A Spinorial proof of the positivity of Quasi-Local mass

Puskar Mondala,b and Shing-Tung Yaua,b,c

a Center of Mathematical Sciences and Applications, Harvard University,
Cambridge, Massachusetts 02138, USA
b Department of Mathematics, Harvard University, Cambridge, Massachusetts
02138, USA
c Yau Mathematical Sciences Centre, Tsinghua University
30 Shuangqing Rd, Beijing 100190, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We provide spinor-based proof of the positivity of the Wang-Yau [13] quasi-
local mass. More precisely we prove that the gravitational mass bounded by a
spacelike topological $2-$sphere is non-negative in a generic spacetime
verifying dominant energy condition and vanishes only if the surface is
embedded in the Minkowski space. This proof is purely quasi-local in nature
and in particular does not rely on Bartanik’s gluing and asymptotic extension
construction [23] and subsequent application of the Schoen-Yau [1, 2] positive
mass theorem.

###### Contents

  1. 1 Introduction
  2. 2 Extrinsic Spin Geometry
  3. 3 Dirac equation and spin inequality
    1. 3.1 Solvability of the Dirac equation
    2. 3.2 An important spin inequality
  4. 4 Wang-Yau Quasi-local mass and its positivity
    1. 4.1 Definition of Wang-Yau Quasi-local Energy
    2. 4.2 Proof of the main theorem
  5. 5 Conclusions
  6. 6 Acknowledgement

##  1 Introduction

In the framework of general relativity, the pure gravitational notion of local
mass/energy density is absent due to the equivalence principle. In simpler
terms, one can always choose a geodesic normal coordinate chart to make the
spacetime metric trivial (Minkowski) at a point. Instead one may ask a quasi-
local definition of energy. Penrose [3] listed a set of major open problems in
mathematical general relativity in 1982 that contained the following problem:
find a quasi-local definition of energy-momentum in the context of pure
gravity. To be more precise, one asks to evaluate the energy content in a
spacelike region bounded by a topological $2-$sphere. Some desirable physical
properties of this definition of energy should include non-negativity under
the assumption of an appropriate energy condition on the spacetime and
rigidity i.e., it should vanish for a topological $2-$sphere embedded in the
Minkowski space. In addition, it should encode the information about pure
gravitational energy (Weyl curvature effect) as well as the stress-energy
tensor of any source fields present in the spacetime. Based on a Hamilton-
Jacobi analysis, Brown-York [8, 9] and Liu-Yau [10, 11] defined a quasi-local
mass by isometrically embedding the $2-$surface into the reference Euclidean
$3-$ space and comparing the extrinsic geometry (the formulation relied on the
embedding theorem of Pogorelov [32] i.e., the topological $2-$sphere needed to
posses everywhere non-negative sectional curvature). However, [12] discovered
surfaces in Minkowski space that do have strictly positive Brown-York and Liu-
Yau mass. It appeared that these formulations lacked a prescription of
momentum information. This led Wang and Yau [13, 14] to define the most
consistent notion of the quasi-local mass associated with a space-like
topological $2-$sphere (we should mention that the second author together with
Alae and Khuri recently provided a new definition of quasi-local mass bounded
by a class of surfaces verifying certain topological conditions in [15], we
should also mention that very recently John Lott [16] defined a new notion of
quasi-local mass using spinor method). It involves isometric embedding of the
topological $2-$sphere bounding a space-like domain in the physical spacetime
satisfying dominant energy condition (energy can not flow into a past light
cone of an arbitrary point in spacetime; essentially finite propagation speed)
into the Minkowski spacetime instead of Euclidean $3-$space. This formulation
relies on a weaker condition on the sectional curvature of the $2-$ surface of
interest and solvability of Jang’s equation [17] with prescribed Dirichlet
boundary data. The Wang-Yau quasi-local mass is then defined as the infimum of
the Wang-Yau quasi-local energy among all physical observers that is found by
solving an optimal isometric embedding equation. This Wang-Yau quasi-local
mass possesses several good properties that are desired on physical ground.
This mass is strictly positive for $2-$surfaces bounding a space-like domain
in a curved spacetime that satisfies the dominant energy condition and it
identically vanishes for any such $2-$surface in Minkowski spacetime. In
addition, it coincides with the ADM mass at space-like infinity [18] and
Bondi-mass at null infinity [19] and reproduces the time component of the Bel-
Robinson tensor (a pure gravitational entity) together with the matter stress-
energy at the small sphere limit, that is when the $2-$sphere of interest is
evolved by the flow of its null geodesic generators and the vertex of the
associated null cone is approached [20]. In addition, explicit conservation
laws were also discovered at the asymptotic infinity [21].

One of the important questions that arise in the context of quasi-local mass
is the proof of its non-negativity and rigidity properties. [22] provided a
proof of the positivity of the Brown-York mass under the assumption of the
non-negativity of the scalar curvature of the bulk $\Omega$ with strictly
convex boundary $\Sigma:=\partial\Omega$. They analyzed a non-linear parabolic
equation that arises due to the use of Bartnik’s gluing and extension
construction [23]. This entails suitable infinite asymptotically flat
extension of $\Omega$ with non-negative scalar curvature by deforming the
exterior of the embedding of the boundary $\Sigma\subset\mathbb{R}^{3}$ of
$\Omega$ to have the same mean curvature as $\Sigma$ in $\Omega$ and to glue
it to the manifold $\Omega$ along its boundary $\Sigma$. Through this
construction, they extend $\Sigma$ along the radial direction to the spacelike
infinity and apply the Schoen-Yau positive mass theorem [1, 2] and non-
increasing property of the Brown-York mass as the radius of the boundary
$\Sigma$ is increased. Later Liu-Yau [10, 11] and Wang-Yau [13] used this
construction in more general settings to prove the non-negativity of Liu-Yau
and Wang-Yau mass. One natural question that arises is the following: can one
prove the non-negativity of the quasi-local mass by using data on the compact
hypersurface $\Omega$ and its boundary $\Sigma$ in the spacetime without the
extension construction of [22] and use of positive mass theorem. Richard
Schoen [24] suggested the following problem find a quasi-local proof of
nonnegativity for the Brown-York and related quasi-local masses. Very
recently, Montiel [25] provided a positive solution to this problem for the
Brown-York mass associated with a surface $\Sigma$ under the assumption that
the bulk $\Omega$ (such that $\partial\Omega=\Sigma)$ has non-negative scalar
curvature and the boundary $\Sigma$ has strictly positive mean curvature using
Spinor method. Here we give spinorial proof of the non-negativity of the Wang-
Yau quasi-local mass assuming the dominant energy condition on the physical
spacetime. As a special case, the non-negativity of the Liu-Yau mass follows.
This essentially completes the quasi-local proof of the non-negativity of the
quasi-local masses modulo the new mass defined by [15]. Below we describe the
main theorem and idea of the proof. For this, we need to introduce the
following basic notations.

Let us assume that the mean curvature vector $\mathbf{H}$ of $\Sigma$ is
space-like. Let $\mathbf{J}$ be the reflection of $\mathbf{H}$ through the
future outgoing light cone in the normal bundle of $\Sigma$. The data that
Wang and Yau use to define the quasi-local energy is the triple
($\sigma,|\mathbf{H}|_{\hat{g}},\alpha_{\mathbf{H}}$) on $\Sigma$, where
$\sigma$ is the induced metric on $\Sigma$ by the Lorentzian metric $\hat{g}$
on the physical spacetime $N$, $|\mathbf{H}|_{\hat{g}}$ is the Lorentzian norm
of $\mathbf{H}$, and $\alpha_{\mathbf{H}}$ is the connexion $1-$form of the
normal bundle with respect to the mean curvature vector $\mathbf{H}$ and is
defined as follows

$\displaystyle\alpha_{\mathbf{H}}(X):=\hat{g}(\nabla[\hat{g}]_{X}\frac{\mathbf{J}}{|\mathbf{H}|},\frac{\mathbf{H}}{|\mathbf{H}|}).$
(1.1)

Choose a basis pair ($e_{3},e_{4}$) for the normal bundle of $\Sigma$ in the
spacetime that satisfy $\hat{g}(e_{3},e_{3})=1,\hat{g}(e_{4},e_{4})=-1$, and
$\hat{g}(e_{3},e_{4})=0$. Now embed the $2-$surface $\Sigma$ isometrically
into the Minkowski space with its usual metric $\eta$ i.e., the embedding map
$X:x^{a}\mapsto X^{\mu}(x^{a})$ satisfies $\sigma(\frac{\partial}{\partial
x^{a}},\frac{\partial}{\partial x^{b}})=\eta(\frac{\partial X}{\partial
x^{a}},\frac{\partial X}{\partial x^{b}})$, where $\\{x^{a}\\}_{a=1}^{2}$ are
the coordinates on $\Sigma$. Now identify a basis pair ($e_{30},e_{40}$) in
the normal bundle of $X(\Sigma)$ in the Minkowski space that satisfies the
exact similar property as $(e_{3},e_{4})$. In addition the time-like unit
vector $e_{4}$ is chosen to be future directed i.e.,
$\hat{g}(e_{4},\partial_{t})<0$. Let $\tau:=-\langle
X,\partial_{t}\rangle_{\eta}$, a function on $\Sigma$ be the time function of
the embedding $X$. The Wang-Yau quasi-local energy is defined as follows

$\displaystyle 8\pi
M^{wy}:=\inf_{\tau}\left(\underbrace{\int_{\Sigma_{t}}\left(-\sqrt{1+|\nabla\tau|^{2}_{\sigma}}\langle\mathbf{H}_{0},e_{30}\rangle-\langle\nabla[\eta]_{\nabla\tau}e_{30},e_{40}\rangle\right)\mu_{\Sigma}}_{I:=Contribution~{}from~{}the~{}Minkowski~{}space}\right.$
(1.2)
$\displaystyle\left.-\underbrace{\int_{\Sigma_{t}}\left(-\sqrt{1+|\nabla\tau|^{2}_{\sigma}}\langle\mathbf{H},e_{3}\rangle-\langle\nabla[\hat{g}]_{\nabla\tau}e_{3},e_{4}\rangle\right)\mu_{\Sigma}}_{II:=contribution~{}from~{}the~{}physical~{}spacetime}\right).$

Technically speaking, there is still a boost-redundancy left in the normal
bundle of $\Sigma$ in $N$. [13] minimizes the physical space contribution $II$
over all possible boost-angles ($II$ is a convex function of the boost-angle
in the normal bundle of $\Sigma$). However, we will prove the non-negativity
of $I-II$ for all admissible $\tau$ and therefore $M^{wy}$ will be non-
negative since subtracting minimum of $II$ over boost angles only increases
the value of the mass.

######  Theorem 1.1 (Main Theorem)

Let $(\Sigma,\sigma)$ be a spacelike topological $2-$sphere that bounds a
spacelike topological ball $\Omega$ in the physical spacetime $N$ verifying
dominant energy condition. Moreover, assume $\Sigma$ is not an apparent
horizon and its mean curvature vector is spacelike. Let $\tau$ be a function
on $\Sigma$ that verifies the following admissibility criterion

$\displaystyle K+(1+|\nabla\tau|^{2}_{\sigma})^{-1}\det(\nabla^{2}\tau)>0$
(1.3)

where $K$ is the Gauss curvature of $(\Sigma,\sigma)$ and $\nabla^{2}\tau$ is
the hessian of $\tau$ with respect to the Levi-Civita connection
$\nabla[\sigma]$ on $(\Sigma,\sigma)$. Then the Wang-Yau quasi-local energy
$M^{wy}$ is non-negative and it vanishes if the spacetime $N$ is Minkowski
along $\Sigma$.

The main idea behind the proof is solving the Dirac equation on $\Omega$ with
Atyiah-Patodi-Singer (APS) boundary condition [26, 27, 28] on $\Sigma$ along
with the application of an important spin inequality 3.1. Under the assumption
of the dominant energy condition on the spacetime and $\Sigma$ not being an
apparent horizon, we prove the solvability of the Dirac equation with APS
boundary condition. This in turn yields the vital spin inequality 3.1. In the
next step, we reduce the expression for the Wang-Yau quasi-local energy to an
appropriate form so that the application of the spin inequality 3.1 becomes
possible. This involves solving Jang’s equation on the physical spacetime. We
utilize the result established in the original article by Wang-Yau [13]
regarding the solvability of Jang’s equation. The dominant energy condition on
the spacetime $N$ yields a curvature condition on the bulk spacelike
topological ball $\Omega$. In addition, the assumption that $\Sigma$ is not an
apparent horizon yields the appropriate condition on the mean curvature of
$\Sigma$. Putting together all these ingredients along with the solution of
the Dirac equation with the APS boundary condition verifying the spin
inequality 3.1 completes the proof. The rigidity of the Wang-Yau quasi-local
mass essentially follows from the fact that it becomes zero when $\Omega$
admits a parallel spinor and hence Ricci flat. Ricci flatness in $3$
dimensions implies flatness and due to being simply connected, a domain in
$\mathbb{R}^{3}$. This together with the dominant energy condition yields the
vanishing momentum property of $\Omega$ which through the structure equation
of the embedding $\Omega\hookrightarrow N$ implies $N$ is Minkowski along
$\Omega$.

##  2 Extrinsic Spin Geometry

Let $(\Omega,g)$ be a compact connected oriented 3 dimensional $C^{\infty}$
Riemannian manifold with boundary $\partial\Omega=\Sigma$. Let
$\widehat{\nabla}$ be the metric compatible connexion on $(\Omega,g)$ and
$\nabla$ is the induced connexion on $\Sigma$. We denote the spin bundle on
$\Omega$ by $S^{\Omega}$ while that of $\Sigma$ is denoted by $S^{\Sigma}$. In
our case, $\Omega$ is a topological ball, and naturally, $\Sigma$ is a
topological $2-$sphere. On the spin bundle $S^{\Omega}$ there is a natural
Hermitian inner product $\langle\cdot,\cdot\rangle$ that is compatible with
the connexion $\widehat{\nabla}$ (we denote the Levi-Civita connexion and its
spin connexion by the same symbol) and Clifford multiplication
$c:Cl(\Omega)\to\text{End}(S^{\Omega})$. We denote a spinor a section of the
spin bundle by $\psi\in\Gamma(S^{\Omega})$. Naturally, its restriction to
$\Sigma$ is a section of $S^{\Sigma}$. Let us consider $\\{e_{i}\\}_{i=1}^{3}$
as an orthonormal frame on $\Sigma$ where $e_{3}=\nu$ is the interior pointing
normal to the boundary $\Sigma$. The following relation holds $\forall
X,Y\in\Gamma(T\Omega)$

$\displaystyle
X\langle\psi_{1},\psi_{2}\rangle=\langle\widehat{\nabla}_{X}\psi_{1},\psi_{2}\rangle+\langle\psi_{1},\widehat{\nabla}_{X}\psi_{2}\rangle$
(2.1)
$\displaystyle\widehat{\nabla}_{X}(c(Y)\psi)=c(\widehat{\nabla}_{X}Y)\psi+c(Y)\widehat{\nabla}_{X}Y$
(2.2) $\displaystyle\langle
c(X)\psi_{1},\psi_{2}\rangle=-\langle\psi_{1},c(X)\psi_{2}\rangle$ (2.3)
$\displaystyle\langle
c(X)\psi_{1},c(X)\psi_{2}\rangle=g(X,X)\langle\psi_{1},\psi_{2}\rangle.$ (2.4)

These are the intrinsic identities that are verified by the geometric entities
on $\Omega$. The boundary map $i:\Sigma\hookrightarrow\Omega$, induces a
connexion on $\Sigma$ that verifies

$\displaystyle\widehat{\nabla}_{X}Y=\nabla_{X}Y+\mathcal{H}(X,Y)\nu,$ (2.5)

where $X,Y\in\Gamma(T\Sigma)$ and $\mathcal{H}$ is the second fundamental form
of $\Sigma$ while viewed as an embedded surface in $\Omega$. Now this split
extends to the spinor bundles. One can consider the restricted bundle
$S^{\Omega}|_{\Sigma}$ which is isomorphic to $S^{\Sigma}$ since dimension of
$\Sigma$ is even. The spin connexion splits as follows

$\displaystyle\widehat{\nabla}_{X}\psi=\nabla_{X}\psi+\frac{1}{2}c(\mathcal{H}(X,\cdot))c(\nu)\psi$
(2.6)

for $X\in\Gamma(T\Sigma)$. With this split, we can write the hypersurface
Dirac operator in terms of the bulk Dirac operator and the second fundamental
form. The hypersurface Dirac operator is defined as follows

$\displaystyle
D\psi=\sum_{i=1}^{2}c(e_{i})c(\nu)\nabla_{e_{i}}\psi=\sum_{i=1}^{2}c(e_{i})c(\nu)\left(\widehat{\nabla}_{e_{i}}\psi-\frac{1}{2}c(\mathcal{H}(e_{i},\cdot))c(\nu)\psi\right)$
(2.7)
$\displaystyle=\sum_{i=1}^{2}c(e_{i})c(\nu)\widehat{\nabla}_{e_{i}}\psi-\frac{1}{2}\sum_{i=1}^{2}c(e_{i})c(\nu)c(\mathcal{H}(e_{i},\cdot))c(\nu)\psi$

Now observe the following $(\mathcal{H}(e_{i},\cdot)\in T^{*}\Sigma$ and
therefore $g(\mathcal{H}(e_{i},\cdot),\nu)=0$ and therefore

$\displaystyle
D\psi=\sum_{i=1}^{2}c(e_{i})c(\nu)\widehat{\nabla}_{e_{i}}\psi+\frac{1}{2}\sum_{i=1}^{2}c(e_{i})c(\nu)c(\nu)c(\mathcal{H}(e_{i},\cdot))\psi$
(2.8)
$\displaystyle=\sum_{i=1}^{2}c(e_{i})c(\nu)\widehat{\nabla}_{e_{i}}\psi+\frac{1}{2}H\psi=-\sum_{i=1}^{2}c(\nu)c(e_{i})\widehat{\nabla}_{e_{i}}\psi+\frac{1}{2}H\psi$
$\displaystyle=-c(\nu)\sum_{i=1}^{3}c(e_{i})\widehat{\nabla}_{e_{i}}\psi+c(\nu)c(\nu)\widehat{\nabla}_{\nu}\psi+\frac{1}{2}H\psi$
$\displaystyle=-c(\nu)\widehat{D}\psi-\widehat{\nabla}_{\nu}\psi+\frac{1}{2}H\psi,$

where $H=\text{tr}\left[\mathcal{H}\right]$. Therefore we have the following
relation between the hypersurface Dircal operator and the bulk Dirac operator

$\displaystyle
D\psi=-c(\nu)\widehat{D}\psi-\widehat{\nabla}_{\nu}\psi+\frac{1}{2}H\psi.$
(2.9)

Now we derive the following identity

###### Proposition 2.1

The following Lichnerowicz type identity is verified by $\psi$

$\displaystyle\int_{\Sigma}\left(\langle
D\psi,\psi\rangle-\frac{1}{2}H|\psi|^{2}\right)\mu_{\Sigma}=\frac{1}{4}\int_{\Omega}R^{\Omega}|\psi|^{2}\mu_{\Omega}-\int_{\Omega}|\widehat{D}\psi|^{2}\mu_{\Omega}+\int_{\Omega}|\widehat{\nabla}\psi|^{2}\mu_{\Omega}$
(2.10)

_Proof._ First, recall the Lichnerowicz identity on $\Omega$

$\displaystyle\widehat{D}^{2}\psi=-\widehat{\nabla}^{2}\psi+\frac{1}{4}R^{\Omega}\psi,$
(2.11)

for a spinor $\psi\in\Gamma(S^{\Omega})$. Let us consider an orthonormal frame
$\\{e_{i}\\}_{i=1}^{3}$ on $\Omega$ with $e_{3}$ being the unit inside
pointing normal $\nu$ of the boundary surface $\partial\Omega=\Sigma$. Now we
evaluate the following using the compatibility property of the Hermitian inner
product $\langle\cdot,\cdot\rangle$ on the spin bundle $S^{\Omega}$

$\displaystyle\widehat{\nabla}_{i}\langle
c(e_{i})\widehat{D}_{i}\psi,\psi\rangle+\widehat{\nabla}_{i}\langle\widehat{\nabla}_{i}\psi,\psi\rangle=\langle
c(e_{i})\widehat{\nabla}_{i}\widehat{D}_{i}\psi,\psi\rangle+\langle
c(e_{i})\widehat{D}_{i}\psi,\widehat{\nabla}_{i}\psi\rangle+\langle\widehat{\nabla}^{2}\psi,\psi\rangle+\langle\widehat{\nabla}_{i}\psi,\widehat{\nabla}_{i}\psi\rangle$
$\displaystyle=\langle\widehat{D}^{2}\psi,\psi\rangle-\langle\widehat{D}_{i}\psi,c(e_{i})\widehat{\nabla}_{i}\psi\rangle+\langle\widehat{\nabla}^{2}\psi,\psi\rangle+\langle\widehat{\nabla}_{i}\psi,\widehat{\nabla}_{i}\psi\rangle$
$\displaystyle=\langle\widehat{D}^{2}\psi,\psi\rangle-\langle\widehat{D}_{i}\psi,\widehat{D}_{i}\psi\rangle+\langle\widehat{\nabla}^{2}\psi,\psi\rangle+\langle\widehat{\nabla}_{i}\psi,\widehat{\nabla}_{i}\psi\rangle,$

where we have used the property 2.1 and 2.3 along with the fact that
$\widehat{\nabla}_{i}(c(e_{i})\widehat{D}_{i}\psi)=c(e_{i})\widehat{\nabla}_{i}\widehat{D}_{i}\psi$.
Now we integrate over $\Omega$ to yield

$\displaystyle\int_{\Omega}\left(\widehat{\nabla}_{i}\langle
c(e_{i})\widehat{D}_{i}\psi,\psi\rangle+\widehat{\nabla}_{i}\langle\widehat{\nabla}_{i}\psi,\psi\rangle\right)\mu_{\Omega}$
(2.12)
$\displaystyle=\int_{\Omega}\left(\langle\widehat{D}^{2}\psi,\psi\rangle-\langle\widehat{D}_{i}\psi,\widehat{D}_{i}\psi\rangle+\langle\widehat{\nabla}^{2}\psi,\psi\rangle+\langle\widehat{\nabla}_{i}\psi,\widehat{\nabla}_{i}\psi\rangle\right)\mu_{\Omega}$
$\displaystyle\implies-\int_{\Sigma}\left(\langle
c(\nu)\widehat{D}_{\nu}\psi,\psi\rangle+\langle\widehat{\nabla}_{\nu}\psi,\psi\rangle\right)\mu_{\Sigma}=\int_{\Omega}\left(\frac{1}{4}R^{\Omega}|\psi|^{2}-|\widehat{D}\psi|^{2}+|\widehat{\nabla}\psi|^{2}\right)\mu_{\Omega}$
$\displaystyle\implies\int_{\Sigma}\left(\langle
D\psi,\psi\rangle-\frac{1}{2}H|\psi|^{2}\right)\mu_{\Sigma}=\int_{\Omega}\left(\frac{1}{4}R^{\Omega}|\psi|^{2}-|\widehat{D}\psi|^{2}+|\widehat{\nabla}\psi|^{2}\right)\mu_{\Omega},$

where we have used the Lichnerowicz idenity 2.11 and the relation 2.9 between
the Dirac operators on the boundary $\Sigma$ and the bulk $\Omega$. This
concludes the proof of the spin identity that we will frequently use in this
article. $\square$

##  3 Dirac equation and spin inequality

The Lichnerowicz type identity obtained in proposition 2.1 is one of the main
ingredients for the proof of the theorem 1.1. Notice that in the identity
2.10, the right-hand side contains a non-positive definite term involving the
action of the Dirac operator on the spinor $\psi$ in the bulk $\Omega$. To get
rid of this term, we want to impose that $\widehat{D}\psi=0$ on $\Omega$ with
a specified boundary condition on $\Sigma:=\partial\Omega$. This issue is non-
trivial since in dimensions greater than or equal to 2, the Dirac operator on
a compact manifold with a boundary generally has an infinite-dimensional
kernel. A simple example would be solving the Dirac equation on a disk in
$\mathbb{C}$. This would essentially be finding the holomorphic and anti-
holomorphic functions that form infinite dimensional vector space. To fix this
problem, one imposes a so-called elliptic boundary condition for the spinor on
the boundary of the compact manifold $\Omega$. The elliptic boundary condition
for the boundary value problem associated with the bulk Dirac operator
$\widehat{D}$ is a pseudo-differential operator $P:L^{2}(S^{\Sigma})\to
L^{2}(V)$, where $V$ is a hermitian vector bundle on $\Sigma$ such that the
boundary value problem

$\displaystyle\widehat{D}\psi=\Psi~{}\text{on}~{}\Omega$ (3.1) $\displaystyle
P(\psi|_{\Sigma})=\alpha~{}\text{on}~{}\Sigma=\partial\Omega$ (3.2)

has a unique solution given smooth data $\Psi$ and $\alpha$ modulo finite
dimensional kernel. Here we choose to work with the Atyiah-Patodi-Singer (APS)
boundary condition [26, 27, 28] that was first utilized to prove the index
theorem for compact manifolds with boundary. Here the choice of the pseudo-
differential operator $P_{\lambda}$ is simply the $L^{2}$ orthogonal
projection onto the subspace spanned by eigenvectors of the boundary Dirac
operator $D$ whose eigenvalues are not smaller than $\lambda$ (note that
$\Sigma$ is closed and therefore the boundary operator $D$ is self-adjoint).
[26, 27, 28] showed that the operator $P_{\lambda}$ is a zero order pseudo-
differential operator. We invoke the following well-known proposition from
[26, 27, 28, 29] regarding the ellipticity of the APS boundary condition.

###### Proposition 3.1

[26, 27, 28, 29] Let $(\Omega,g)$ be a compact connected oriented Riemannian
$3-$ manifold with smooth boundary $\Sigma:=\partial\Omega$. Then $APS$
boundary condition is elliptic and moreover, for $\lambda=0$, the spectrum of
the Dirac operator is unbounded discrete.

First, we note the following properties of the Sobolev spaces on $\Omega$ and
$\Sigma$. A well known fact is that all functions belonging to
$H^{s}(\Omega),~{}s\in\mathbb{N}$ have traces on the smooth boundary
$\Sigma=\partial\Omega$ that belong to $H^{s-\frac{1}{2}}(\Sigma)$. With a bit
of technicality this passes to the case of spinors due to the compatibility of
the spin connexion with the hermitian inner product
$\langle\cdot,\cdot\rangle$ (in fact this generalizes to sections of smooth
vector bundles where fiber metrics are compatible with the bundle connexions).
More precisely the trace operator $\text{Tr}:H^{s}(\Omega)\to
H^{s-\frac{1}{2}}(\Sigma)$ is bounded and surjective. The following
proposition completes the discussion of the solvability of the Dirac equation
on $\Omega$ with the APS boundary condition and the regularity of the
solutions.

###### Proposition 3.2

Let $\Omega$ be a connected oriented $3-$Riemannian manifold with connected
smooth boundary $\Sigma$ and $X$ be a smooth vector field on $\Omega$. Assume
that the scalar curvature $R^{\Omega}$ of $\Omega$ verifies $R^{\Omega}\geq
2|X|^{2}-2\text{div}X$ and the mean curvature $H$ of $\Sigma$ while viewed as
an embedded surface in $\Omega$ verifies $H-\langle X,\nu\rangle>0$. Then the
inhomogeneous Dirac equation

$\displaystyle\widehat{D}\psi=\Psi~{}\text{on}~{}\Omega~{}$ (3.3)
$\displaystyle\psi_{\geq\lambda}=\alpha_{\geq\lambda}~{}\text{on}~{}\Sigma$
(3.4)

with $\Psi\in H^{s-1}(S^{\Omega})$ and $\alpha_{\geq\lambda}\in
H^{s-\frac{1}{2}}(S^{\Sigma}),s\geq 2$ has a unique solution
$\psi_{\lambda}\in H^{s}(S^{\Omega})\cap H^{s-\frac{1}{2}}(S^{\Sigma})$ for
every $\lambda\leq 0$. We have an estimate of the type

$\displaystyle||\psi_{\lambda}||_{H^{s}(\Omega)}\lesssim||\Psi||_{H^{s-1}(\Omega)}+||\alpha_{\geq\lambda}||_{H^{s-\frac{1}{2}}(\Sigma)},~{}s\geq
2,$ (3.5)

where $A\lesssim B$ denotes $A\leq CB$, for a constant $C$ dependent on
background geometry of $\Omega$ and $\Sigma$.

The previous proposition works with $s\geq 1$. Here we choose $s\geq 2$ to
ensure continuity which will be important while obtaining the spin inequality
3.1. We prove this by showing the isomorphism property of the Dirac operator
between appropriate Sobolev spaces on $\Omega$ with the APS boundary condition
on $\Sigma:=\partial\Omega$.

###  3.1 Solvability of the Dirac equation

Now we prove a vital proposition that essentially guarantees the solvability
of the Dirac equation $\widehat{D}\psi=0$ on $\Omega$ with prescribed boundary
condition on $\Sigma$ i.e., the proof of the proposition 3.2.

###### Proposition 3.3

Let $X$ be a smooth vector field on $\Omega$ and the Scalar curvature
$R^{\Omega}$ of $\Omega$ verifies the following point-wise bound

$\displaystyle R^{\Omega}\geq 2|X|^{2}-2\text{div}X,$ (3.6)

and the mean curvature $H$ of $\Sigma=\partial\Omega$ while viewed as an
embedded surface in $\Omega$ verifies

$\displaystyle H-\langle X,\nu\rangle>0,$ (3.7)

where $\nu$ is the unit spacelike unit outward directed normal to $\Sigma$.
Then the homogeneous boundary value problem

$\displaystyle\widehat{D}\psi=0~{}\text{on}~{}\Omega,~{}\psi_{\geq\lambda}=0~{}\text{on}~{}\Sigma:=\partial\Omega$
(3.8)

has no non-trivial smooth solution i.e., $Ker(\widehat{D})=0$.

_Proof._ First note that $\psi_{\lambda}=\psi_{\geq\lambda}+\psi_{<\lambda}$
due to $L^{2}$ orthogonality of the projection. We want to show that
$\psi_{<\lambda}=0$ first. Use the identity 2.10 with $\psi_{\lambda}$ to
yield

$\displaystyle\int_{\Sigma}\left(\langle
D\psi_{\lambda},\psi_{\lambda}\rangle-\frac{1}{2}H|\psi_{\lambda}|^{2}\right)\mu_{\Sigma}=\frac{1}{4}\int_{\Omega}R^{\Omega}|\psi_{\lambda}|^{2}\mu_{\Omega}+\int_{\Omega}|\widehat{\nabla}\psi_{\lambda}|^{2}\mu_{\Omega}.$
(3.9)

Now addition of $\frac{1}{2}\int_{\Sigma}\langle
X,\nu\rangle|\psi_{\lambda}|^{2}\mu_{\Sigma}$ to the both side along with the
use of the integration by parts identity

$\displaystyle\int_{\Sigma}\langle
X,\nu\rangle|\psi_{\lambda}|^{2}=\int_{\Omega}\left(\text{div}X|\psi_{\lambda}|^{2}+X(|\psi_{\lambda}|^{2})\right)\mu_{\Omega}$
(3.10)

yields

$\displaystyle\int_{\Sigma}\left(\langle
D\psi_{\lambda},\psi_{\lambda}\rangle-\frac{1}{2}(H-\langle
X,\nu\rangle)|\psi_{\lambda}|^{2}\right)\mu_{\Sigma}=\frac{1}{4}\int_{\Omega}(R^{\Omega}+2\text{div}X)|\psi_{\lambda}|^{2}\mu_{\Omega}+\int_{\Omega}|(|\widehat{\nabla}\psi_{\lambda}|^{2}+\frac{1}{2}X(|\psi_{\lambda}|^{2}))\mu_{\Sigma}$
$\displaystyle\geq\frac{1}{4}\int_{\Omega}(R^{\Omega}+2\text{div}X-2|X|^{2})|\psi_{\lambda}|^{2}\mu_{\Omega}+\frac{1}{2}\int_{\Omega}|\widehat{\nabla}\psi|^{2}\mu_{\Omega}\geq
0$

due to the fact that
$|\widehat{\nabla}\psi_{\lambda}|^{2}+\frac{1}{2}X(|\psi_{\lambda}|^{2})=|\widehat{\nabla}\psi_{\lambda}|^{2}+\frac{1}{2}(\langle\widehat{\nabla}_{X}\psi_{\lambda},\psi_{\lambda}\rangle+\langle\psi_{\lambda},\widehat{\nabla}_{X}\psi_{\lambda}\rangle)\geq\frac{1}{2}|\widehat{\nabla}\psi_{\lambda}|^{2}-\frac{1}{2}|X|^{2}|\psi_{\lambda}|^{2}$.
Therefore we have the following

$\displaystyle\int_{\Sigma}\left(\langle
D\psi_{\lambda},\psi_{\lambda}\rangle-\frac{1}{2}(H-\langle
X,\nu\rangle)|\psi_{\lambda}|^{2}\right)\mu_{\Sigma}\geq 0.$ (3.11)

But on the boundary $\Sigma$, $\psi_{\geq\lambda}=0$. Set $\lambda\leq 0$ to
yield

$\displaystyle-\left(\int_{\Sigma}\sum_{i}\lambda^{2}_{i}|\psi_{i}|^{2}+\frac{1}{2}(H-\langle
X,\nu\rangle)|\psi_{<0}|^{2}\right)\mu_{\Sigma}\geq 0$ (3.12)

which yields $\psi_{<0}=0$ since $H-\langle X,\nu\rangle>0$. Therefore
$\psi_{\lambda}=0$ on $\Sigma$. Now since $\widehat{D}\psi_{\lambda}=0$ on
$\Omega$, using Lichnerowicz formula
$\widehat{D}^{2}=-\widehat{\nabla}^{2}+\frac{1}{4}R^{\Omega}$ we obtain

$\displaystyle\int_{\Omega}|\widehat{\nabla}\psi_{\lambda}|^{2}\mu_{\Omega}+\frac{1}{4}R^{\Omega}|\psi_{\lambda}|^{2}\mu_{\Omega}=0$
(3.13)

due to $\psi_{\lambda}=0$ on $\Sigma$. Now we add
$\frac{1}{2}\int_{\Sigma}\langle X,\nu\rangle|\psi_{\lambda}|^{2}$. Since
$\psi_{\lambda}=0$ on $\Sigma$, it does not alter anything. This yields
through integration by parts

$\displaystyle\int_{\Omega}\left(|\widehat{\nabla}\psi_{\lambda}|^{2}\mu_{\Omega}+\frac{1}{4}R^{\Omega}|\psi_{\lambda}|^{2}+\frac{1}{2}\text{div}X|\psi|^{2}+\frac{1}{2}X(|\psi_{\lambda}|^{2})\right)=0$
(3.14)

or

$\displaystyle\frac{1}{4}\int_{\Omega}(R^{\Omega}+2\text{div}X-2|X|^{2})|\psi_{\lambda}|^{2}\mu_{\Omega}+\frac{1}{2}\int_{\Omega}|\widehat{\nabla}\psi_{\lambda}|^{2}\mu_{\Sigma}\leq
0$ (3.15)

yielding $\psi_{\lambda}=0$ on $\Omega$ since
$R^{\Omega}+2\text{div}X-2|X|^{2}\geq 0$. This follows from the fact that
$|\psi_{\lambda}|^{2}$ is either identically zero or constant on $\Omega$ with
vanishing boundary value. Therefore it must have zero length throughout
$\Omega$ due to continuity. Also, due to an argument of [30], the Hausdorff
dimension of the zero sets of the solution of the Dirac equation on $\Omega$
is at most $1-$ or the solution is identically 0. But $\psi_{\lambda}=0$ on
$\Sigma:=\partial\Omega$. So $\psi_{\lambda}$ must be identically $0$ on
$\Omega$. This concludes the proof of the proposition. $\square$

Following the previous proposition and the triviality of the co-kernel (i.e.,
solution of the problem $D\psi=0$ on $\Omega$ and $\psi_{>\lambda}=0$ on
$\Sigma$) under the mean curvature condition $H-\langle X,\nu\rangle>0$, we
conclude that there is a unique solution to the Dirac equation on $\Omega$
with the prescribed APS boundary condition if the spinors are chosen to lie in
appropriate function spaces. With the help of this proposition, we will obtain
a crucial inequality in the next section that will be vital to proving the
positivity of the Wang-Yau quasi-local mass.

###  3.2 An important spin inequality

Now consider a vector field $X$ that verifies $R^{\Omega}\geq
2|X|^{2}-2\text{div}X$. We prove the following theorem.

###### Theorem 3.1

Let $\Omega$ be a spacelike topological ball in the spacetime $N$. Let $X$ be
a vector field on $\Omega$ that verifies the following point-wise inequality

$\displaystyle R^{\Omega}\geq 2|X|^{2}-2\text{div}X,$ (3.16)

where $R^{\Omega}$ is the scalar curvature of $\Omega$. Let the boundary of
$\Omega$ be $\Sigma$ with mean curvature $H$ while viewed as an embedded
surface in $\Omega$ verifying

$\displaystyle H-\langle X,\nu\rangle>0.$ (3.17)

Then for every spinor $\alpha\in H^{s}(S^{\Sigma}),~{}s\geq\frac{3}{2}$ we
have the following inequality

$\displaystyle\int_{\Sigma}\langle
D\alpha,\alpha\rangle\mu_{\Sigma}\geq\frac{1}{2}\int_{\Sigma}(H-\langle
X,\nu\rangle)|\alpha|^{2}\mu_{\Sigma}.$ (3.18)

The equality holds only if $X=0$, and $\alpha$ is the boundary condition of a
parallel spinor $\psi$ on $\Omega$ solving the Dirac equation. In such case

$\displaystyle D\alpha=\frac{1}{2}H\alpha.$ (3.19)

_Proof._ The proof relies on the proposition 2.1 and solving the Dirac
equation on $\Omega$ with the Atyiah-Singer-Patodi (APS) boundary condition.
Now suppose, we have $\widehat{D}\psi=0$, then

$\displaystyle\int_{\Sigma}\left(\langle
D\psi,\psi\rangle-\frac{1}{2}H|\psi|^{2}\right)\mu_{\Sigma}=\frac{1}{4}\int_{\Omega}R^{\Omega}|\psi|^{2}\mu_{\Omega}+\int_{\Omega}|\widehat{\nabla}\psi|^{2}\mu_{\Omega}$
(3.20)

We want to solve the equation $\widehat{D}\psi=0$ on $\Omega$ with Atyiah-
Singer boundary condition on $\Sigma$. Let us denote by $\psi_{\geq\lambda}$
the spinor with eigenvalues $\geq\lambda$. We solve the following boundary
value problem

$\displaystyle\widehat{D}\psi=0~{}\text{on}~{}\Omega$ (3.21)
$\displaystyle\psi_{\geq\lambda}=\alpha_{\geq\lambda}~{}\text{on}~{}\Sigma.$
(3.22)

In the end, we want to take the limit $\lambda\to-\infty$. First, recall the
Sobolev trace inequality

$\displaystyle||\psi_{\geq\lambda}||_{H^{s}(\Sigma)}\lesssim||\psi_{\geq\lambda}||_{H^{s+\frac{1}{2}}(\Omega)},~{}s>\frac{1}{2}$
(3.23)

assuming smooth $\Sigma$. For the point-wise convergence to hold, we need to
work in the following function space $\psi\in H^{s}(\Omega),~{}s\geq 2$ and
from trace inequality $\alpha_{\geq\lambda}\in H^{s-\frac{1}{2}}(\Sigma)$ and
the proposition 3.2 we have

$\displaystyle||\psi_{\lambda}||_{H^{s}(\Omega)}\lesssim||\alpha_{\geq\lambda}||_{H^{s-\frac{1}{2}}(\Sigma)}\lesssim||\alpha||_{H^{s-\frac{1}{2}}(\Sigma)}~{}\forall\lambda\leq
0.$ (3.24)

Due to $\alpha_{\geq\lambda}$ is $H^{s-\frac{1}{2}}(\Sigma),~{}s\geq 2$, we
have $H^{s-\frac{1}{2}}(\Sigma)\hookrightarrow C^{0}(\Sigma)$ and therefore
point-wise convergence as well i.e.,

$\displaystyle\psi|_{\Sigma}:=\lim_{\lambda\to-\infty}\psi_{\geq\lambda}|_{\Sigma}=\lim_{\lambda\to-\infty}\alpha_{\geq\lambda}=\alpha.$
(3.25)

Therefore, we have a unique solution to the boundary value problem following
proposition 3.2

$\displaystyle\widehat{D}\psi=0~{}\text{on}~{}\Omega$ (3.26)
$\displaystyle\psi|_{\partial\Omega=\Sigma}=\alpha$ (3.27)

Now we do the following manipulations to the identity 3.20. First we add
$\frac{1}{2}\int_{\Sigma}\langle X,\nu\rangle\mu_{\Sigma}|\psi|^{2}$ to the
both sides of 3.20. This yields

$\displaystyle\int_{\Sigma}\left(\langle
D\psi,\psi\rangle-\frac{1}{2}H|\psi|^{2}\right)\mu_{\Sigma}+\frac{1}{2}\int_{\Sigma}\langle
X,\nu\rangle|\psi|^{2}\mu_{\Sigma}=\frac{1}{4}\int_{\Omega}R^{\Omega}|\psi|^{2}\mu_{\Omega}+\int_{\Omega}|\widehat{\nabla}\psi|^{2}\mu_{\Omega}+\frac{1}{2}\int_{\Sigma}\langle
X,\nu\rangle|\psi|^{2}\mu_{\Sigma}.$

Now recall the following integration by parts identity

$\displaystyle\int_{\Sigma}\langle
X,\nu\rangle|\psi|^{2}=\int_{\Omega}\left(\text{div}X|\psi|^{2}+X(|\psi|^{2})\right)\mu_{\Omega}$
(3.28)

to yield

$\displaystyle\int_{\Sigma}\left(\langle
D\psi,\psi\rangle-\frac{1}{2}(H-\langle
X,\nu\rangle)|\psi|^{2}\right)\mu_{\Sigma}=\frac{1}{4}\int_{\Omega}(R^{\Omega}+2\text{div}X)|\psi|^{2}\mu_{\Omega}+\int_{\Omega}|(|\widehat{\nabla}\psi|^{2}+\frac{1}{2}X(|\psi|^{2}))\mu_{\Sigma}.$

Now note the following calculations

$\displaystyle|\widehat{\nabla}\psi|^{2}+\frac{1}{2}X(|\psi|^{2})=|\widehat{\nabla}\psi|^{2}+\frac{1}{2}(\langle\widehat{\nabla}_{X}\psi,\psi\rangle+\langle\psi,\widehat{\nabla}_{X}\psi\rangle)\geq\frac{1}{2}|\widehat{\nabla}\psi|^{2}-\frac{1}{2}|X|^{2}|\psi|^{2}$
(3.29)

and therefore we have

$\displaystyle\int_{\Sigma}\left(\langle
D\psi,\psi\rangle-\frac{1}{2}(H-\langle
X,\nu\rangle)|\psi|^{2}\right)\mu_{\Sigma}\geq\frac{1}{4}\int_{\Omega}(R^{\Omega}+2\text{div}X-2|X|^{2})|\psi|^{2}\mu_{\Omega}+\frac{1}{2}\int_{\Omega}|\widehat{\nabla}\psi|^{2}\mu_{\Omega}$
$\displaystyle\geq\frac{1}{4}\int_{\Omega}(R^{\Omega}+2\text{div}X-2|X|^{2})|\psi|^{2}\mu_{\Omega}\geq
0$

by the assumption $R^{\Omega}\geq 2|X|^{2}-2\text{div}X$. Now equality holds
only when

$\displaystyle\widehat{\nabla}\psi=0,$ (3.30) $\displaystyle
R^{\Omega}+2\text{div}X-2|X|^{2}=0.$ (3.31)

Now $\widehat{\nabla}\psi=0$ implies $\psi$ is a parallel Dirac spinor on
$\Omega$. Therefore $\Omega$ must be Ricci flat and therefore flat. Moreover,
the identity 2.9
$D\psi=-c(\nu)\widehat{D}\psi-\widehat{\nabla}_{\nu}\psi+\frac{1}{2}H\psi$
yields

$\displaystyle D\psi=\frac{1}{2}H\psi~{}\text{on}~{}\Sigma.$ (3.32)

Therefore

$\displaystyle\int_{\Sigma}\langle X,\nu\rangle|\psi|^{2}\mu_{\Sigma}=0.$
(3.33)

which after integration by parts and parallelity condition
$\widehat{\nabla}\psi=0$ implies

$\displaystyle\int_{\Omega}\text{div}X|\psi|^{2}\mu_{\Omega}=0$ (3.34)

Moreover, flatness implies $R^{\Omega}=0$ yielding

$\displaystyle\text{div}X-|X|^{2}=0$ (3.35)

and therefore

$\displaystyle\int_{\Omega}|X|^{2}|\psi|^{2}\mu_{\Omega}=0\implies X=0.$
(3.36)

This concludes the proof of the theorem. Note that the calculations 3.32-3.36
are not necessary as the saturation of 3.29 yields $X=0$. $\square$

###### Remark 1

In the case of equality in the previous theorem, $\Omega$ admits parallel
spinor and hence it is Ricci flat and hence flat in dimension three. Due to
being simply connected, $\Omega$ is a domain in $\mathbb{R}^{3}$.

##  4 Wang-Yau Quasi-local mass and its positivity

We first define the Wang-Yau quasi-local mass. The details can be found in the
original article of Wang $\&$ Yau [13]. Here we state the necessary ideas that
lead to the construction of the Wang-Yau quasi-local mass. The aim is to
define the gravitational energy contained in a spacelike region bounded by a
topological $2-$sphere in a $3+1$ dimensional physical spacetime
$(N,\widehat{g})$ verifying dominant energy condition. We assume that the
spacetime $M$ is globally hyperbolic, that is, it can be expressed as the
product $M\times\mathbb{R}$ where $M$ is a Cauchy hypersurface. We write the
spacetime metric $\widehat{g}$ in the following form

$\displaystyle\widehat{g}:=-\beta^{2}dt\otimes
dt+g_{ij}(dx^{i}+\gamma^{i}dt)\otimes(dx^{j}+\gamma^{j}dt),$ (4.1)

in a local chart $\\{t,x^{i}\\}_{i=1}^{3}$. Here $\beta$ and
$\gamma:=\gamma^{i}\frac{\partial}{\partial x^{i}}$ are the lapse function and
the shift vector field, respectively. Here we will be interested in Einstein’s
constraint equations primarily. Note that the Gauss and Codazzi equations for
the spacetime $N$ foliated by $M$ yield the constraint equations on each
$t=\text{constant}$ slice

$\displaystyle R(g)-P_{ij}P^{ij}+(\text{tr}_{g}P)^{2}=2\mu$ (4.2)
$\displaystyle\widehat{\nabla}^{j}k_{ij}-\nabla_{i}\text{tr}_{g}P=J_{i},$
(4.3)

where $\mu$ and $J:=J_{i}dx^{i}$ are the local energy and momentum density of
the matter (or radiation) sources present in the spacetime. The dominant
energy condition states $\mu\geq\sqrt{g(J,J)}$. $P_{ij}$ is the extrinsic
curvature or the second fundamental form of $M$ in $N=M\times\mathbb{R}$. Once
we are equipped with this information about the spacetime $N$ (which is
assumed to be Einsteinian spacetime i.e., the spacetime metric verifies
Einstein’s constraint and evolution equations), we can define the energy
contained within a membrane $\Sigma\subset M$ i.e., a spacelike topological
$2-$sphere. The Wang-Yau quasi-local mass precisely quantifies this energy
through a Hamilton-Jacobi analysis (see [13, 14] for the detail on how such
energy is obtained).

###  4.1 Definition of Wang-Yau Quasi-local Energy

First, we recall the following definition of a generalized mean curvature as
given in [13].

######  Definition 1 (Wang-Yau)

[13] Suppose $i:\Sigma\hookrightarrow N$ is an embedded space-like two-surface
with induced metric $\sigma$. Given a smooth function $\tau$ on $\Sigma$ and a
space-like normal $e_{3}$, the generalized mean curvature
$h(\Sigma,i,\tau,e_{3})$ associated with these data is defined to be

$\displaystyle
h(\Sigma,i,\tau,e_{3}):=-\sqrt{1+|\nabla\tau|^{2}}\langle\mathbf{H},e_{3}\rangle-\alpha_{e_{3}}(\nabla\tau),$
(4.4)

where $\mathbf{H}$ is the mean curvature vector of $\Sigma$ in $N$ and and
$\alpha_{e_{3}}$ is the connexion form (see Definition 1.1) of the normal
bundle of $\Sigma$ in $N$ determined by $e_{3}$ and the future-directed time-
like unit normal $e_{4}$ orthogonal to $e_{3}$.

Now we consider the isometric embedding of $(\Sigma,\sigma)$ into the
Minkowski space $(\mathbb{R}^{1,3},\eta)$ with prescribed mean curvature in a
fixed time direction. Let us denote the standard Lorentzian inner product on
$\mathbb{R}^{1,3}$ by $\langle\cdot,\cdot\rangle$ and let $T_{0}$ be a
constant timeline unit vector in $\mathbb{R}^{1,3}$. Wang $\&$ Yau [13] proved
the following existence and uniqueness theorem for such an isometric embedding

######  Theorem 4.1 (Wang-Yau)

[13] Let $\chi$ be a function on $(\Sigma,\sigma)$ with
$\int_{\Sigma}\chi\mu_{\Sigma}=0$. Let $\tau$ be a potential function of
$\chi$ i.e., $\Delta_{\sigma}\tau=\chi$. Suppose the Gaussian curvature $K$ of
$(\Sigma,\sigma)$ verifies the inequality
$K+(1+|\nabla\tau|^{2}_{\sigma})^{-1}\det(\nabla^{2}\tau)>0$, then there
exists a space-like isometric embedding $i_{0}:\Sigma\to\mathbb{R}^{3}$ such
that the mean curvature $\textbf{H}_{0}$ of the embedded surface
$(i_{0}(\Sigma),\sigma)$ verifies $\langle\textbf{H}_{0},T_{0}\rangle=-\chi$.

The main idea behind the proof of this theorem is to consider the projection
of $\widehat{i}_{0}:\Sigma\to\mathbb{R}^{3}$ onto the orthogonal complement of
$T_{0}$ i.e., $i_{0}:=i-\tau T_{0}$. Let us denote the projected surface by
$\widehat{\Sigma}$. The induced metric on $\widehat{\Sigma}$ is given by
$\widehat{\sigma}:=\sigma+d\tau\otimes d\tau$ and the Gauss curvature
$\widehat{K}$ as
$(1+|\nabla\tau|^{2}_{\sigma})^{-1}(K+(1+|\nabla\tau|^{2}_{\sigma})^{-1}\det(\nabla^{2}\tau))$
which is positive by the assumption of the theorem 4.1. Therefore, the
existence of $\widehat{i}_{0}$ follows by the classic theorem of Nirenberg and
Pogolerov [32]. Now existence of $i_{0}$ can be established as follows. One
starts with a metric $\sigma$ and a function $\chi$ and solves for
$\Delta_{\sigma}\tau=\lambda$ (unique solution exists since $\Sigma$ is
compact, the kernel of $\Delta_{\sigma}$ is trivial modulo constants). Now one
constructs the new metric $\widehat{\sigma}=\sigma+d\tau\otimes d\tau$ and
finds that its Gauss curvature $\widehat{K}$ is again
$(1+|\nabla\tau|^{2}_{\sigma})^{-1}(K+(1+|\nabla\tau|^{2}_{\sigma})^{-1}\det(\nabla^{2}\tau))$
which is strictly positive by the hypothesis. Therefore the embedding into
$\mathbb{R}^{3}$ is established. The desired embedding $i_{0}$ is found as
$i_{0}=\widehat{i}_{0}+\tau T_{0}$.

Now the generalized mean curvature of the surface $(i_{0}(\Sigma),\sigma)$ in
the Minkowski space is defined in a similar fashion

$\displaystyle
h(\Sigma,i_{0},\tau,(e_{3})_{0}):=-\sqrt{1+|\nabla\tau|^{2}}\langle\mathbf{H}_{0},(e_{3})_{0}\rangle-\widehat{\alpha}_{(e_{3})_{0}}(\nabla\tau)$
(4.5)

where $\widehat{\alpha}$ is connection of the normal bundle of $i_{0}(\Sigma)$
induced by the flat connection $\nabla^{\mathbb{R}^{1,3}}$ of the Minkowski
space, $(e_{3})_{0}$ is the spacelike unit normal vector to
$(i_{0}(\Sigma),\sigma)$, and $\mathbf{H}_{0}$ is the mean curvature of
$i_{0}(\Sigma)$ as an embedded spacelike surface in the Minkowski space. The
Wang-Yau quasi-local energy is defined to be the following

$\displaystyle 8\pi
M^{wy}:=\inf_{\tau}\int_{\Sigma}\left(h(\Sigma,i_{0},\tau,(e_{3})_{0})-h(\Sigma,i,\tau,e_{3})\right)\mu_{\Sigma}.$
(4.6)

Strictly speaking, in the definition of $h(\Sigma,i,\tau,e_{3})$, there is
still redundancy left due to the boost transformation of the normal bundle of
$\Sigma$ in the physical spacetime $N$. In fact, [13] minimizes
$h(\Sigma,i,\tau,e_{3})$ over the boost-angle ($h(\Sigma,i,\tau,e_{3})$ is a
convex function of the boost angle). Here, we aim to prove the positivity of
the entity $h(\Sigma,i_{0},\tau,(e_{3})_{0})-h(\Sigma,i,\tau,e_{3})$ for any
$\tau$ verifying the condition of theorem 4.1. This would suffice to show the
positivity of $M^{wy}$ since subtracting the minimum of
$h(\Sigma,i,\tau,e_{3})$ only increases the mass. So from now on if there are
no confusions, we will denote
$\int_{\Sigma}\left(h(\Sigma,i_{0},\tau,(e_{3})_{0})-h(\Sigma,i,\tau,e_{3})\right)\mu_{\Sigma}$
by $8\pi M^{wy}$. The main idea is to convert the integrals
$\int_{\Sigma}h(\Sigma,i_{0},\tau,(e_{3})_{0})\mu_{\Sigma}$ and
$\int_{\Sigma}h(\Sigma,i,\tau,e_{3})\mu_{\Sigma}$ to integrals of mean
curvatures of surfaces in $\mathbb{R}^{3}$ and in the Cauchy slice of the
physical spacetime. Then we can use the theorem 3.1 to prove the positivity.

The most important observation one makes is the following two reductions
obtained by Wang-Yau [13]. They proved that the Minkowski entity
$\int_{\Sigma}h(\Sigma,i_{0},\tau,(e_{3})_{0})\mu_{\Sigma}$ can be re-written
as the integral of the mean curvature of the projection $\widehat{\Sigma}$ of
$\Sigma$ onto the compliment of $T_{0}$ i.e., onto $\mathbb{R}^{3}$. First
consider an embedding $i_{0}:\Sigma\hookrightarrow\mathbb{R}^{1,3}$ is an
embedding and $\tau:=\langle i_{0},T_{0}\rangle$ is the restriction of the
time function associated with the time vector field $T_{0}$. The outward unit
normal vector $\widehat{\nu}$ of $\widehat{\Sigma}$ in $\mathbb{R}^{3}$ and
$T_{0}$ form a basis of the normal bundle of $\widehat{\Sigma}$ in
$\mathbb{R}^{1,3}$. One can parallel transport $\widehat{\nu}$ along the
integral curves of $T_{0}$ to make it a vector field in $\mathbb{R}^{1,3}$ and
denote this vector field as $\widehat{e}_{3}$. Let the mean curvature of
$\widehat{\Sigma}$ in $\mathbb{R}^{3}$ be $\widehat{k}$. Then the following
proposition holds due to [13].

###### Proposition 4.1

The following identity is verified by $h(\Sigma,i_{0},\tau,(e_{3})_{0})$

$\displaystyle\int_{\Sigma}h(\Sigma,i_{0},\tau,(e_{3})_{0})\mu_{\Sigma}=\int_{\widehat{\Sigma}}\widehat{k}.$
(4.7)

Here the induced metric on $\widehat{\Sigma}$ is $\sigma+d\tau\otimes d\tau$
and $\sigma$ is the metric on $\Sigma$.

Now one hopes to find a similar expression for the physical spacetime entity
$\int_{\Sigma}h(\Sigma,i,\tau,e_{3})\mu_{\Sigma}$. But in the physical
spacetime, one needs to account for the momentum information (i.e., the second
fundamental form of the hypersurface $\Omega$ in $N$). [13] derives an almost
similar identity by projecting $\Sigma$ onto a constant time hypersurface with
extra terms accounting for the momentum information. As we shall see, the
appearance of this extra momentum contribution would necessitate invoking the
dominant energy condition. Consider the initial data set
$(\Omega,g_{ij},P_{ij})$, where $P_{ij}$ is the second fundamental form of
$\Omega$ in the physical spacetime $N$ and plays the role of momentum for the
metric $g_{ij}$. Now consider the product $\Omega\times\mathbb{R}$ and extend
$P_{ij}$ by parallel translation along the $\Omega$-normal field $e_{4}$. We
denote this entity by $P_{ij}$ as well. Now we seek a hypersurface
$\widetilde{\Omega}$ in $\Omega\times\mathbb{R}$ defined as the graph of a
function $f$ over $\Omega$, such that the mean curvature of
$\widetilde{\Omega}$ in $\Omega\times\mathbb{R}$ is the same as the trace of
the restriction of $P$ to $\widetilde{\Omega}$. This is accomplished in [13]
by solving Jang’s equation verified by $f$ (Jang’s equation enjoys a rich
history: Jang’s equation was proposed by Jang [17] in an attempt to solve the
positive energy conjecture. Schoen and Yau came up with different geometric
interpretations, studied the equation in full, and applied them to their proof
of the positive mass theorem [1, 2]). First, pick an orthonormal basis
$\\{\widetilde{e}_{\alpha}\\}_{\alpha=1}^{4}$ for the tangent space of
$\Omega\times\mathbb{R}$ along $\tilde{\Omega}$ such that
$\\{\widetilde{e}_{i}\\}_{i=1}^{3}$ is tangent to $\widetilde{\Omega}$ and
$\widetilde{e}_{4}$ is the downward unit normal to $\widetilde{\Omega}$. If
the induced metric on $\Omega$ is $g_{ij}$, then the induced metric on
$\widetilde{\Omega}$ is
$\widetilde{g}_{ij}=g_{ij}+\partial_{i}f\partial_{j}f$. Jang’s equations
states

$\displaystyle\sum_{i=1}^{3}\langle\widetilde{\nabla}_{\widetilde{e}_{i}}\widetilde{e}_{4},\widetilde{e}_{i}\rangle=\sum_{i=1}^{3}P(\widetilde{e}_{i},\widetilde{e}_{i})$
(4.8)

or more explicitly

$\displaystyle\sum_{i,j=1}^{3}\left(g^{ij}-\frac{f^{i}f^{j}}{1+|\widehat{\nabla}f|^{2}}\right)\left(\frac{\widehat{\nabla}_{i}\partial_{j}f}{(1+|\widehat{\nabla}f|^{2})^{\frac{1}{2}}}-P_{ij}\right)=0$
(4.9)

where $\widetilde{\nabla}$ is the usual Levi-Civita connexion of the product
spacetime $\Omega\times\mathbb{R}$. Now let $\tau$ be a smooth function on
$\Sigma$. We consider a solution $f$ to Jangs equation 4.8 in
$\Omega\times\mathbb{R}$ that verifies the Dirichlet boundary condition
$f=\tau$ on $\Sigma$. Let $\widetilde{\Sigma}$ be the graph of $\tau$ over
$\Sigma$ and $\tilde{\Omega}$ be the graph of $f$ over $\Omega$ such that
$\partial\widetilde{\Omega}=\widetilde{\Sigma}$. From now on we denote
$\widetilde{\Sigma}$ by $\widehat{\Sigma}$ since they are isometric. More
precisely the metric on $\widetilde{\Sigma}$ is $\sigma+d\tau\otimes d\tau$.
Jang’s equation is a quasi-linear elliptic PDE for $f$ and typically its
solution blows up in the presence of an apparent horizon. Here we assume that
such an apparent horizon is absent or
$\widetilde{k}-\text{tr}_{\widehat{\Sigma}}P>0$. Under such an assumption, a
solution to Jang’s equation exists and the foregoing construction makes sense.
Under such circumstances, one can relate the generalized total mean curvature
of $\Sigma$ in $\mathbb{R}^{1,3}$ to that of $\widehat{\Sigma}$ in
$\widetilde{\Omega}$ together with the momentum information. In particular,
the following proposition holds due to [13]

Figure 1: Schematics of the isometric embedding and reduction procedure in
Wang-Yau construction. Isometric surfaces are denoted by the same symbol
(e.g., $\widehat{\Sigma}$). Jang’s equation is solved on $\Omega$ with
boundary data $f=\tau$ on $\Sigma:=\partial\Omega$. $\widetilde{\Omega}$ is
the graph of $f$ over $\Omega$ and
$\widehat{\Sigma}:=\partial\widetilde{\Omega}$ is the graph of $\tau$ over
$\Sigma$. $i_{0}$ denotes the isometric embedding of the surface
$(\Sigma,\sigma)$ in $(N,\widehat{g})$ into $(\mathbb{R}^{1,3},\eta)$. Image
of this embedded surface is projected onto $T_{0}$ (constant unit timeline
vector field in $(\mathbb{R}^{1,3},\eta)$) orthogonal hyperplane, and the
projection is denoted by $\widehat{\Sigma}$ as well since it is isometric to
$\partial\widetilde{\Omega}$. Wang-Yau energy definition is then reduced to a
form that depends on the data on $\widehat{\Sigma}$ and
$\partial\widetilde{\Omega}$ (also denoted by $\widehat{\Sigma}$) and
spacetime momentum information.

###### Proposition 4.2

Let $\tau$ be a smooth function on $\Sigma=\partial\Omega$. We consider a
solution $f$ of Jang’s equation in $\Omega\times\mathbb{R}$ that satisfies the
Dirichlet boundary condition $f=\tau$ on $\Sigma$. Denote the graph of $\tau$
over $\Sigma$ by $\widehat{\Sigma}$. Then there exists a unit space-like
normal field $e^{{}^{\prime}}_{3}$ to $\Sigma$ in $N$ such that the following
holds

$\displaystyle\int_{\widehat{\Sigma}}\left(\widetilde{k}-\langle\widetilde{\nabla}_{\widetilde{e}_{4}}\widetilde{e}_{4},\widetilde{e}_{3}\rangle+P(\widetilde{e}_{4},\widetilde{e}_{3})\right)\mu_{\widehat{\Sigma}}=\int_{\Sigma}\left(-\sqrt{1+|\nabla\tau|^{2}}\langle\mathbf{H},e^{{}^{\prime}}_{3}\rangle-\alpha_{e^{{}^{\prime}}_{3}}(\nabla\tau)\right)\mu_{\Sigma}$
(4.10)

where $\widetilde{k}$ is the mean curvature of $\widehat{\Sigma}$ in
$\tilde{\Omega}$ and $\widetilde{e}_{4}$ and $\widetilde{e}_{3}$ are the basis
of the normal bundle of $\widehat{\Sigma}$ in $N$.

###  4.2 Proof of the main theorem

Following the two propositions 4.1 and 4.2, the expression of the quasi-local
energy reduces to the following

$\displaystyle 8\pi
M^{wy}:=\int_{\widehat{\Sigma}}\widehat{k}\mu_{\widehat{\Sigma}}-\int_{\widehat{\Sigma}}\left(\widetilde{k}-\langle\widetilde{\nabla}_{\widetilde{e}_{4}}\widetilde{e}_{4},\widetilde{e}_{3}\rangle+P(\widetilde{e}_{4},\widetilde{e}_{3})\right)\mu_{\widehat{\Sigma}},$
(4.11)

where induced metric on $\widehat{\Sigma}$ is simply $\sigma+d\tau\otimes
d\tau$. This form of the mass $M^{wy}$ is conducive to the application of the
theorem 3.1. Let us first consider the surface $\widehat{\Sigma}$ in
$\mathbb{R}^{3}$. From theorem 3.1, we know that for this surface embedded in
$\mathbb{R}^{3}$ the equality is satisfied and the following holds

$\displaystyle D\alpha=\frac{1}{2}\widehat{k}\alpha$ (4.12)

for a spinor $\alpha$ in $S^{\widehat{\Sigma}}$. Now denote the vector dual to
the $1-$form
$-P(\widetilde{e}_{4},\cdot)+\langle\widetilde{\nabla}_{\widetilde{e}_{4}}\widetilde{e}_{4},\cdot\rangle$
by $X$. Therefore the integral
$\int_{\widehat{\Sigma}}\left(\widetilde{k}-\langle\widetilde{\nabla}_{\widetilde{e}_{4}}\widetilde{e}_{4},\widetilde{e}_{3}\rangle+P(\widetilde{e}_{4},\widetilde{e}_{3})\right)\mu_{\widehat{\Sigma}}$
is nothing but $\int_{\widehat{\Sigma}}\left(\widetilde{k}-\langle
X,\widetilde{e}_{3}\rangle\right)\mu_{\widehat{\Sigma}}$, where
$\widetilde{e}_{3}$ is the unit normal to $\widehat{\Sigma}$ parallel to a
constant time hypersurface $\widetilde{\Omega}$. Now according to the theorem
3.1, the following entity

$\displaystyle M_{\alpha}:=\int_{\widehat{\Sigma}}\langle
D\alpha,\alpha\rangle\mu_{\widehat{\Sigma}}-\frac{1}{2}\int_{\widehat{\Sigma}}\left(\widetilde{k}-\langle
X,\widetilde{e}_{3})\right)|\alpha|^{2}\mu_{\widehat{\Sigma}}$ (4.13)

is non-negative. Now substitute $D\alpha=\frac{1}{2}\widehat{k}\alpha$ and set
$|\alpha|^{2}=1$ (since 3.2 holds) to yield

$\displaystyle 0\leq
M_{1}=\frac{1}{2}\int_{\widehat{\Sigma}}\widehat{k}-\frac{1}{2}\int_{\widehat{\Sigma}}\left(\widetilde{k}-\langle
X,\widetilde{e}_{3})\right)\mu_{\widehat{\Sigma}}=4\pi M^{wy}$ (4.14)

and therefore $M^{wy}\geq 0$. This concludes the proof of the theorem. Now the
only thing that remains to be justified is the validity of the curvature
condition $R^{\widetilde{\Omega}}\geq 2|X|^{2}-2\text{div}X$. Now recall the
geometry of $\widetilde{\Omega}$ in the product $\Omega\times\mathbb{R}$. On
$\widetilde{\Omega}$ the induced metric is
$\widetilde{g}_{ij}:=g_{ij}+\partial_{i}f\partial_{j}f$ and let us denote its
second fundamental form by $h_{ij}$ (note that
$h_{\mu\nu}=\frac{1}{2}(\widetilde{\nabla}_{\mu}\widetilde{n}_{\mu}+\widetilde{\nabla}_{\nu}\widetilde{n}_{\mu})$,$\mu,~{}\nu=0,1,2,3,4$,
where $\widetilde{n}=\widetilde{e}_{4}$ is the unit downward normal of
$\widetilde{\Omega}$ in $\Omega\times\mathbb{R}$). Similarly, let $P_{ij}$ be
the second fundamental form of $\Omega$ in $\Omega\times\mathbb{R}$. On
$\widetilde{\Omega}$, one has the following inequality verified by the scalar
curvature $R^{\widetilde{\Omega}}$ if the Hamiltonian and the momentum
constraints are satisfied on $\widetilde{\Omega}$

$\displaystyle 2(\mu-\sqrt{\widetilde{g}(J,J)})\leq
R^{\tilde{\Omega}}-\sum_{i,j}(h_{ij}-P_{ij})^{2}-2\sum_{i}(h_{i4}-P_{i4})^{2}+2\sum_{i}\widetilde{\nabla}_{i}(h_{i4}-P_{i4})$
(4.15)

But since the spacetime verifies the dominant energy condition, we have
$\mu\geq\sqrt{\widetilde{g}(J,J)}$, one has

$\displaystyle
R^{\tilde{\Omega}}\geq\sum_{i,j}(h_{ij}-P_{ij})^{2}+2\sum_{i}(h_{i4}-P_{i4})^{2}-2\sum_{i}\widetilde{\nabla}_{i}(h_{i4}-P_{i4})$
(4.16) $\displaystyle\geq
2\sum_{i}(h_{i4}-P_{i4})^{2}-2\sum_{i}\widetilde{\nabla}_{i}(h_{i4}-P_{i4}).$

But note that $h_{i4}-P_{i4}$ is nothing but the $\widetilde{e}_{i}$th
components of the $1-$form
$-P(\widetilde{e}_{4},\cdot)+\langle\widetilde{\nabla}_{\widetilde{e}_{4}}\widetilde{e}_{4},\cdot\rangle$.
Therefore $\sum_{i}(h_{i4}-P_{i4})^{2}=|X|^{2}$ and
$\sum_{i}\widetilde{\nabla}_{i}(h_{i4}-P_{i4})=\text{div}X$. Therefore, under
the assumption of the dominant energy condition, we have

$\displaystyle R^{\tilde{\Omega}}\geq 2|X|^{2}-2\text{div}X.$ (4.17)

In addition, the assumption of no apparent horizon gives the existence of a
bounded solution of Jang’s equation and $\widetilde{k}-\langle
X,\widehat{e}_{3}\rangle>0$. This concludes the proof the proof of the main
theorem. One vital property to note is the non-negativity holds for any
admissible $\tau$, that is, a $\tau$ that verifies the condition stated in the
theorem 4.1. In addition to the non-negativity of the Wang-Yau quasi-local,
one obtains a rigidity property as well.

######  Corollary 4.1 (Rigidity)

Under the assumption of the theorem 4.1 and if the set of admissible $\tau$ is
non-empty, then the Wang-Yau quasi-local energy bounded by the spacelike
topological $2-$sphere $\Sigma$ is non-negative. Moreover, if $\Sigma$ is a
spacelike topological $2-$sphere embedded in $\mathbb{R}^{1,3}$, then Wang-Yau
quasi-local energy $M^{wy}$ is zero. Also, if $M^{wy}$ is zero then the
physical spacetime $N$ is Minkowski along $\Sigma$.

_Proof._ The proof of the non-negativity is trivial if the admissible set of
$\tau$ is non-empty and essentially a consequence of the proof of the main
theorem stated earlier in this section. If the topological $2-$sphere $\Sigma$
is embedded in $\mathbb{R}^{1,3}$, then one can comply take the embedding
$i_{0}$ to be $i$ and therefore the vanishing of $M^{wy}$ follows trivially.
Now if $M^{wy}$ vanishes, then we have $\widetilde{\Omega}$ to be Ricci flat
and therefore flat and more specifically $R^{\tilde{\Omega}}=0$. Now if the
equality holds in 4.14, then according to the theorem 3.1, we have $X=0$.
Therefore $h_{i4}=P_{i4}$ which yields together with $R^{\tilde{\Omega}}=0$

$\displaystyle 0\leq 2(\mu-|J|)\leq-\sum_{i,j}(h_{ij}-P_{ij})^{2}\leq 0$
(4.18)

and therefore $\mu=|J|$ and $h_{ij}=P_{ij}$. Therefore $\widetilde{\Omega}$
coincides with $\Omega$ and $f=0$ is a solution to Jang’s equation yielding
$\text{tr}_{g}P=0$ and $R^{\Omega}=0$. Now write the following for
$P_{ij}=P^{tr}_{ij}+\frac{1}{3}\text{tr}_{g}Pg_{ij}$ together with
$R^{\Omega}=0$ and substitute in the Hamiltonian constraint to yield

$\displaystyle-|P^{tr}|^{2}_{g}=2\mu$ (4.19)

but $\mu\geq 0$ (since $\mu\geq\sqrt{g(J,J)}$) yielding $P_{ij}=0=\mu$ and
therefore $J=0$ as well. Therefore, $N$ is a vacuum spacetime i.e.,
$\mathbf{Ric}[\widehat{g}]=0$ Now recall the structure equations for the for
$\Omega$ embedded in $N$

$\displaystyle
R_{ij}-P_{im}P^{m}_{j}+P_{ij}\text{tr}_{g}P=E_{ij}+\textbf{Ric}_{ij}$ (4.20)
$\displaystyle\nabla_{i}P_{jm}-\nabla_{j}P_{km}=\epsilon_{ij}~{}^{l}B_{lm},$
(4.21)

where $E_{ij}$ and $B_{ij}$ are the electric and magnetic fields that
completely determine the Riemann curvature (Weyl curvature since $N$ is
vacuum). It follows trivially now that $E_{ij}=B_{ij}=0$ since $R_{ij}=0$ due
to Ricci flatness of $\Omega$ and $\mathbf{Ric}_{ij}=0$ in vacuum, and
therefore $\textbf{Riem}[\widehat{g}]=0$ implying $N$ is Minkowski along
$\Omega$ due to being simply connected. This completes the proof. $\square$

We have the obvious corollary

###### Corollary 4.2

Liu-Yau mass is non-negative.

_Proof._ Set $\tau=\text{constant}$. Then the Wang-Yau quasi-local energy
reduces to Liu-Yau energy. The embedding $i_{0}$ is in $\mathbb{R}^{3}$. Then
one needs a stronger condition of strict positivity of the Gauss curvature of
$\Sigma$ i.e., $K>0$. Now the result follows from the main theorem. $\square$

##  5 Conclusions

We provided a complete quasi-local proof of the non-negativity of the Wang-Yau
quasi-local energy and its rigidity property by using spinors that solve the
Dirac equation on the bulk with Atyiah-Patodi-Singer (APS) boundary condition.
The analogous result for the Brown-York mass is recently provided by [25]. In
the asymptotically flat setting, the Wang-Yau quasi-local mass approaches ADM
mass at spacelike infinity and its positivity proof using spinor (after
Schoen-Yau [1, 2] proof was announced) in that circumstance was provided by
Witten [4] which was subsequently made rigorous by Parker-Taubes [31]. We
would like to understand if the spinors can be used to define quasi-local mass
contained within a compact spacelike hypersurface with a disconnected
boundary, such as a topologically spherical annulus. Another interesting
problem would be to obtain a spinorial quasi-local proof of the positivity and
rigidity of the new quasi-local mass defined by Alaee-Khuri-Yau[15] where the
boundary surface $\Sigma$ is allowed to have non-trivial topology. These
problems are under investigation.

##  6 Acknowledgement

This work was supported by the Center of Mathematical Sciences and
Applications (CMSA), Department of Mathematics at Harvard University.

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