### Quasi-local Mass

### KWONG, Kwok Kun

### A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of

### Doctor of Philosophy in

### Mathematics

### The Chinese University of Hong Kong

### July 2011

### Professor Au Kwok Keung Thomas (Chair) Professor Tam Luen Fai (Thesis Supervisor) Professor Wan Yau Heng Tom (Committee Member)

### Professor Shi Yuguang (External Examiner)

## Abstract

### The concept of quasi-local masses was proposed by physicists about forty years ago to measure the energy of a given compact region by a closed spacelike 2-surface. There are several natural conditions which we expect a quasi-local mass to satisfy ([39]):

### 1. A quasi-local mass must be non-negative in general and zero when, and only when the ambient spacetime of the surface is the Minkowski spacetime in the asymptotically flat case (or hyperbolic space in the asymptotically hyperbolic case). These are called the positivity and rigidity conditions.

### 2. Also, the ADM mass should be recovered as the surfaces tend to the spacial infinity.

### In this thesis, we will report some results about the limiting

### behaviors and positivity of some quasi-local masses, both in the

### asymptotically flat case and in the asymptotically hyperbolic case.

## d d dd d d

### ddddddddddddddddddddddddddd dddddddddddddddddddddddddddd ddd dd([39])d

### 1. dddddddddd dddddddddddddddd

### ddddddddddMinkowskidddddd(ddddd

### d ddddddddd)d

### 2. d d d d d d d d d d d d d d d d d d d d d(ADM mass) d

### d ddddddddddddddddddddddddddd

### ddddddddddddddddddddd

## Acknowledgements

No words can express my gratitude towards my advisor, Prof. Luen-Fai Tam.

Throughout my Ph.D. program, he has been very patient to me answering all my questions and has generously shared with me a lot of insights and experiences.

This work would not have been possible without his guidance and constant en- couragement.

I would also like to thank the following mathematicians for teaching me math- ematics and/or helping me during various stages of my studies: Prof. Thomas Kwok-Keung Au, Prof. Tom Wan, Prof. Yuguang Shi, Prof. Conan Leung, Prof.

Pengzi Miao, Prof. Robert Bartnik, Prof. Yuanlong Xin, Prof. Feng Luo and Prof. Haibao Duan. I have learned a lot from them.

I have enjoyed numerous useful and delighting discussions with my friends and colleagues including Xuqian Fan, Naqing Xie, Chengjie Yu, Kwok-Wai Chan, Changzheng Li, Chor-Yin Ho, Guoyin Li, Xiaoliu Wang, Chenglong Yu, Siu- Cheong Lau, Ka-Shing Ng, Sin-Wa Yau, Wai-Kit Chan, Wei Yao, Matthew Wong, Dan Wu, Xin Liu, Yong Li, Fu-Man Lau, Hung-Ming Tsoi, Jack Lo, Chun-Yin Hui, Martin Li, Pun-Wai Tong, Lok-Hang So, Chit-Yu Ng, Chun-Kit Lai, Ting- Kam Wong, Kim-Hong Chiu, Kit-Ho Mak, Kai-Leung Chan, Sin-Tsun Fan, Chi- Kwong Fok, Chun-Kit Suen, Yin-Tat Lee, Yunxia Chen, Kin-Hei Mak, Chun- Ngai Cheung, Man-Shun Ma, Yat-Tin Chow, Hon-Leung Lee, King-Leung Lee, Cheung-Yu Leung, Kin-Wai Chan, Yat-Ming Cheung, Ka-Fai Li, Jun Luo, Chit Ma, Ding Ma, Lijiang Wu, Wen Yang, Yi Zhang, Fa-Wai Yiu, Weifeng Wo, Zhenyu Zhang and Weiwei Ao. I would also like to express my special thanks to my secondary school mathematics teacher, Mr. Yuk-Cheung Siu, for always guiding me in the right direction.

Lastly, but certainly not least, I want to offer my deepest thanks to my family and friends for their unfailing help and support over the years, especially to my mother, without whose love and understanding I would surely be lost.

### Abstract i

### Acknowledgements iii

### Introduction 1

### 1 Brown-York mass in AF manifolds 8

### 1.1 Asymptotically flat manifolds . . . . 8 1.2 Brown-York mass of revolution surfaces . . . . 11 1.2.1 Reduction to the Schwarzschild case . . . . 13 1.2.2 Estimates for the curvatures and embeddings

### of S

_{a}

### . . . . 16 1.2.3 Proof of Proposition 1.8 . . . . 21

### 2 Quasi-local mass in AH manifolds 24

### 2.1 Asymptotically hyperbolic (AH) manifolds . . . . 24 2.2 Quasi-local mass integral of AH manifolds . . . . 25 2.2.1 Curvature estimates . . . . 28 2.2.2 Inscribed and circumscribed geodesic spheres . 33

iv

### 2.2.3 Normalized embedding of (S

_{r}

### , γ

_{r}

### ) . . . . 37

### 2.2.4 Proof of Theorem 2.3 . . . . 41

### 3 Positivity of quasi-local mass 43 3.1 Preliminaries . . . . 45

### 3.2 A Positive mass theorem . . . . 49

### 3.2.1 Killing spinors on (H

^{n}

_{−k}

^{2}

### , g

^{0}

### ) . . . . 50

### 3.2.2 The hypersurface Dirac operator . . . . 53

### 3.2.3 Proof of Theorem 3.7 . . . . 57

### 3.3 Positivity of Shi-Tam mass . . . . 61

### Bibliography 66

## Introduction

As is well known, by the equivalence principle in general relativity, the concept of gravitational energy at a point is not well defined. The local effects of gravity can be removed by using a freely falling frame. The object centered at the origin of such a frame will not experience any gravitational acceleration.

On the other hand, when there is asymptotic symmetry (asymptotically flat or hyperbolic), the concepts of total energy and momentum can be well defined.

In the asymptotically flat case, these are the so called ADM energy momentum [2] and the Bondi energy-momentum when the system is viewed from spatial infinity and null infinity, respectively. It was proved by Bartnik [3] that in an asymptotically flat manifold, these concepts are well-defined, i.e. independent of the coordinates chosen. These concepts are fundamental in general relativity and have been proven to be natural. Moreover, the works of Schoen-Yau [31, 32], Witten [41] show that they satisfy the important positivity condition. These kinds of results are now known as positive mass theorems. However, when the physical system is not isolated, or the asymptotic symmetry fails, there would be limitations to these concepts. It was proposed about forty years ago to measure the energy of a system by enclosing a region with a “membrane”, i.e. a closed spacelike 2-surface, and define on it an energy-momentum 4-vector. This is the motivation behind the definition of quasi-local masses of surfaces.

There are several natural conditions which we expect a quasi-local mass to satisfy (see for example [39]):

1. Most importantly, a quasi-local mass must be non-negative in general and zero when, and only when the ambient spacetime of the surface is the Minkowski spacetime (or hyperbolic space in the asymptotically hyperbolic case). These are called the positivity and rigidity conditions.

2. Also, the ADM mass or Bondi mass should be recovered as the surfaces

tend to the spacial or null infinity.

There is still no universal agreement on the definition of the quasi-local mass, and many other definitions have been proposed, for example from Hawking [13]

and Penrose [27]. A promising one was given by Brown-York [7], motivated by Hamiltonian formulation. Shi and Tam [34] proved that it is positive in the time symmetric case, but in general it is not positive. Later on, Wang and Yau [38]

proposed the notion of Wang-Yau mass and proved its positivity and rigidity.

The study of these quasi-local masses and their relations is now a subject under intense study.

In this thesis, we will establish some results about the limiting behaviors and positivity of some quasi-local masses in asymptotically flat (AF) or asymptotically hyperbolic (AH) manifolds.

In Chapter 1, we will discuss the limiting behaviors of the Brown-York quasi- local mass of some family of surfaces. As mentioned before, we expect that the quasi-local mass of the boundary of exhausting domains tends to the ADM mass.

Indeed, many people have proved that the Brown-York quasi-local mass of the coordinate sphere tends to the ADM mass in an AF manifold, see for example the works of Brown-York [8], Hawking-Horowitz [14], Baskaran-Lau-Petrov [4], Shi-Tam [34] and also Fan-Shi-Tam [12]. Shi-Wang-Wu [36] also proved that the same result is true even for surfaces which are not necessarily coordinate spheres, but are nearly round near infinity.

The motivation of investigating the Brown-York mass for some general class of surfaces is as follows. In [3], Bartnik proved the following important result (see Theorem 1.3 for a more precise statement):

Theorem 0.1. Suppose (M, γ) is an AF manifold with integrable scalar curva-
ture. Let {D_{k}} be an exhaustion of M by closed sets such that the set S_{k} = ∂D_{k}
are connected C^{1} surfaces (not necessarily coordinate spheres) satisfying some

reasonable conditions. Then
m_{ADM}(M, γ) = lim

k→∞

1 16π

Z

Sk

3

X

i,j=1

(γ_{ij,i}− γ_{ii,j}) ν^{j}dσ^{0}.

That is, the ADM mass is independent of the sequence of {S_{k}}. (Note that ADM
mass is defined exactly as the R.H.S. of the above equation, except that S_{k} are
coordinate spheres. )

Because of this result, it is natural to ask if the Brown-York quasi-local mass of some general family of surfaces, other than those which are close to the coordinate spheres, will tend to the ADM mass in some AF manifolds. We will see in this chapter that this is true for certain kinds of revolution surfaces, for example a family of expanding ellipsoids, which are not close to the coordinate spheres.

More precisely we will prove the following

Theorem A. [Theorem 1.6, Limiting behaviors in AF case] If (N^{3}, g) is an
asymptotically Schwarzchild manifold and S is a given closed revolution surface
S. Then there is an ε > 0 such that for any family of revolution surfaces Sa with
Gaussian curvatures of order O(a^{−2}), mean curvatures of order O(a^{−1}) and radial
distances of order O(a), if the rescaled surfaces a^{−1}Sa are ε-close to S, then the
Brown-York masses of the surfaces will tend the ADM mass:

a→∞lim m_{BY}(S_{a}) = m_{ADM}(N, g).

This partly generalizes the results of [6, 34, 12].

In Chapter 2, we will work in the asymptotically hyperbolic (AH) case. The
motivation of this chapter is quite similar to the previous chapter. In particular,
this is partly motivated by the positive mass theorem proved by X.D. Wang
[40] in an AH manifold. Let us first recall the positive mass theorem in an AF
manifold: if we are given a complete asymptotically flat initial data set (M^{3}, g, h)
for the Einstein equations, we can then define the total 4-momentum (E, P ) of
(M^{3}, g, h), where P ∈ R^{3}. The positive mass theorem of Schoen-Yau [31, 33, 32]

then states that

Theorem 0.2. Let (M^{3}, g, h) be an asymptotically flat initial data set satisfying
the dominated energy condition (e.g. non-negative scalar curvature when h = 0),
then E ≥ |P |.

This can be interpreted as the 4-momentum being a future directed non-
spacelike vector in R^{3,1}. Later on, this result was reproved by Witten [41] (under
spin condition) using a different proof involving spinors. The spinor method turns
out to be very useful in proving positive mass type theorems. In particular we
have the following result of X.D. Wang, which can be regarded as the analogue
of Schoen-Yau’s result in the AF case:

Theorem 0.3. [40, Theorem 2.5] If (M^{n}, g) is spin, asymptotically hyperbolic
and the scalar curvature R ≥ −n(n − 1), then the total mass defined by (see
Theorem 2.2 for precise definitions)

( Z

S^{n−1}

tr_{g}_{0}(h)dµ_{g}_{0},
Z

S^{n−1}

tr_{g}_{0}(h)xdµ_{g}_{0}) ∈ R^{n,1}
is a future-directed non-spacelike vector.

In an AH manifold, we can define a quasi-local mass integral which is similar
to the Brown-York mass in the AF case. Let (Ω, g) be a three dimensional
compact manifold with smooth boundary Σ homeomorphic to sphere. Under
certain conditions, Σ can be uniquely embedded into H^{3} ⊂ R^{3,1}. Then the quasi-
local mass integral of Ω is defined as:

Z

Σ

(H_{0}− H)X (1)

where H_{0} is the mean curvature of Σ in H^{3}and X = (x^{0}, x^{1}, x^{2}, x^{3}) is the position
vector in R^{3,1}.

The motivation of this definition is as follows. In [35], Shi and Tam proved that if the scalar curvature of Ω satisfies R ≥ −6, then the vector R

Σ(H_{0}− H)W
is a future directed non-spacelike vector for W (x^{0}, x^{1}, x^{2}, x^{3}) = (αx^{0}, x^{1}, x^{2}, x^{3})
with α > 1 depending on the geometry of Σ. (This is exactly Theorem C when

n = 3. ) Hence W is close to the position vector. It is also conjectured by Shi and Tam that the same result is true if W is replaced by the position vector X.

It is therefore natural to ask if the quasi-local mass integral defined as in (1) for coordinate spheres will tend to the total mass as defined in Theorem 0.3. We will give a positive answer to this question. Namely, we will show that

Theorem B. [Theorem 2.3, Limiting behaviors in AH case] In an asymptotically
hyperbolic manifold (M^{3}, g), for a coordinate sphere S_{r} which is close enough to
infinity, we can associate with it a quasi-local mass expression (as a vector in
R^{3,1}), which will tend to the total mass of (M^{3}, g) defined by Theorem 0.3 when
S_{r} approaches infinity.

Whereas the first two chapters deals with the limiting behaviors of the quasi- local masses, in Chapter 3 we will look at the positivity of a quasi-local mass.

This chapter is also closely related to Chapter 2. As mentioned before, Witten [41] (see also [26, 3]) gave a simplified proof of the positive mass theorem using the spinor method. Since then the method of spinor has been adopted by many people to prove positive mass type theorems or some rigidity results [34, 1, 23, 38].

For example, M. T. Wang and Yau [38] developed a quasi-local mass for a three dimensional manifold with boundary whose scalar curvature is bounded from below by some negative constant. Using spinor method, they were able to prove that this mass is non-negative. Later on, Shi and Tam [35] also proved a similar result in the three dimensional case, but with a simpler definition of the mass.

More precisely, they proved the following:

Theorem 0.4. ([35] Theorem 3.1) Let (Ω, g) be a compact orientable 3-dimensional manifold with smooth boundary Σ = ∂Ω, homeomorphic to a 2-sphere. Assuming the following conditions:

1. The scalar curvature R of (Ω, g) satisfies R > −6k^{2} for some k > 0,
2. Σ is a topological sphere with Gaussian curvature K > −k^{2} and mean

curvature H > 0, so that Σ can be isometrically embedded into H^{3}−k^{2} with
mean curvature H_{0}.

Then there is a future directed time-like vector-valued function W on Σ such that the vector

Z

Σ

(H_{0}− H)W dΣ

is time-like. Here W = (x_{1}, x_{2}, x_{3}, αt) for some α > 1 depending only on the
intrinsic geometry of Σ, with X = (x_{1}, x_{2}, x_{3}, t) being the future-directed unit
normal vector of H^{3}−k^{2} (defined in (3.1)) in R^{3,1}.

In this chapter, we will prove an analogous result in higher dimension for spin manifolds (note that three-dimensional orientable manifolds are spin) as follows.

Theorem C. [Theorem 3.16, Positivity of Shi-Tam mass] Let n ≥ 3 and (Ω, g) be a compact spin n-manifold with smooth boundary Σ such that

1. The scalar curvature R of (Ω, g) satisfies R > −n(n − 1)k^{2} for some k > 0,
2. Σ is topologically a (n − 1)-sphere with sectional curvature K > −k^{2}, mean
curvature H > 0 and Σ can be isometrically embedded uniquely into H^{n}−k^{2} ⊂
R^{n,1} with mean curvature H_{0}.

Under these conditions, we can define on Σ a quasi-local mass introduced by Shi and Tam [35]:

m_{ST}(Σ) =
Z

Σ

(H_{0} − H)W ∈ R^{n,1}

where W = (x_{1}, x_{2}, · · · , x_{n}, αt) with α > 1 depending on the geometry of Σ and
(x_{1}, x_{2}, · · · , x_{n}, t) is the position vector of Σ in R^{n,1}.

Then the mass is positive in the sense that m_{ST}(Σ) is a future directed non-
spacelike vector in R^{n,1}.

There are two important ingredients in establishing the main result. One is a monotonicity formula (Lemma 3.6) for the mass expression, and the other is a

positive mass type theorem (Theorem 3.7). The later is particularly important.

This theorem was originally proved by M.T. Wang and Yau [38] in the three dimensional case. Here we will give a proof in general dimension. In particular, the Killing spinor fields play an important role in the proof, and we will give a detailed study on them. What is new in the proof of the theorem in higher dimension are two identities involving Killing spinors on the hyperbolic space (Proposition 3.9 and 3.10).

Theorem A, Theorem B and Theorem C, which are the main results of this thesis, will be proved in Chapter 1, 2 and 3 respectively.

## Brown-York mass in AF manifolds

### 1.1 Asymptotically flat manifolds

In this chapter, we will discuss the limiting behaviors of the quasi-local mass of a family of surfaces in an asymptotically flat manifold. Let us first recall some definitions. We will adopt the following definition of asymptotically flat manifolds.

Definition 1.1. A complete three dimensional manifold (M, γ) is said to be
asymptotically flat (AF) of order τ (with one end) if there is a compact sub-
set K such that M \ K is diffeomorphic to R^{3}\ B_{R}(0) for some R > 0 and in the
standard coordinates in R^{3}, the metric γ satisfies:

γ_{ij} = δ_{ij} + σ_{ij} (1.1)

with

|σ_{ij}| + r|∂σ_{ij}| + r^{2}|∂∂σ_{ij}| + r^{3}|∂∂∂σ_{ij}| = O(r^{−τ}), (1.2)
for some constant ^{1}_{2} < τ ≤ 1.

Here r and ∂ as the Euclidean distance and the standard derivative operator
on R^{3} respectively, δ is the usual Euclidean metric.

8

A coordinate system of M near infinity so that the metric tensor in this system satisfy the above decay conditions is said to be admissible. In such a coordinate system, we can define the ADM mass as follows.

Definition 1.2. The Arnowitt-Deser-Misner (ADM) mass (see [2]) of an asymp- totically flat manifold (M, γ) is defined as:

mADM(M, γ) = lim

r→∞

1 16π

Z

Sr

(γij,i− γii,j) ν^{j}dσ^{0}, (1.3)
where S_{r} is the Euclidean sphere, dσ^{0} is the area element of S_{r} induced by the
Euclidean metric, ν is the outward unit normal of S_{r} in R^{3} and the derivative is
the ordinary partial derivative.

To see that this gives a reasonable definition of mass, let us look at the
Schwarzschild metric. On a three dimensional slice of Schwarzschild spacetime,
corresponding to time = constant, the metric is given by (1 + _{2r}^{m})^{4}δ (using the
convention that G = c = 1), where m is the mass of a star (as r → ∞, its limit
becomes the Newtonian model of a point mass with mass m). It is easily calcu-
lated that the integral on the R.H.S. of (1.3) also tends to m as r → ∞. Thus
the ADM mass gives a reasonable definition of mass, at least in this case.

We always assume that the scalar curvature is in L^{1}(M ) so that the limit
exists in the definition. In [3], Bartnik showed that the ADM mass is a geometric
invariant. More precisely, he proved the following theorem (see [3, Proposition
4.1] for a more general setting):

Theorem 1.3. Suppose (M, γ) is an AF manifold with scalar curvature R(γ) ∈
L^{1}(M ). Let {D_{k}}^{∞}_{k=1} be an exhaustion of M by closed sets such that the set
S_{k}= ∂D_{k} are connected C^{1} surfaces without boundary in R^{3} such that

rk= inf{|x|, x ∈ Sk} → ∞ as k → ∞
r^{−2}_{k} Area(Sk) is bounded as k → ∞.

Then

m_{ADM}(M, γ) = lim

k→∞

1 16π

Z

Sk

(γ_{ij,i}− γ_{ii,j}) ν^{j}dσ^{0}.

That is, the ADM mass is independent of the sequence of {S_{k}}.

Next, let us recall the definition of the Brown-York quasi-local mass. Suppose (Ω, γ) is a compact three dimensional manifold with smooth boundary ∂Ω, if moreover ∂Ω has positive Gauss curvature, then the Brown-York mass of ∂Ω is defined as (see [7, 8]):

Definition 1.4.

m_{BY} (∂Ω) = 1
8π

Z

∂Ω

(H_{0} − H)dσ (1.4)

where H is the mean curvature of ∂Ω with respect to the outward unit normal
and the metric γ, dσ is the area element induced on ∂Ω by γ and H_{0} is the mean
curvature of ∂Ω when embedded in R^{3}.

The existence of an isometric embedding in R^{3} (Weyl’s embedding theorem)
for ∂Ω was proved by Nirenberg [25], the uniqueness of the embedding was given
by [15, 30, 29], so the Brown-York mass is well-defined.

It can be proved that the Brown-York mass and the Hawking quasi-local mass [13] of the coordinate spheres tends to the ADM mass in some AF manifolds, see [8, 14, 6, 4, 34, 12], and even of nearly round surfaces [36]. It is therefore natural to ask whether the quasi-local mass of a more general class of surfaces tends to the ADM mass.

In the coming sections, we will consider a special class of AF manifolds, called asymptotically Schwarzschild manifolds, which is defined as follows:

Definition 1.5. (N,eg) is called an asymptotically Schwarzschild manifold if N =
R^{3}\ K, K is a compact set containing the origin, and

eg_{ij} = φ^{4}δ_{ij} + b_{ij}, φ = 1 + m

2r, m > 0,
where |b_{ij}| + r|∂b_{ij}| + r^{2}|∂∂b_{ij}| + r^{3}|∂∂∂b_{ij}| = O (r^{−2}) .

Clearly, (N,eg) is an AF manifold. For b = 0, (N,eg) is called a Schwarzschild manifold. In this case, we always denote eg as g. Note that the scalar curvature

of (N, g) is zero [19] (page 283) and that of (N,eg) is in L^{1}(N ), so in both cases
the ADM mass is well defined.

### 1.2 Brown-York mass of revolution surfaces

In this section, we will study the limiting behaviors of Brown-York mass on some family of convex revolution surfaces in an asymptotically Schwarzschild manifold.

Our main result is the following:

Theorem 1.6. [11] Let (N,eg) be an asymptotically Schwarzschild manifold and
S be a C^{6,α} (0 < α < 1) closed convex revolution surface parametrized by

(w(ϕ) cos θ, w(ϕ) sin θ, h(ϕ)), 0 ≤ θ ≤ 2π and 0 ≤ ϕ ≤ l. (1.5)
Then there exists ε > 0 such that for any family of C^{5,α} closed convex revolution
surfaces S_{a} in (R^{3}, δ) satisfying the following conditions:

(i)

K ≥ C1

a^{2} (1.6)

where K is the Gaussian curvature of S_{a} with induced Euclidean metric.

(ii)

0 < H ≤ C_{2}

a (1.7)

where H is the mean curvature of S_{a} with induced Euclidean metric.

(iii)

C_{3}a ≤ min

x∈Sa

r(x) ≤ max

x∈Sa

r(x) ≤ C_{4}a, (1.8)

where C_{i} > 0 are independent of a for i = 1, 2, 3, 4.

Suppose also that (by applying a rotation if necessary) S_{a} is parametrized by
(aw_{a}(ϕ) cos θ, aw_{a}(ϕ) sin θ, ah_{a}(ϕ)) , 0 ≤ θ ≤ 2π and 0 ≤ ϕ ≤ l
such that

|w_{a}− w|_{C}^{4} + |h_{a}− h|_{C}^{4} < ε for sufficiently large a. (1.9)

Then

a→∞lim m_{BY}(S_{a}) = m_{ADM}(N,eg).

From this result, one has

Corollary 1.7. Let (N,eg) be an asymptotically Schwarzschild manifold. Let {S_{i}}
be a family of C^{7} closed convex revolution surfaces in (R^{3}, δ) satisfying (1.6)-(1.8)
and is parametrized as:

(aiwi(ϕ) cos θ, aiwi(ϕ) sin θ, aihi(ϕ)) , 0 ≤ θ ≤ 2π and 0 ≤ ϕ ≤ l
for some constant l > 0, here a_{i} are positive numbers with lim

i→∞a_{i} = +∞. If there
is a constant c such that

|w_{i}|_{C}^{7} + |h_{i}|_{C}^{7} ≤ c

for all i, then there is a subsequence {S_{i}_{k}} of {S_{i}} such that

k→∞lim m_{BY} (S_{i}_{k}) = m_{ADM}(N,eg).

To prove Theorem 1.6, we will show that we can actually reduce the case to which the ambient space is Schwarzschild. The main proposition is the following:

Proposition 1.8. Let (N, g) be a Schwarzschild manifold. Suppose {S_{a}}_{a>0} is
a family of closed convex surfaces of revolution in (R^{3}, δij) with the rotation axis
passing through the origin, satisfying (1.6)-(1.8). Then

a→∞lim m_{BY}(S_{a}) = m_{ADM}(N, g).

Remark 1.9. The conditions (i) and (ii) in Theorem 1.6 imply that the principal
curvature λ of S_{a} in (R^{3}, δ) satisfy _{C}^{C}^{1}

2a ≤ λ ≤ ^{C}_{a}^{2} for any a. For, if 0 < λ_{1} ≤ λ_{2}
are the principal curvatures, then (1.7) implies λ_{2} ≤ ^{C}_{a}^{2}. Together with (1.6),
λ_{1} ≥ _{λ}^{C}^{1}

2a^{2} ≥ _{C}^{C}^{1}

2a.

Remark 1.10. By condition (i) of Theorem 1.6 and the Gauss-Bonnet theorem,
the Euclidean area of S_{a} is of order O(a^{2}).

We will first show in Subsection 1.2.1 how our main result follows from Propo-
sition 1.8 by a perturbation argument. We will then prove our Proposition 1.8
in Subsection 1.2.3. To do this, we need some estimates for the embeddings and
the various curvatures of S_{a}, which will be done in Subsection 1.2.2.

One example of surfaces satisfying the conditions in Theorem 1.6 is the family of ellipsoids:

S_{a} =

(x^{1})^{2}+ (x^{2})^{2}+ (x^{3})^{2}
4 = a^{2}

,

which is not nearly round [36]. In contrast, the Hawking mass of this family of ellipsoids in (N, g) does not tend to the ADM mass of (N, g), indeed one can check that the Hawking mass [13] of this family tends to negative infinity as a → ∞.

### 1.2.1 Reduction to the Schwarzschild case

In this subsection, we will reduce the case of Theorem 1.6 to the Schwarzschild
case. Let us first compare the mean curvatures of S_{a} under different metrics.

Lemma 1.11. For the surfaces S_{a} satisfying the conditions in Theorem 1.6, we
have

| eH − H| ≤ Ca^{−3}

for some constant C independent of a, where eH and H are the mean curvatures
of S_{a} with respect to eg and g respectively.

Proof. We claim that

| eA − A|_{g} = O a^{−3}

(1.10) where A and eA are the second fundamental forms with respect to g and eg respec- tively.

Let ρ(x) defined on N to be the distance from x to S_{a} with respect toeg. We
will use the fact [18, (7.10)]:

A(X, Y ) − |∇ρ|e _{g}A(X, Y ) =

Γ^{k}_{ij}− eΓ^{k}_{ij}

X^{i}Y^{j}ρ_{k} (1.11)

for any tangent vectors X, Y of S_{a}. For completeness, we prove it here. We
proceed as in [36] Lemma 2.6. First of all, we have

A(X, Y ) = g

∇_{X}

∇ρ

|∇ρ|_{g}

, Y

= g(∇_{X}(∇ρ), Y )

|∇ρ|_{g}

= X(Y (ρ)) − (∇XY )(ρ)

|∇ρ|_{g}

= X^{i}Y^{j}ρ_{ij} − X^{i}Y^{j}Γ^{k}_{ij}ρ_{k}

|∇ρ|_{g} ,

(1.12)

here the subscript denotes ordinary derivative and Γ^{k}_{ij} are the Christoffel symbols
with respect to g, with the indices i, j, k = 1, 2, 3. Denote eΓ^{k}_{ij} to be the Christoffel
symbols with respect toeg. Then since the eg gradient | e∇ρ|_{e}_{g} = 1, we also have

A(X, Y ) = Xe ^{i}Y^{j}ρ_{ij} − X^{i}Y^{j}eΓ^{k}_{ij}ρ_{k}.
Combining this with (1.12), we can get (1.11).

Note that |Γ^{k}_{ij}− eΓ^{k}_{ij}| = O (r^{−3}) by the assumptions of the metrics. By asymp-
totic flatness, 1 =eg^{ij}ρ_{i}ρ_{j} ≥ CX

ρ^{2}_{i}, so |ρ_{i}| is uniformly bounded. The condition
egij = gij + bij implies |eg^{ij} − g^{ij}| = O (r^{−2}), so

||∇ρ|^{2}_{g} − 1| = |(g^{ij} −eg^{ij})ρ_{i}ρ_{j}| = O r^{−2}
which implies

|∇ρ|_{g} = 1 + O r^{−2} .

Finally, the principal curvatures λi in Euclidean metric are of order O (a^{−1}) by
Remark 1.9, the principal curvatures λ_{i} with respect to g are related to λ_{i} by
([19] Lemma 1.4): λi = φ^{−2}λi + 2φ^{−3}n(φ) where n is the unit outward normal
with respect to δ. In particular, as n(φ) = O(a^{−2}),

|A|_{g} = O(a^{−1}).

Combining all these together with (1.11), it is easy to see that (1.10) holds.

Combining (1.10) and the metric conditions of g and eg in Definition 1.5, this implies the lemma.

Let (S_{a}, des^{2}) , (S_{a}, ds^{2}) denote the surface S_{a} with metric des^{2}, ds^{2} induced
fromeg, g respectively. By Lemma 1.15, for a >> 1, the Gaussian curvatures on
(S_{a}, des^{2}) and (S_{a}, ds^{2}) are both positive, which implies that they can be isomet-
rically embedded into (R^{3}, δ) uniquely. Now let us compare the mean curvature
after embedding:

Lemma 1.12. Under the same notations and conditions of Theorem 1.6. Let
He_{0}, H_{0} be the mean curvature of the embedded surfaces of (S_{a}, dse^{2}) and (S_{a}, ds^{2})
in (R^{3}, δ) respectively, as a >> 1, we have | eH_{0}− H_{0}| ≤ C_{5}a^{−3} for some constant
C_{5}(S).

Proof. We can set ˆϕ = ^{π}_{l}ϕ, so it suffices to show that the lemma holds for l = π.

Also, by identifying S and S_{a} with the sphere S^{2}, we can regard all the metrics
here (ds^{2} etc.) to be metrics on S^{2}. We will denote w_{a} as w and h_{a} as h. Similar
to (1.22), one has

ds^{2} = a^{2}((w^{02}+ h^{02}) dϕ^{2}+ w^{2}dθ^{2}) ,
ds^{2}_{S} =

w^{02}+ h^{02}

dϕ^{2}+ w^{2}dθ^{2}

which are the metrics on S_{a}and S induced from the Euclidean metric respectively.

By definition,

ds^{2} = φ^{4}ds^{2}, des^{2} = ds^{2}+ b, where b = bijdx^{i}dx^{j}

Sa on Sa.

From (1.9), w and its derivatives up to forth order are uniformly bounded for
a >> 1, the same holds for h. By the conditions of b_{ij}, it is easy to see that the
followings hold:

ka^{−2}des^{2}− a^{−2}ds^{2}k_{C}^{3} = a^{−2}kbk_{C}^{3} ≤ C_{6}a^{−2}, (1.13)
ka^{−2}ds^{2}− a^{−2}ds^{2}k_{C}^{3} = a^{−2}k(φ^{4}− 1)ds^{2}k_{C}^{3} ≤ C_{6}a^{−1} (1.14)
for some constant C_{6}(S). By (1.9), we have

ka^{−2}ds^{2}− ds^{2}_{S}k_{C}^{3} ≤ C_{7}ε (1.15)

for some constant C_{7}(S). So for a >> 1, by (1.14) and (1.15), we have
ka^{−2}ds^{2}− ds^{2}_{S}k_{C}^{3} ≤ (C_{6}+ C_{7}) ε.

By the result of [24] Lemma 5.3, if we choose some 0 < ε < δ

π^{1−α}(C_{6}+ C_{7}) such
that

ka^{−2}ds^{2}− ds^{2}_{S}k_{C}^{2,α} < δ

for sufficiently large a, where δ is the one given by [24] Lemma 5.3, then there
are isometric embeddings eX and X of (S^{2}, a^{−2}dse^{2}) and (S^{2}, a^{−2}ds^{2}) respectively,
such that by (1.13), for sufficiently large a,

k eX − Xk_{C}^{2,α} ≤ C8ka^{−2}des^{2}− a^{−2}ds^{2}k_{C}^{2,α} = O a^{−2}

for some constant C_{8}(S). Since a eX, aX are the isometric embeddings of (S^{2}, des^{2})
and (S^{2}, ds^{2}) respectively. Hence | eH_{0}− H_{0}| = O (a^{−3}) . The lemma holds.

Now we can prove Theorem 1.6.

Proof of Theorem 1.6. By Proposition 1.8, we know that

a→∞lim 1 8π

Z

Sa

(H_{0}− H) dσ = m_{ADM}(N, g).

Since the ADM mass of (N, g) is equal to that of (N,eg), combining with Lemma 1.11 and Lemma 1.12, we can get the result.

### 1.2.2 Estimates for the curvatures and embeddings of S

_{a}

For simplicity, from now on to the end of this chapter, we use O a^{k} to denote a
quantity which is bounded by Ca^{k} for some constant C independent of a as a is
sufficiently large. We will first compute the mean curvature of S_{a} in (N, g) and
of the embedded surface of the Euclidean space respectively.

From the assumptions of S_{a}, we can assume that S_{a} is parametrized by
(aw_{a}(ϕ) cos θ, aw_{a}(ϕ) sin θ, ah_{a}(ϕ)), 0 ≤ ϕ ≤ l_{a}, 0 ≤ θ ≤ 2π,

w_{a}(ϕ), h_{a}(ϕ) being smooth functions for ϕ ∈ [0, l_{a}] (i.e. w_{a}, h_{a} can be extended
smoothly on a slightly larger interval ). Moreover,

(i)

h_{a}(0) > h_{a}(l_{a})
C_{3} ≤p

w^{2}_{a}+ h^{2}_{a} ≤ C_{4}
w_{a} > 0 on (0, l_{a}),

(1.16)

(ii) The generating curve (w_{a}(ϕ), h_{a}(ϕ)) is parameterized by arc length. i.e.

w_{a}^{02}+ h^{02}_{a} = 1. (1.17)

(iii) w_{a} is anti-symmetric about 0 and l_{a}, h_{a} is symmetric about 0 and l_{a}, i.e.

w_{a}(−ϕ) = −w_{a}(ϕ), w_{a}(l_{a}+ ϕ) = −w_{a}(l_{a}− ϕ),

h_{a}(−ϕ) = h_{a}(ϕ), h_{a}(l_{a}+ ϕ) = h_{a}(l_{a}− ϕ) for ϕ ∈ [0, ε).

(1.18)

This implies

w_{a}(0) = w_{a}(l_{a}) = h^{0}_{a}(0) = h^{0}_{a}(l_{a}) = 0. (1.19)
Since S_{a}is convex in (R^{3}, δ) and the Gaussian curvature K of S_{a}with the induced
metric ds^{2} is

K = h^{0}_{a}(w_{a}^{0}h^{00}_{a}− w_{a}^{00}h^{0}_{a})
a^{2}wa

for ϕ ∈ (0, l_{a}).

So h^{0}_{a} < 0 for ϕ ∈ (0, la) by (1.16).

Let φ_{a} be the function φ restricted on S_{a}, note that in (ϕ, θ) coordinates,
φa = φa(ϕ) is independent of θ. We have the following lemma:

Lemma 1.13. The functions w_{a}

h^{0}_{a} and φ^{0}_{a}

h^{0}_{a} can be extended continuously to the
whole [0, l_{a}]. Moreover there exists a constant C independent of a such that for
all a,

w_{a}
h^{0}_{a}

≤ C,

φ^{0}_{a}
h^{0}_{a}

≤ C a. Proof. We first show that the limits lim

ϕ→0

wa

h^{0}_{a} and lim

ϕ→la

wa

h^{0}_{a} exist and are uniformly
bounded.

The Gaussian curvature K of the point (0, 0, ah_{a}(0)) on S_{a} with induced
Euclidean metric is equal to K = ^{h}^{00}^{a}_{a}^{(0)}2 ^{2}. (This can be seen by noting that for
an arc-length parametrized plane curve (w_{a}(ϕ), h_{a}(ϕ)), its curvature is given by

−w_{a}^{00}h^{0}_{a}+ h^{00}_{a}w^{0}_{a}. ) So at (0, h_{a}(0)), its curvature is h^{00}_{a}(0).

As K ≥ C1

a^{2} by (1.6), |h^{00}_{a}(0)| ≥ √

C_{1} > 0. By L’Hospital rule,

ϕ→0lim
w_{a}

h^{0}_{a} = w_{a}^{0}(0)
h^{00}_{a}(0)

which is finite and is bounded by some C > 0 by (1.6) and (1.17). The same applies to lim

ϕ→la

w_{a}
h^{0}_{a}.

Next, observe that one of the principal curvatures of S_{a} in (R^{3}, δ) is − h^{0}_{a}
aw_{a} (
[10] p.162, (10)). So by Remark 1.9, we have

wa

h^{0}_{a}

≤ C on the whole [0, l_{a}] for all
a.

By differentiating φ_{a} = 1 + m

2apw^{2}_{a}+ h^{2}_{a}, ^{φ}_{h}^{0}^{a}0

a = − ^{m}

2a(w^{2}_{a}+h^{2}_{a})^{3}^{2}(w^{0}_{a}^{w}_{h}0^{a}

a + h_{a}) which
can be extended to [0, l_{a}] by the above, and is of order O (a^{−1}) by (1.16), (1.17).

We have the following estimates

Lemma 1.14. Regarding φ_{a} = φ_{a}(ϕ) as functions on S_{a}, we have φ^{0}_{a} = O(a^{−1})
and φ^{00}_{a} = O(a^{−1}).

Proof. Let A = w_{a}^{2}+h^{2}_{a}. As φ_{a}= 1+ m
2a√

A, we only have to prove (A^{−}^{1}^{2})^{0} = O(1)
and (A^{−}^{1}^{2})^{00} = O(1). By direct computations and (1.16), (1.17),

|(A^{−}^{1}^{2})^{0}| = |A^{−}^{3}^{2}(w_{a}w^{0}_{a}+ h_{a}h^{0}_{a})| ≤ A^{−}^{3}^{2}(w_{a}^{2}+ h^{2}_{a})^{1}^{2}(w^{02}_{a} + h^{02}_{a})^{1}^{2} = O(1).

|(A^{−}^{1}^{2})^{00}| =
3

2A^{−}^{5}^{2}(w_{a}w_{a}^{0} + h_{a}h^{0}_{a})^{2}− A^{−}^{3}^{2}(1 + w_{a}w^{00}_{a}+ h_{a}h^{00}_{a})

≤ 3

2A^{−}^{5}^{2}(w_{a}^{2}+ h^{2}_{a}) + A^{−}^{3}^{2}(1 + (w_{a}^{2}+ h^{2}_{a})^{1}^{2}(w^{002}_{a} + h^{002}_{a} )^{1}^{2}).

The two principal curvatures of S_{a}with induced Euclidean metric are − h^{0}_{a}
aw_{a} and
a^{−1}(w_{a}^{002}+ h^{002}_{a} )^{1}^{2} ([10] p.162, (10)), hence by Remark 1.9, |(A^{−}^{1}^{2})^{00}| = O(1).

From now on, we will drop the subscript a and denote w_{a} by w, h_{a} by h, φ_{a}
by φ and l_{a} by l. We also denote ds^{2} to be the metric on S_{a} induced from g.

Lemma 1.15. The Gaussian curvature K of (S_{a}, ds^{2}) is positive for sufficiently
large a. In particular, there exists a unique isometric embedding of (S_{a}, ds^{2}) into
(R^{3}, δ) for sufficiently large a.

Proof. Let ds^{2} and ds^{2} be the metrics on S_{a} induced by δ and g respectively.

ds^{2} = a^{2}(dϕ^{2}+ w^{2}dθ^{2}) = Edϕ^{2}+ Gdθ^{2},
ds^{2} = φ^{4}ds^{2} = Edϕ^{2}+ Gdθ^{2},

implies

E = E + O(a), E_{ϕ} = E_{ϕ}+ O(a), E_{ϕϕ}= E_{ϕϕ}+ O(a)
E_{θ} = E_{θ}+ O(a), E_{θθ} = E_{θθ}+ O(a).

Similar result holds for G. By the formula K = − ^{1}

2√ EG

Eθ

√ EG

θ

+ _{G}

√φ

EG

φ

and the corresponding formula for K, one can get K = K + O (a^{−3}). Hence the
lemma holds.

Now let us compute the mean curvature of a revolution surface in (R^{3}, δ).

Lemma 1.16. For a smooth revolution surface S in (R^{3}, δ) parametrized by
(au(ϕ) cos θ, au(ϕ) sin θ, av(ϕ)), 0 < ϕ < l, 0 < θ < 2π,

its mean curvature H with respect to δ is
H = u^{00}

aT v^{0} − T^{0}u^{0}

aT^{2}v^{0} − v^{0}

aT u where T =√

u^{02}+ v^{02}.
Proof. The mean curvature H of S with respect to δ is computed to be

H = v^{0}u^{00}− u^{0}v^{00}
aT^{3} − v^{0}

aT u.
Differentiating T^{2} gives u^{0}u^{00}+ v^{0}v^{00} = T T^{0}. This implies

v^{0}u^{00}− u^{0}v^{00}= v^{0}u^{00}+u^{02}u^{00}− u^{0}T T^{0}

v^{0} = (u^{02}+ v^{02})u^{00}− u^{0}T T^{0}

v^{0} = T^{2}u^{00}− T T^{0}u^{0}

v^{0} .

So we have H = _{aT v}^{u}^{00}0 − _{aT}^{T}^{0}2^{u}v^{0}^{0} − _{aT u}^{v}^{0} .

Lemma 1.17. The mean curvature H of S_{a} with respect to g is
H = w^{00}

aφ^{2}h^{0} − h^{0}

aφ^{2}w + 4φ^{−3}n(φ) (1.20)
where n is the outward unit normal vector of Sa with respect to δ.

Proof. By Lemma 1.16, the mean curvature of S_{a} with respect to δ is H =

w^{00}

ah^{0} − _{aw}^{h}^{0}. The mean curvature H of Sa with respect to g is ([31], p. 72) H =
φ^{−2} H + 4φ^{−1}n (φ). The result follows.

Lemma 1.18. For sufficiently large a, there is an isometric embedding of (Sa, ds^{2})
into (R^{3}, δ) which is given by

(x^{1}, x^{2}, x^{3}) = (au(ϕ) cos θ, au(ϕ) sin θ, av(ϕ)), ϕ ∈ [0, l], θ ∈ [0, 2π] (1.21)
where

u = φ^{2}w, v^{0} = φ^{2}h^{0}

1 − ^{2φ}_{h}^{0}^{ww}02 ^{0} + O (a^{−2})

,
u^{02}+ v^{02} = φ^{4}.

Proof. The existence has already been proved in Lemma 1.15.

In (ϕ, θ) coordinates, the metric on S_{a} induced by g can be written as:

ds^{2} = a^{2}φ^{4}dϕ^{2}+ a^{2}φ^{4}w^{2}dθ^{2}. (1.22)
We can regard (S_{a}, ds^{2}) as S^{2}, the sphere with the metric ds^{2}. Now we want
to find two functions u, v such that the surface written as the form (1.21) is an
embedded surface S_{a}^{e} of S_{a} into (R^{3}, δ). First of all, the induced metric by the
Euclidean metric on the surface which is of the form (1.21) can be written as:

ds^{2}_{e} = a^{2} u^{02}+ v^{02} dϕ^{2}+ a^{2}u^{2}dθ^{2}.
Comparing this with (1.22), one can choose

u = φ^{2}w. (1.23)

Consider

φ^{4}− u^{02}= φ^{2}(φ^{2}− (2φ^{0}w + φw^{0})^{2}) = φ^{2}(φ^{2}(w^{02}+ h^{02}) − (2φ^{0}w + φw^{0})^{2})

= φ^{2}(φ^{2}h^{02}− 4φφ^{0}ww^{0}− 4φ^{02}w^{2})

= φ^{4}h^{02}

1 −4φ^{0}ww^{0}

φh^{02} −4φ^{02}w^{2}
φ^{2}h^{02}

.

(1.24)

By Lemma 1.13 and Lemma 1.14, the functions φ^{0}ww^{0}

φh^{02} ,φ^{02}w^{2}

φ^{2}h^{02} can be extended
continuously on [0, l] with φ^{0}ww^{0}

φh^{02} = O(a^{−1}),φ^{02}w^{2}

φ^{2}h^{02} = O(a^{−2}). So 1 − 4φ^{0}ww^{0}
φh^{02} −
4φ^{02}w^{2}

φ^{2}h^{02} > 0 for sufficiently large a. For these a, we can take
v^{0} = φ^{2}h^{0}

1 − 4φ^{0}ww^{0}

φh^{02} − 4φ^{02}w^{2}
φ^{2}h^{02}

^{1}_{2}

so that u^{02}+ v^{02}= φ^{4}. Note that by (1.18), v^{0} is an odd function for ϕ ∈ [−l, l]. By
choosing an initial value, one can get an even function v. By the above argument,
one has

v^{0} = φ^{2}h^{0}

1 − 2φ^{0}ww^{0}

h^{02} + O a^{−2}

.

From (1.23) and (1.24), near ϕ = 0, u, v can be extended naturally to (−ε, ε)
for some ε > 0. Since u is an odd function in ϕ , v is an even function in ϕ, and
u^{02}+ v^{02}= T^{2} > 0, the generating curve in {x^{2} = 0} is symmetric with respect to
x^{3}-axis, and is smooth at ϕ = 0. Similarly, it is also smooth at ϕ = l. Hence the
revolution surface determined by the choice of u, v as above, can be extended
smoothly to a closed revolution surface, which is an embedded surface of Sa. This
completes the proof of the lemma.

### 1.2.3 Proof of Proposition 1.8

Now we are ready to prove Proposition 1.8.

Proof of Proposition 1.8. Let u, v be defined as in Lemma 1.18. Recall that

u = φ^{2}w, v^{0} = φ^{2}h^{0}

1 − ^{2φ}_{h}^{0}^{ww}02 ^{0} + O (a^{−2})

,
u^{02}+ v^{02}= φ^{4} = T^{2} where T = φ^{2}.

(1.25)

By Lemma 1.14, we have

T^{0} = 2φ^{0}+ O (a^{−2}) , u^{0} = φ^{2}w^{0} + O (a^{−1}) ,
u^{00} = φ^{2}w^{00}+ 4φ^{0}w^{0} + 2φ^{00}w + O (a^{−2}) .

(1.26)

By Lemma 1.16 and Lemma 1.18,
H0 = u^{00}

aT v^{0} − T^{0}u^{0}

aT^{2}v^{0} − v^{0}

aT u. (1.27)

Combining with Lemma 1.17,
H_{0} − H =

u^{00}

aT v^{0} − w^{00}
aφ^{2}h^{0}

− T^{0}u^{0}
aT^{2}v^{0} −

v^{0}

aT u− h^{0}
aφ^{2}w

− 4φ^{−3}n(φ). (1.28)
Using (1.25) and (1.26),

u^{00}

aT v^{0} − w^{00}

aφ^{2}h^{0} = w^{00}

aφ^{2}h^{0} + 4φ^{0}w^{0}

ah^{0} + 2φ^{00}w

ah^{0} + 2φ^{0}ww^{0}w^{00}

ah^{03} − w^{00}

aφ^{2}h^{0} + O a^{−3}

= 4φ^{0}w^{0}

ah^{0} +2φ^{00}w

ah^{0} +2φ^{0}ww^{0}w^{00}

ah^{03} + O a^{−3} .

(1.29)

By (1.25) and (1.26),

− T^{0}u^{0}

aT^{2}v^{0} = −2φ^{0}w^{0}

ah^{0} + O a^{−3} . (1.30)

By (1.25),

− v^{0}

aT u+ h^{0}

aφ^{2}w = − h^{0}

aφ^{2}w+ 2φ^{0}w^{0}

ah^{0} + h^{0}

aφ^{2}w + O a^{−3} = 2φ^{0}w^{0}

ah^{0} + O a^{−3} .
(1.31)
Summing (1.29), (1.30) and (1.31) and comparing with (1.28), we have

H_{0}− H = 4φ^{0}w^{0}

ah^{0} +2φ^{00}w

ah^{0} +2φ^{0}ww^{0}w^{00}

ah^{03} − 4φ^{−3}n(φ) + O a^{−3} .
As w^{0}w^{00} = −h^{0}h^{00} by (1.17), so

H_{0}− H = 4φ^{0}w^{0}

ah^{0} +2φ^{00}w

ah^{0} −2φ^{0}wh^{00}

ah^{02} − 4φ^{−3}n(φ) + O a^{−3} .

Denote the Euclidean area element of S_{a} by dσ_{0}, the area element of (S_{a}, ds^{2}) by
dσ. Note that H_{0}− H = O (a^{−2}), dσ − dσ_{0} = O (a^{−1}) dσ_{0} and

Z

Sa

dσ_{0} = O a^{2}.

To prove the result, it suffices to show

a→∞lim 1 8π

Z

Sa

(H0− H) dσ0 = m.

The Euclidean area element is computed to be dσ_{0} = a^{2}wdϕdθ. By (1.19) and
Lemma 1.13,

Z

Sa

(4φ^{0}w^{0}

ah^{0} +2φ^{00}w

ah^{0} −2φ^{0}wh^{00}

ah^{02} )dσ_{0} =2πa
Z l

0

(4φ^{0}ww^{0}

h^{0} + 2φ^{00}w^{2}

h^{0} − 2φ^{0}w^{2}h^{00}
h^{02} )dϕ

=2πa Z l

0

d dϕ

2φ^{0}w^{2}
h^{0}

dϕ

=0.

Since the norm of the Euclidean gradient of φ has |∇_{0}φ| = O(r^{−2}), one has
n(φ) = O(a^{−2}). So

1 8π

Z

Sa

(H_{0}− H) dσ_{0} = − 1
2π

Z

Sa

φ^{−3}n(φ)dσ_{0}+ O a^{−1}

= − 1 2π

Z

Sa

n(φ)dσ_{0}+ O a^{−1} .

By the result of [3] (Proposition 4.1), the definition of the ADM mass of N can be taken as

a→∞lim 1 16π

Z

Sa

X

i,j

(g_{ij,i}− g_{ii,j})n^{j}dσ_{0} = m. (1.32)
where n is the unit outward normal of S_{a} with respect to δ. By a direct compu-
tation,

X

i,j

(g_{ij,i}− g_{ii,j})n^{j} = −8φ^{3}X

j

n^{j} ∂φ

∂x^{j} = −8n(φ) + O a^{−3} . (1.33)
Combining (1.32) and (1.33), we have

m = − lim

a→∞

1 2π

Z

Sa

n(φ)dσ_{0}.
Therefore

a→∞lim 1 8π

Z

Sa

(H_{0}− H)dσ = lim

a→∞

1 8π

Z

Sa

(H_{0}− H)dσ_{0} = m.

We are done.

## Quasi-local mass in AH manifolds

It is known that in an asymptotically flat manifold, the Brown-York quasi-local mass of the coordinate spheres will converge to the ADM mass of the manifold [12, 36, 11]. In this chapter, we will show an analogous result for asymptotically hyperbolic (AH) manifolds.

### 2.1 Asymptotically hyperbolic (AH) manifolds

First we give the meanings of mass of an AH manifold and quasi-local mass. In this chapter, all manifolds are assumed to be connected and orientable.

We will follow X. D. Wang [40] to define asymptotically hyperbolic manifolds as follows:

Definition 2.1. A complete noncompact Riemannian manifold (M^{n}, g) is said
to be asymptotically hyperbolic (AH) if M is the interior of a compact manifold
M with boundary ∂M such that:

(i) there is a smooth function r on M with r > 0 on M and r = 0 on ∂M such
that g = r^{2}g extends as a smooth Riemannian metric on M ;

(ii) |dr|g = 1 on ∂M ;

24