde{R}(\nabla u,\partial_{1},\partial_{1},\nabla u)=$
$\displaystyle R(\nabla u,\partial_{1},\partial_{1},\nabla u)+k(\nabla
u,\nabla u)k(\partial_{1},\partial_{1})-k^{2}(\nabla u,\partial_{1})$ (83)
$\displaystyle=$ $\displaystyle-|\nabla u|^{4}F_{x_{1}x_{1}},$ (84)
$\displaystyle\tilde{R}(\nabla u,\partial_{2},\partial_{2},\nabla u)=$
$\displaystyle R(\nabla u,\partial_{1},\partial_{1},\nabla u)+k(\nabla
u,\nabla u)k(\partial_{2},\partial_{2})-k^{2}(\nabla u,\partial_{2})$ (85)
$\displaystyle=$ $\displaystyle-|\nabla u|^{4}F_{x_{2}x_{2}},$ (86)
$\displaystyle\tilde{R}(\nabla u,\partial_{1},\partial_{2},\nabla u)=$
$\displaystyle-|\nabla u|^{4}F_{x_{1}x_{2}}.$

According to Theorem 4.2 in [10], we have $|\nabla u|=1+O_{1}(|x|^{-2\tau})$.
Combining this with the asymptotics for $g$ and $k$ in (1), we obtain
$F_{x_{1}x_{1}}=O(|x|^{-\tau-2})$ and $F_{x_{2}x_{2}}=O(|x|^{-\tau-2})$.
Therefore, we can follow the proof of Lemma 3.5 to conclude that $F$ is a
linear function with respect to $x_{1}$, $x_{2}$. Thus, $\tilde{g}$ is flat
which finishes the proof.

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