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Title The geometry of vacuum static spaces and deformations of scalar curvature

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Author Yuan, Wei

Publication Date 2015

Peer reviewed|Thesis/dissertation

eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA SANTA CRUZ

THE GEOMETRY OF VACUUM STATIC SPACES AND DEFORMATIONS OF SCALAR CURVATURE A dissertation submitted in partial satisfaction of the requirements for the degree of

DOCTOR OF PHILOSOPHY

MATHEMATICS

Wei Yuan

June 2015

The Dissertation of Wei Yuan is approved:

Professor Jie Qing, Chair

Professor Richard Montgomery

Professor Longzhi Lin

Dean Tyrus Miller Vice Provost and Dean of Graduate Studies Copyright c by

Wei Yuan

2015 Contents

Abstract v

Dedication vii

Acknowledgments viii

1 Introduction and preliminaries 1 1.1 Introduction to vacuum static spaces ...... 1 1.2 Notations and conventions ...... 7 1.3 Statements of main theorems ...... 8

2 Classiﬁcations of vacuum static spaces 25 2.1 Bach ﬂatness and vanishing of D-tensor ...... 25 2.2 The geometry of level surfaces ...... 31 2.3 3-dimensional cases ...... 40

3 Local rigidity phenomena 44 3.1 Local scalar curvature rigidity in general ...... 44 3.2 Local scalar curvature rigidity of space forms ...... 56 3.2.1 Euclidean spaces ...... 56 3.2.2 Hyperbolic spaces ...... 58 3.2.3 Hemispheres ...... 61 3.3 Conformal rigidity of vacuum static space ...... 62

4 Brown-York mass and compactly conformal deformations 68 4.1 Brown-York mass and conformal rigidity ...... 68 4.2 Non-rigidity phenomena ...... 74

Bibliography 87

iii A Variations of curvature 95 A.1 Metrics ...... 96 A.2 Christoﬀel symbols ...... 97 A.3 Riemannian curvature tensor ...... 98 A.4 Ricci tensor ...... 101 A.5 Scalar curvature ...... 104

B Conformal transformations of Riemannian metric 108 B.1 Metric ...... 109 B.2 Christoﬀel symbols ...... 109 B.3 Hessian and Laplacian ...... 110 B.4 Second fundamental form and mean curvature ...... 110 B.5 Riemann curvature tensor ...... 111 B.6 Ricci curvature tensor ...... 114 B.7 Scalar curvature ...... 114 B.8 Traceless Ricci tensor ...... 116 B.9 Schouten tensor ...... 117 B.10 Weyl tensor ...... 117 B.11 Cotten Tensor ...... 118 B.12 Bach tensor ...... 120

C Curvature in warped product metrics 123 C.1 Metric ...... 124 C.2 Christoﬀel symbols ...... 124 C.3 Hessian and Laplacian ...... 125 C.4 Second fundamental form and mean curvature ...... 127 C.5 Riemann curvature tensor ...... 128 C.6 Ricci curvature tensor ...... 130 C.7 Scalar curvature ...... 131 C.8 Traceless Ricci tensor ...... 132 C.9 Schouten tensor ...... 133 C.10 Weyl tensor ...... 134 C.11 Cotten Tensor ...... 136

iv Abstract

The geometry of vacuum static spaces and deformations of scalar curvature

Wei Yuan

In this dissertation we mainly study the geometric structure of vacuum static spaces and some related geometric problems.

In particular, we have made progress in solving the classiﬁcation problem raised in [29] of vacuum static spaces and in proving the conjecture made in [5] about manifolds admitting solutions to the critical point equation in general dimensions.

We obtain even stronger results in dimension 3.

We also extend the local scalar curvature rigidity result in [10] to a small domain on general vacuum static spaces, which conﬁrms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the paper [20]. We obtain the local scalar curvature rigidity of bounded domains in hyperbolic spaces. We also obtain the global scalar curvature rigidity for conformal deformations of metrics in the domains, where the lapse functions are positive, on vacuum static spaces with positive scalar curvature, and show such domains are maximal, which generalizes the work in [30].

As for generic Riemannian manifolds, we ﬁnd a connection between Brown-

York mass and the ﬁrst Dirichlet Eigenvalue of a Schrödinger type operator. In

v particular, we prove a local positive mass type theorem for metrics conformal to the background one with suitable presumptions. As applications, we investigate compactly conformal deformations which either increase or decrease scalar curvature. We ﬁnd local conformal rigidity phenomena occur in both cases for small domains and as for manifolds with nonpositive scalar curvature it is even more rigid in particular. On the other hand, such deformations exist for closed or a type of non-compact manifolds with positive scalar curvature. These results together give an answer to a question arises naturally in [20, 40].

vi To my parents, grannies and Barbara McDowell (1918 - 2012)

vii Acknowledgments

I would like to express my deepest appreciations to my adviser Professor Jie

Qing in Department of Mathematics. It is my honor to be one of his Ph. D students. In the past seven years, I learned a lot from him. It includes how to be a excellent mathematician, a fantastic adviser and a great teacher, but most important of all, a nice person. Especially, I would like to thank him for giving me valuable suggestions and advise when I was in confusions of how to make choices in jobs and life. Without his support, I may not make it.

Also I would like to thank my parents for their support and taking good care of everything in the family so that I can pursue my dream without worrying much.

Although they never got any higher education, they know what does it mean to their son.

Finally, I would like to thank Professor Richard Montgomery, Professor Justin

Corvino, Professor Pengzi Miao, Professor Longzhi Lin, Dr. David DeConde, Dr.

Yi Fang and Dr. Yueh-Ju Lin for their inspiring discussions and help in writing this dissertation. Especially for those long conversations with Dr. Yi Fang, they have provided me conﬁdence and inspirations.

viii Chapter 1

Introduction and preliminaries

1.1 Introduction to vacuum static spaces

In 1915, A.Einstein published the article "Die Feldgleichungen der Gravita- tion" ([26]), in which he proposed a ﬁeld equation for vacuum space-time:

Gµν = κTµν,

1 where Gµν = Rµν − 2 Rgµν is the Einstein tensor, κ is a constant and Tµν is the Energy-Momentum-Stress tensor. It is the ﬁrst time in the history people introduced a geometric theory for gravitation. Now this theory is popularly known as "general relativity".

As one of the most successful theories of physics in the twenty century, general relativity has revealed the fundamental interplay between physics and the

1 geometry of space-time. During the past 100 years, it has been one of the most active research ﬁelds in both physics and mathematics. It has become one of the fundamental tools for people to get to know the past and the future of our uni- verse. Besides its essential role in the theoretical study, general relativity has also gained great success in engineering when applying to our daily life. For example, general relativistic eﬀect has to be considered to reduce the systemic errors when engineers try to design the Global Positioning System (GPS)( for more details please refer to [45]).

After being proposed, searching for diﬀerent solutions to Einstein’s ﬁeld equations became one of the most important problems.

The most obvious solution is the Minkowski’s space-time, which is the four- dimensional Euclidean space R4 equipped with Lorentzian metric

2 g = −dt + δR3 ,

3 where δR3 is the Euclidean metric on R . In fact, this is one of the solution to vacuum Einstein’s ﬁeld equations:

Rµν = 0, which corresponds to a ﬂat space-time.

The ﬁrst non-trivial solution was published in 1916 by K.Schwartzchild ([53]),

2 which is now known as the Schwartzchild’s solution:

−1 2GM 2 2GM 2 2 g = − 1 − dt + 1 − dr + r g 2 , r r S

2 where gS2 is the canonical spherical metric on S .

Other well-know solutions were founded later as time went by, such as de

Sitter/anti-de Sitter space-times, Kerr space-time, Friedmann-Lemaître-Robertson-

Walker space-time, etc.

Among all those solutions, static space-times are special and important global solutions. The Minkowski space-time, Schwartzchild space-time and de Sitter/anti- de Sitter space-times are typical examples of static space-time. One of the com- mon features is that the time direction after a proper rescaling can be written as a Killing ﬁeld, when evolving along the integral curve of which, the spatial slices remain invariant. This is the reason we call them "static" according to Newton’s viewpoint on space and time.

When working in a more abstract frame, we can consider static space-times that carry a perfect fluid matter field as introduced in [32, 35]. One may include a cosmological constant to maintain mass-energy density to be nonnegative and consider a Lorentzian manifold Mˆ = R × M with a static space-time metric gˆ = −f 2dt2 + g which satisfies the Einstein’s field equation

1 Ricˆ − Rˆgˆ + Λˆg = 8πGT, (1.1.1) 2

3 for the energy-momentum-stress tensor T = µf 2dt2 +pg of a perfect ﬂuid, where f

is independent of time and called a lapse function, µ and p are nonnegative, time-

independent mass-energy density and pressure of the perfect ﬂuid respectively and

G is the gravitational constant. Here we take the speed of light to be c = 1.

In fact, the energy density and pressure can be expressed in terms of the

Riemannian metric g and cosmological constant Λ:

R µ = g − Λ (1.1.2) 2

and n − 1 ∆ f n − 2 p = g − R + Λ, (1.1.3) n f 2(n − 1) g

where ∆g and Rg are the Laplace-Beltrami operator and scalar curvature of g respectively. Thus, by rewriting equation (1.1.1) in terms of g and f, we can see that a static space-time is completely described by the Riemannian manifold

(M n, g) and the smooth function f on it:

R 1 R ∇2f − Ric − g g f − ∆ f + g f g = 0. (1.1.4) g n − 1 n g n − 1

We will refer equation (1.1.4) as the static equation and call a complete Rie-

mannian manifold (M n, g) a static space (with perfect ﬂuid) if there exists a

smooth function f (6≡ 0) on M n such that f solves this equation.

The space-time is said to be vacuum, if the energy-momentum-stress tensor T

vanishes. That is,

µ = p = 0.

4 By equations (1.1.2) and (1.1.3), we have

R ∆ f + g f = 0. (1.1.5) g n − 1

Deﬁnition 1.1.1. A complete Riemannian manifold (M n, g) is said to be a vacuum static space if (1.1.4) reduces to

R ∇2f − Ric − g g f = 0. (1.1.6) g n − 1

It is very interesting to notice that the vacuum static equation (1.1.6) are also considered by Fischer and Marsden [29] in their study of the surjectivity of scalar curvature function from the space of Riemannian metrics (cf. [36, 54, 20]).

In fact, let gt = g + th be a deformation of g, where h ∈ S2(M) is a symmetric

2-tensor and t ∈ (−, ). The linearization of scalar curvature is

d 2 γgh := R(gt) = −∆g(trh) + δg h − Ric · h, dt t=0

2 where δg = −divg. And its L -formal adjoint is

∗ γg f = Hessgf − g∆gf − fRic, where f ∈ C∞(M).

∗ Suppose a smooth non-vanishing function f ∈ Ker γg . i.e. f 6≡ 0 solves

Hessgf − g∆gf − fRic = 0.

By taking trace, R ∆ f + g f = 0. g n − 1

5 Hence f solves the equation

R Hess f = Ric − g g f. g g n − 1

That means we can also deﬁne a Riemannian manifold to be vacuum static, if

∗ Ker γg 6= {0}.

After being deﬁned, we give some typical examples of vacuum static spaces as

follows:

• Ricci ﬂat manifold.

By taking f = constant 6= 0.

n • Sphere (S , gSn )

n n+1 2 2 2 Let S = {x ∈ R : x1 + x2 + ··· + xn+1 = 1}

and take f = xn+1|Sn , i.e. the height function.

n • Hyperbolic space (H , gHn )

Let Hn = {(x, t) ∈ Rn+1 : t2 − |x|2 = 1, t > 0, x ∈ Rn}

and take f = t| , i.e. the height function. Hn

As a type of special and important global solutions to the Einstein equation,

vacuum static spaces possess many interesting geometric properties and these

features play a crucial part in understanding the geometry of such spaces and their

fundamental role in general relativity. In particular, we are going to discuss a few

6 fundamental problems about vacuum static spaces and some related geometric

problems at the same time.

1.2 Notations and conventions

We adopt standard notations and conventions in this dissertation, but note

that, some of the conventions we adopted in the main part may be diﬀerent from

the one in the appendices.

In particular, in the main part of the dissertation, the Riemann curvature

tensor is deﬁned to be

i Rijkl = Rjkl (1.2.1)

and Ricci tensor is

k kl Rij = Rikj = g Rikjl. (1.2.2)

Also Schouten tensor is deﬁned to be

1 S = R − Rg , (1.2.3) ij ij 2(n − 1) ij

Cotten tensor is

Cijk = ∇iSjk − ∇jSik (1.2.4)

7 and Bach tensor is

1 B = −∇iC + SilW . (1.2.5) jk n − 2 ijk ijkl

1.3 Statements of main theorems

We will discuss the following three topics in this dissertation.

(i). Classiﬁcations of vacuum static spaces and Besse Conjecture.

For static spaces, in [35], Kobayashi and Obata (cf. [39] for n = 3) showed

that, nearby the hypersurface f −1(c) for a regular value c, a static metric g is

isometric to a warped metric of a constant curvature metric, provided that g is

locally conformally ﬂat. In [29], Fischer and Marsden raised the possibility of

identifying all compact vacuum static spaces. In fact, now one knows that in

dimension 3, besides ﬂat tori T 3 and round spheres S3, S1 × S2 is also a compact

1 2 vacuum static space. Later, in [27], other warped metrics on S ×r S were found to

be vacuum static. The open conjecture is that those, possibly moduli some ﬁnite

group, are all the compact vacuum static spaces. Please refer to [36, 54, 37, 38]

for progresses made in solving the classifying problem raised in [29]. In short the

classifying problem is solved [36, 38] for locally conformally ﬂat static spaces. But

1 √ 1 n−1 n−1 an easy calculation shows that S ( n−2 ) × E for Einstein manifolds E with scalar curvature (n − 1)(n − 2) are compact vacuum static spaces, which are not locally conformally ﬂat and therefore not accounted in [36], when n > 3.

8 The critical point equation is introduced for the Hilbert-Einstein action on the

space of conformal classes represented by Riemannian metrics with unit volume

and constant Ricci scalar curvature in [5] in an attempt to more eﬃciently identify

Einstein metrics in two steps. Formally the Euler-Lagrangian equation of Hilbert-

Einstein action on the space of Riemannian metrics with unit volume and constant

Ricci scalar curvature is

1 1 Ric − Rg = ∇2f − (Ric − Rg)f. n n − 1

It may look more apparent that it is related to the static equations (1.1.4) and

(1.1.6) if we replace f by f − 1 and consider the equation

1 1 ∇2f − (Ric − Rg)f − Rg = 0. (1.3.1) n − 1 n(n − 1)

A complete Riemannian manifold (M n, g) (n ≥ 3) of constant Ricci scalar

curvature is said to be CPE if it admits a smooth solution f (6≡ 0) to the critical

point equation (1.3.1) (cf. [5, 33, 13, 14]). In [5] it conjectured that a CPE metric

is always Einstein.

Conjecture 1.3.1. A CPE metric is always Einstein.

It is clear that (M n, g) is Einstein if it admits a trivial solution f ≡ −1.

Other CPE metrics with constant function f are Ricci ﬂat metrics. g is isometric to a round sphere metric if it is a Einstein CPE metric with a non-constant function f. Hence Conjecture 1.3.1 really says that a CPE metric with a non- constant solution f to (1.3.1) is isometric to a round sphere metric. Lafontaine in

9 [38] veriﬁed Conjecture 1.3.1 when assuming metrics are locally conformally ﬂat.

Recently Chang, Hwang, and Yun in [14] veriﬁed Conjecture 1.3.1 for metrics of

harmonic curvature.

Recently in [18, 17] the authors studied Bach ﬂat gradient Ricci solitons. Based

on the similar idea from [18, 17] we are able to solve the classifying problem raised

in [29] for Bach ﬂat vacuum static spaces in general dimensions. It is worth to

1 √ 1 n−1 mention that we will include in our list the vacuum static spaces S ( n−2 )×E that were not accounted in the lists given in [35, 36] when n > 3. In the mean

time, we are also able to verify Conjecture 1.3.1 for Bach ﬂat CPE metrics.

Particularly in dimension 3, we establish an intriguing integral identity

Z p Z f pC = − f p|C|2 (1.3.2) M 3 4 M 3

ijk where C = Cijk, is the complete divergence and Cijk is the Cotton tensor, on a

compact 3-manifold (M 3, g) admitting non-constant solution f to the equation

fRij = fi,j + Φgij (1.3.3)

for some function Φ. Therefore we are able to obtain stronger results for both

static metrics and CPE metrics in dimension 3. For vacuum static spaces, based

on the solutions to the corresponding ODE given in [36], we are able to solve the

classifying problem raised in [29].

Theorem 1.3.2. Suppose that (M 3, g) is compact vacuum static space with no

10 boundary with nonnegative complete divergence C of the Cotton tensor. Then it must be one of the following up to a ﬁnite quotient:

• Flat 3-manifolds;

• S3;

• S1 × S2;

1 2 2 2 • S ×r S for g = ds + r (s)gS2 , where r(s) is a periodic function given in

Eample 4 in [36].

Regarding Conjecture 1.3.1, based on [38, 14], we prove the following:

Theorem 1.3.3. Conjecture 1.3.1 holds for compact Riemannian 3-manifold with

no boundary with nonnegative complete divergence C of the Cotton tensor.

(ii). Local rigidity phenomena of vacuum static spaces.

The positive mass theorem [51, 52, 58] is a fascinating theorem that has been

pivotal in mathematical relativity. The global scalar curvature rigidity of the

Euclidean space Rn is at the core of the positive mass theorem for asymptotically

ﬂat manifolds. Analogously the global scalar curvature rigidity of hyperbolic

space Hn is at the core of the positive mass theorem for asymptotically hyperbolic

manifolds [43, 2, 57, 19, 1]. This led Min-Oo in [44] to conjecture that some global

n scalar curvature rigidity should also hold for the round hemisphere S+.

11 In the paper [29], Fischer and Marsden studied the deformations of scalar curvature and introduced the notion of static spaces that incidentally is the same notion of vacuum static spaces introduced in [32, 35] in mathematical relativity.

Fischer and Marsden showed the local surjectivity for the scalar curvature as a map from the space of metrics to the space of functions at a non-static metric on a closed manifold. Corvino in [20] considered compactly supported deformations of metrics and extended the local surjectivity result. In [29], Fischer and Marsden also observed the local scalar curvature rigidity of closed flat manifolds. Their local scalar curvature rigidity states that, on a closed flat manifold, any metric with nonnegative scalar curvature that is sufficiently close to a flat metric has to be isometric to the flat metric. This dichotomy of local surjectivity and local rigidity about scalar curvature seems extremely intriguing. One wonders if such dichotomy holds in general based on the work of Corvino [20].

Min-Oo’s conjecture attracted a lot of attentions among geometric analysts.

It was remarkable that Brendle, Marques and Neves in [9] (see also [24] for a later developement) discovered that there is even no local scalar curvature rigidity of the round hemispheres and constructed counter-examples to Min-Oo’s conjecture.

Later, in a subsequent paper [10], Brendle and Marques established the local scalar curvature rigidity of round spherical caps of some appropriate size (cf. [21, 42] for a better estimate on the size). These developments inspire us to study the local

12 scalar curvature rigidity of general vacuum static spaces.

Theorem 1.3.4. Suppose that (M n, g,¯ f) (n ≥ 3) is a vacuum static space. Let p ∈ M and f(p) 6= 0. Then there exist r0 > 0 such that, for each geodesic ball

Br(p), there exists ε0 > 0 such that, for any metric g on Br(p) satisfying,

• g =g ¯ on ∂Br(p);

• Rg ≥ Rg¯ in Br(p);

• Hg ≥ Hg¯ on ∂Br(p);

• ||g − g¯|| 2 < ε , C (Br(p)) 0

there exists a diﬀeomorphism ϕ : Br(p) → Br(p) such that ϕ|∂Br(p) = id and

∗ ϕ g =g ¯ in Br(p), provided that r < r0.

This conﬁrms the dichotomy of local surjectivity and local rigidity about scalar curvature in general in the light of the local surjectivity work of Corvino in [20].

In fact Theorem 1.3.4 is stronger than a local rigidity, because it allows the metric g diﬀer from g¯ up to the boundary as long as the mean curvature is not less pointwisely on the boundary. Therefore Theorem 1.3.4 is a local rigidity of bounded domains in vacuum static spaces generalizing the scalar curvature rigidity of bounded domains in particular vacuum static spaces established in [41, 50, 10].

Space forms are the special vacuum static spaces. In Section 3.2 we will discuss the local scalar curvature rigidity of each space form. In the Euclidean cases, we

13 are able to obtain the local scalar curvature rigidity, which may be considered as a local version of the rigidity results of [41, 50]. In the hyperbolic cases, it seems that our local scalar curvature rigidity is new and addresses the rigidity problem of the positive mass theorem for metrics with corners that is raised in [6].

H Theorem 1.3.5. For n ≥ 3, let Br be the geodesic ball with radius r > 0 in

n H hyperbolic space H . There exists an ε0 > 0, such that, for any metric g on Br satisfying

H • g = gHn on ∂Br ;

H • Rg ≥ −n(n − 1) in Br ;

H • Hg ≥ Hg n on ∂B ; H r

• ||g − g n || 2 < ε0, H C (BrH)

H H ∗ H there is a diﬀeomorphism ϕ : Br → Br such that ϕ g = gHn in Br and ϕ = id

H on ∂Br .

In the spherical cases, the local scalar curvature rigidity is established in [10,

21]. It remains intriguing to ﬁnd out whether one can identity the size of the spherical cap on which the local scalar curvature rigidity ﬁrst fails to be valid.

More interestingly it remains open that whether there is global scalar curvature rigidity for bounded domains in the hyperbolic cases and bounded domains of appropriate size in the spherical cases.

14 It has been noticed that it is an interesting intermediate step to consider scalar curvature rigidity among conformal deformations. Hang and Wang in [30], for instance, obtained the global scalar curvature rigidity among conformal metrics for the round hemisphere (a weaker version of Min-Oo’s conjecture). They in fact also showed that the local scalar curvature rigidity even among conformal metrics is no longer true for the round metric on any spherical cap bigger than the hemisphere. We observe that the rigidity for conformal deformations in [30] can be extended for general vacuum static spaces with positive scalar curvature.

Theorem 1.3.6. Let (M n, g,¯ f) be a complete n-dimensional vacuum static space

+ n with Rg¯ > 0 (n ≥ 2). Assume the level set Ω = {x ∈ M : f(x) > 0} is a pre-compact subset in M. Then if a metric g ∈g¯ on M satisﬁes that

+ • Rg ≥ Rg¯ in Ω ,

• g and g¯ induced the same metric on ∂Ω+, and

+ • Hg = Hg¯ on ∂Ω , then g =g ¯. On the other hand, for any open domain Ω in M n that contains Ω+, there is a smooth metric g ∈g¯ such that

• Rg ≥ Rg¯ in Ω, Rg > Rg¯ at some point in Ω, and

• supp(g − g¯) ⊂ Ω.

15 Our proof of the rigidity in Theorem 1.3.6 only uses the maximum principles,

which seems to be more straightforward than the proof used in [30] in the case

of the round hemisphere. ( Motivated by a question proposed by Escobar ([28]),

Barbosa, Mirandola and Vitorio found an elegant integral identity and with the

aid of which they proved the rigidity part independently in a more general setting

in [4].) The construction in the second part of Theorem 1.3.6 is based on the

idea in [9]. For a detailed history of the study of the scalar curvature rigidity

phenomena and the solution of Min-Oo’s conjecture, readers are referred to the

excellent survey article [8] of Brendle.

(iii). Brown-York mass and compactly conformal deformations.

We have showed that the conformal rigidity holds in vacuum static spaces. On the other hand, as a comparison, it would be interesting to know what happens if we require deformations decrease scalar curvature instead of increase it as what we discussed previously. In fact, not many works were known with respect to this question to the best of the authors knowledge. Among them, Lohkamp’s result is the most well-known one (see [40]). He showed that there is a generic compact deformation which decrease scalar curvature for an arbitrary Riemannian manifold. However, it was not clear that such a deformation can be realized within conformal classes or not due to Lohkamp’s proof. So we would like to ask the question: does such a conformal deformation exists? Moreover, we can ask

16 similar questions for Corvino’s constructions (see [20]).

Before we answer these questions, we introduce the following well-known notion

due to Brown and York (see [11, 12]). For the purpose of our article, we restrict

metrics in the conformal class of the background metric.

Deﬁnition 1.3.7. For n ≥ 2, let (Ωn, g¯) be an n-dimensional compact Rieman-

nian manifold with smooth boundary ∂Ω. Then for any metric g ∈ [¯g] on Ω with

g =g ¯ on ∂Ω, the Brown-York mass relative to g¯ is deﬁned to be the quantity

Z mBY (∂Ω, g¯; g) = (Hg¯ − Hg) dσg¯, ∂Ω

where Hg¯ and Hg are mean curvatures of g¯ and g of ∂Ω with respect to the outward

normals respectively.

Another notion we will used frequently is the following one:

Deﬁnition 1.3.8. We denote the ﬁrst Dirichlet eigenvalue of the Schrödinger

type operator

Rg¯ Lg¯ := −∆g¯ − n − 1

on the domain Ω ⊂ M as

R Ω ϕ Lg¯ϕ dvg¯ Λ1(Ω, g¯) = inf ϕ∈C∞(Ω) R 2 0 Ω ϕ dvg¯ and φ 6≡ 0 is a corresponding eigenfunction, if

Lg¯φ = Λ1(Ω, g¯)φ.

17 Now we can state our main theorem as follow, which can be viewed as an

analogue of Shi and Tam’s positive mass theorem (cf. [50]):

Theorem 1.3.9. For n ≥ 2, let (M n, g¯) be an n-dimensional compact Riemannian

manifold with smooth boundary. Suppose the ﬁrst Dirichlet eigenvalue of Lg¯ on

M satisﬁes that

Λ1(M, g¯) > 0.

(I) : For any metric g+ ∈ [¯g] with

g+ =g ¯

on ∂M and in addition,

Rg+ ≥ Rg¯ on M,

we have

mBY (∂M, g¯; g+) ≥ 0

and equality holds if and only if when g+ =g ¯ on M.

(II): For any δ ∈ (0, 1) and any metric g− ∈ [¯g] with

 4 n+2  √ (n − 1)δΛ1(M, g¯)  n 1 + when max Rg¯ > 0  maxM Rg¯ M ||g−||C0(M,g¯) < α :=   + ∞, when max Rg¯ ≤ 0 M and

g− =g ¯

18 on ∂M also in addition,

Rg− ≤ Rg¯ on M we have

mBY (∂M, g¯; g−) ≤ 0

and equality holds if and only if when g− =g ¯ on M.

Remark 1.3.10. When g¯ is ﬂat and g is asymptotically ﬂat, the presumption

Rg ≥ Rg¯ = 0 is usually referred as Dominant Energy Condition in general relativity.

If the diameter of the manifold is suﬃciently small, the ﬁrst eigenvalue of

Laplacian can be suﬃciently large. Hence the above theorem always holds on a suﬃciently small domain in an arbitrary Riemannian manifold:

Corollary 1.3.11. For n ≥ 2, let (M n, g¯) be an n-dimensional Riemannian manifold. For any p ∈ M, there exists an r0 > 0, such that for any domain Ω ⊂ M

contains p with Ω ⊂ Br0 (p), the metric g ∈ [¯g] is a conformal deformation of g¯

supported in Ω which satisﬁes either

Rg ≥ Rg¯

Rg ≤ Rg¯

19 with ||g − g¯||C0(M,g¯) suﬃciently small on M. Then we have g ≡ g¯ on M. In

particular, this implies that g¯ is an isolated solution of the problem    R(g) = Rg¯   g ∈ [¯g]     supp(g − g¯) ⊂ Ω, if the diameter of Ω is suﬃciently small.

This corollary shows that in particular there is no conformal perturbation increase or decrease scalar curvature with suﬃciently small support. It suggests that deformations result in [20, 40] is sharp in the sense that the generic deformations out of conformal classes are necessary.

Another interesting conclusion is that Brown-York mass behave perfect on manifolds with nonpositive scalar curvature, which suggests that they are much more rigid in terms of compactly conformal deformations.

Corollary 1.3.12. For n ≥ 2, let (M, g¯) be a Riemannian manifold with scalar curvature Rg¯ ≤ 0. Then for any compactly contained domain Ω & M and any metric g ∈ [¯g] on Ω with

g =g ¯ on ∂Ω.

(I): Suppose

Rg ≥ Rg¯ on Ω,

20 then

mBY (∂Ω, g¯; g) ≥ 0.

(II): Suppose

Rg ≤ Rg¯ on Ω,

then

mBY (∂Ω, g¯; g) ≤ 0.

In either case, the equality holds if and only g =g ¯ on Ω.

In particular, these imply that g =g ¯ is the unique solution to the following

problem    R(g) = Rg¯   g ∈ [¯g]     supp(g − g¯) ⊂ Ω. It is natural to ask what happens to manifolds with positive scalar curvature?

We have already known that if the domain is suﬃciently small, rigidity phenomena occur. So it is interesting to investigate the phenomena in large domains. In fact, we have the following deformation result, which suggests that the vanishing of

Brown-York mass won’t imply the conformal rigidity as it does previously:

Theorem 1.3.13. For n ≥ 2, let (M, g¯) be a Riemannian manifold with scalar curvature Rg¯ > 0. Suppose there exists a compactly contained domain Ω ⊂ M

21 with the ﬁrst Dirichlet eigenvalue of Lg¯ satisﬁes that

Λ1(Ω, g¯) < 0.

Then there are smooth metrics g+, g− ∈ [¯g] such that

supp (¯g − g+), supp (¯g − g−) ⊂ Ω and scalar curvatures satisfy that

Rg+ ≥ Rg¯ and

Rg− ≤ Rg¯ with strict inequality holding inside Ω respectively.

If the manifold is closed, such domains always exist. Hence we can always have the following deformation result:

Corollary 1.3.14. For n ≥ 2, let (M, g¯) be a closed Riemannian manifold with scalar curvature Rg¯ > 0. Then there is a compactly contained domain Ω ⊂ M and smooth metrics g+, g− ∈ [¯g] such that

supp (¯g − g+), supp (¯g − g−) ⊂ Ω and scalar curvatures satisfy that

Rg+ ≥ Rg¯

22 and

Rg− ≤ Rg¯ respectively with strict inequality holding inside Ω.

However, if the manifold (M, g¯) is complete but noncompact, whether the positivity of scalar curvatures implies the existence of such domains is not clear.

But if (M, g¯) has quadratic volume growth, such domains do exists and thus we have the following deformation result:

Corollary 1.3.15. For n ≥ 2, let (M, g¯) be a complete noncompact Riemannian manifold with scalar curvature Rg¯ > Q > 0, where Q is a positive constant.

Suppose that (M, g¯) has quadratic volume growth, then there is a compactly contained domain Ω ⊂ M and smooth metrics g+, g− ∈ [¯g] such that

supp (¯g − g+), supp (¯g − g−) ⊂ Ω and scalar curvatures satisfy that

Rg+ ≥ Rg¯ and

Rg− ≤ Rg¯ respectively with strict inequality holding inside Ω.

The thesis is organized as follow: We are going to discuss the classiﬁcations of vacuum static spaces and Besse conjecture in Chapter 2 and local rigidity

23 phenomena of scalar curvature in Chapter 3, Brown-York mass and compactly conformal deformations in the last Chapter.

24 Chapter 2

Classiﬁcations of vacuum static spaces

2.1 Bach ﬂatness and vanishing of D-tensor

In this section we will use Bach ﬂatness to force the vanishing of the augmented

Cotton tensor D as the authors did for gradient Ricci solitons in [18, 17]. To introduce the Bach curvature tensor of a Riemannian manifold (M n, g), we recall the well known decomposition of Riemann curvature tensor.

1 R = W + (S g − S g − S g + S g ) (2.1.1) ijkl ijkl n − 2 ik jl il jk jk il jl ik where Rijkl is the Riemann curvature tensor, Wijkl is the Weyl curvature tensor,

1 S = R − Rg ij ij 2(n − 1) ij

25 j i is Schouten curvature tensor, Rij = Rijk is Ricci curvature tensor, and R = Ri is the Ricci scalar curvature. Then the Cotton tensor C is given as:

Cijk = Sjk,i − Sik,j. (2.1.2)

The following consequence of Bianchi identity is often useful:

n − 3 W l = − C (2.1.3) ijkl, n − 2 ijk

when n ≥ 4. We are now ready to introduce the Bach curvature tensor on a

Riemannian manifold (M n, g) as follows:

1 1 B = W li + SilW , (2.1.4) jk n − 3 ijkl, n − 2 ijkl when n ≥ 4. Using (2.1.3) we may extend the deﬁnition of Bach tensor in dimensions including 3 as follows:

1 B = (−C i + SilW ). (2.1.5) jk n − 2 ijk, ijkl

Finally, as in [35] and [18, 17], we deﬁne the following augmented Cotton tensor,

which will play an important role in the calculations in this paper.

2 l Dijk = f Cijk − fWijklf . (2.1.6)

It is easy to see that Dijk is anti-symmetric in indices i and j. In fact the following

is a key observation (1.11) in [35]. In order to treat both static equations (1.1.4)

26 and critical point equation (1.3.1) at the same time, we need to rewrite them in a

uniﬁed way. We ﬁrst rewrite the static equation (1.1.4) as follows:

1 n − 2 fS = ∇2f − (∆f − Rf)g. (2.1.7) n 2(n − 1)

We then rewrite the critical point equation (1.3.1) as follows:

Rf R fS = ∇2f + ( − )g. (2.1.8) 2(n − 1) n(n − 1)

In summary we will write both (2.1.7) and (2.1.8) in following form

fS = ∇2f + Φg (2.1.9)

for a function Φ (this Φ is diﬀerent from that in (1.3.3)).

Proposition 2.1.1. Suppose that (M n, g) is a Riemannain manifold admitting a

smooth solution f to the equation (2.1.9). Then

1 Dijk = Alt{(n − 1)fi,kfj + Ψjgik}, (2.1.10) n − 2 i,j where Alt means anti-symmetrizing with the indices i and j, and i,j

l Ψj = −(n − 2)fΦj + fj,lf + nΦfj. (2.1.11)

Proof. It is a straightforward calculation based on the equation (2.1.9) and the deﬁnition of Dijk (cf. [35]). For the convenience of readers we include some

27 calculations here. First we calculate

2 2 f Cijk = f (Sjk,i − Sik,j)

= f(fk,ji − fk,ij) − f(Sjkfi − Sikfj) + f(Φigjk − Φjgik)

Then recall the Ricci identity

l l fk,ji − fk,ij = flR kji = Rijklf and conclude that

1 1 f − f = W f l + (S g − S g )f l + (S f − S f ). k,ji k,ij ijkl n − 2 jl ik il jk n − 2 ik j jk i

Hence we obtain

2 l ˜ (n − 2)(f Cijk − fWijklf ) = Alt{(n − 1)fSikfj + gikΨj} i,j for

˜ l Ψj = −(n − 2)fΦj + fSjlf .

From this, using the equation (2.1.9), we complete the proof of (2.1.10).

Remark 2.1.2. Note that

1 n − 2 Φ = ( Rf − ∆f) and Ψ = f f l − ∆ff (2.1.12) n 2(n − 1) j j,l j for static metrics and

Rf R Φ = − and Ψ = f f l − ∆ff (2.1.13) 2(n − 1) n(n − 1) j j,l j

28 for CPE metrics. It is very intriguing to see that Ψ is the same in both cases.

Then we can rewrite the Bach tensor as follows:

Proposition 2.1.3. Suppose that (M n, g) is a Riemannian manifold admitting a

smooth solution f to the equation (2.1.9). Then

D n − 3 f l f if l (n − 2)B = −∇i( ijk ) + C + W . (2.1.14) jk f 2 n − 2 lkj f ijkl f 2

Proof. It is straightforward to calculate that, from the deﬁnition (2.1.6),

f l 1 (n − 2)B = −C i + SilW = −∇i(W + D ) + SilW jk ijk, ijkl ijkl f f 2 ijk ijkl D f l f i,l f if l = −∇i( ijk ) − W i + W (Sil − + ) f 2 ijkl, f ijkl f f 2 D n − 3 f l f if l = −∇i( ijk ) + C + W . f 2 n − 2 lkj f ijkl f 2

Now, as a consequence of (2.1.14), we can state one of the key identities in

this paper. To state that we introduce some notations. We will denote the level

set

n Mc = {x ∈ M : f(x) = c}

and

n Mc1,c2 = {x ∈ M : c1 < f(x) < c2}.

Proposition 2.1.4. Suppose that (M n, g) is a Riemannian manifold admitting a

smooth solution f to the equation (2.1.9). Let c1 and c2 be two regular values for

29 the function f and two level sets Mc1 and Mc2 be compact. Then, for all p ≥ 2, we have the following integral identity:

Z 1 Z f pB f jf k = f p−2|D|2. (2.1.15) jk 2(n − 1) Mc1,c2 Mc1,c2

Proof. By the anti-symmetries of Wijkl, Cijk and Dijk, from (2.1.14) one gets

1 f jf k B f jf k = − D i . jk n − 2 ijk, f 2

Applying integration by parts, we get

Z Z p j k i p−2 j k (n − 2) f Bjkf f = Dijk∇ (f f f ). Mc1,c2 Mc1,c2 Again, due to the anti-symmetries and trace-free properties of Cotton tensor C and the augmented Cotton tensor D, we arrive at (2.1.15)

Z Z p j k p−2 i,k j (n − 2) f Bjkf f = f Dijkf f Mc1,c2 Mc1,c2 n − 2 Z = f p−2|D|2. 2(n − 1) Mc1,c2

Consequently we obtain the following important initial step to understand the geometric structure of a Riemannain manifold admitting a smooth solution to the equation (2.1.9).

Corollary 2.1.5. The augmented Cotton tensor D vanishes identically on a Bach

ﬂat manifold admitting a smooth non-constant solution f to the equation (2.1.9), provided that each level set f −1(c) is compact for any regular value c.

30 2.2 The geometry of level surfaces

In this section, based on Corollary 2.1.5, we investigate geometric structure

of a Bach ﬂat manifold admitting a smooth non-constant solution f to the equation (1.1.4) or (1.3.1). To facilitate our local calculations we need to choose local frames and set notations.

For a regular value c, we denote the level set f −1(c) as Σ, W := |∇f|2, and

∇f en := |∇f| as the unit normal to Σ. We then choose an orthonormal frame

{e1, e2, ··· , en−1, en}

along Σ. We will use Greek letters to denote the index from 1 to n − 1, while

Latin letters for the index from 1 to n. Then the second fundamental form of Σ

f h = h∇ e , e i = −he , ∇ e i = − α,β , (2.2.1) αβ eα β n β eα n |∇f|

the mean curvature is

αβ − 1 H = g hαβ = W 2 (fn,n − ∆f), (2.2.2)

and the square of the norm of the second fundamental form is

n−1 2 αβ −1 X 2 |A| = hαβh = W |fα,β| . (2.2.3) α,β=1

31 Furthermore

n−1 Σ 2 X 2 |∇ W | = 4W |fn,α| (2.2.4) α=1 and

2 2 |∇nW | = 4W |fn,n| . (2.2.5)

Now we are ready to prove another key identity in this paper.

Proposition 2.2.1. Suppose that (M n, g) is a Riemannian manifold admitting a non-constant solution to either (1.1.4) or (1.3.1). Then the following identity holds:

(n − 1)2 H n − 1 |D|2 = 2 W 2|A − gΣ|2 + |∇ΣW |2. (2.2.6) (n − 2)2 n − 1 2(n − 2)

Proof. By Proposition 2.1.1, we have

2 2 2 2 2 2 2 2 i k,j (n − 2) |D| = 2(n − 1) |∇f| |∇ f| + 2(n − 1)|Ψ| − 2(n − 1) fk,if f fj

i j + 4(n − 1)(∆f∇f · Ψ − fi,jf Ψ ) H = 2(n − 1)2|∇f|4|A − gΣ|2 + 2(n − 1)|∇f|4H2 n − 1 n−1 2 2 X 2 + 2(n − 1) |∇f| |fn,α| α=1

2 i j + 2(n − 1)|Ψ| + 4(n − 1)(∆f∇f · Ψ − fi,jf Ψ )

Because

n−1 n−1 2 2 X 2 X 2 2 |∇ f| = |fα,β| + 2 |fn,α| + |fn,n| . α,β=1 α=1

32 We also calculate, due to Remark 2.1.2,

n−1 2 2 X 2 2 |Ψ| = |∇f| ( |fn,α| + |fn,n − ∆f| ) α=1 and

i j 2 ∆f∇f · Ψ − fi,jf Ψ = −|∇f| (fn,n − ∆f)(fn,n − ∆f)

n−1 2 X 2 − |∇f| |fn,α| α=1

Therefore

(n − 2)2 H n − 2 |D|2 = 2(n − 1)W 2|A − gΣ|2 + |∇ΣW |2 n − 1 n − 1 2

An immediate consequence is following:

Corollary 2.2.2. Suppose that (M n, g) (n ≥ 3) is a Riemannian manifold admitting a non-constant solution to either (1.1.4) or (1.3.1). And suppose that the augmented Cotton tensor D vanishes. Then the level set Σ is umbilical and the mean curvature H is constant.

Proof. By the assumption we know that the solution f can not be a constant.

Therefore it follows from Lemma 2.2.1 that the level set Σ is umbilical and W is a constant along Σ in the light of (2.2.6). In fact

n−1 n−1 Σ 2 X 2 X 2 |∇ W | = |∇αW | = 4W |fn,α| . α=1 α=1

33 Hence, according to the equation (2.1.9), we conclude that Rαn = 0, for α =

1, 2, ··· , n − 1. On the other hand, by contracting the Codazzi equations we get

n − 2 0 = R = ∇ΣH, α = 1, 2, . . . , n − 1. αn n − 1 α

Therefore the mean curvature H is constant along Σ.

Next we show the constancy of R and ∆f along Σ.

Lemma 2.2.3. Suppose that (M n, g) (n ≥ 3) is a static space or a CPE metric

with a non-constant function f. Then

∇ΣR = ∇Σ∆f = 0. (2.2.7)

Proof. The statement of this lemma is obviously true for a CPE metric. For a static metric, taking divergence of the static equation (1.1.4), we have

n d(Rf + (n − 1)∆f) = fdR, (2.2.8) 2

which implies n ( − 1)fdR = Rdf + (n − 1)d∆f. (2.2.9) 2

Taking exterior diﬀerential of the two sides of the above equation, we get df ∧dR =

0. Hence, by Cartan’s lemma, there exists a smooth function φ such that dR = φdf,

Σ Σ which implies ∇α R = ∇αR = φ∇αf = 0, i.e. ∇ R = 0. Consequently, in the light of (2.2.9), one also gets ∇Σ∆f = 0.

34 Consequently we know that the level set Σ is of constant scalar curvature if the augmented Cotton tensor vanishes.

Corollary 2.2.4. Suppose that (M n, g) (n ≥ 3) is a static space or a CPE metric

with a non-constant function f. And suppose that the augmented Cotton tensor

D vanishes identically. Then the level set Σ is of constant scalar curvature.

Proof. Recall Gauss equation

Σ 2 2 R = R − 2Rnn + H − |A| .

Hence it suﬃces to show that Rnn to be constant along Σ in the light of Corollary

2.2.2 and Lemma 2.2.3. To do that we ﬁrst realize that fn,n is constant from

(2.2.2). Then the conclusion follows from the static equation (1.1.4) or critical

point equation (1.3.1).

Working a little bit harder, we can show that in fact the level set Σ is Einstein

when the augmented Cotton tensor vanishes.

Proposition 2.2.5. Suppose that (M n, g) (n ≥ 3) is a static space or a CPE metric with a non-constant function f. And suppose that the augmented Cotton tensor

D vanishes identically. Then the level set Σ is Einstein.

Proof. We start with the assumption that D = 0. Hence, from the deﬁnition

(2.1.6), we have

l i i Wijklf f = fCijkf .

35 On the other hand, from Bach ﬂatness and Proposition 2.1.3, we also have

n − 3 W f if l = − fC f i. ijkl n − 2 ijk

i l Therefore we can conclude that Wijklf f = 0, that is, Wnjkn = 0. Using the

Riemann curvature decomposition we derive

1 1 1 R = W + R + (S − R)g αnβn αnβn n − 2 αβ n − 2 nn 2(n − 1) αβ 1 1 1 = R + (R − R)g . n − 2 αβ n − 2 nn n − 1 αβ

Meanwhile, from the equation (2.1.9), we obtain

∇ ∇ f 1 ∆f R = α β + (R − )g αβ f n f αβ |∇f| 1 ∆f = − h + (R − )g f αβ n f αβ 1 ∆f H |∇f| = ( (R − ) − )g . n f n − 1 f αβ

Finally, using Gauss equation,

Σ γ Rαβ = Rαβ − Rαnβn + Hhαβ − hαγh β

we can conclude that Σ is Einstein by Schur’s lemma when n ≥ 4. Notice that

Corollary 2.2.4 implies the proposition when n = 3. Thus the proof is complete.

We now summarize what we have achieved in the following local splitting result

for the geometric structure of a static metric or a CPE metric (cf. Theorem 3.1

in [35]).

36 Theorem 2.2.6. Suppose that (M n, g) is a static space or CPE manifold with non-constant function f and compact level set f −1(c) for a given regular value c.

And assume it is Bach ﬂat. Then

2 2 g = ds + (r(s)) gE,

−1 df 2 nearby the level set f (c), where ds = |df| , (r(s)) gE = g|f −1(c) and gE is an

Einstein metric.

Consequently, based on the solutions to the corresponding ODE given in [36], one gets the classiﬁcation theorem for Bach ﬂat vacuum static spaces. Notice that the function f and the warping factor r still satisfy the same ODE system:   00 r0 0 R f + (n − 1) r f + n−1 f = 0

 0 0 00 r f − r f = 0 which is (1.9) in [36]. It is remarkable that Kobayashi was able to ﬁnd the integrals and completely solved it. The solutions depend on the constants R,

R a = rn−1r00 + rn, n(n − 1) and R 2a k = (r0)2 + r2 + r2−n. n(n − 1) n − 2

The horizontal slice E is Einstein with Ric = (n − 2)kgE here.

37 Theorem 2.2.7. Let (M n, g, f) be a Bach ﬂat vacuum static space with compact level sets (n ≥ 3). Then up to a ﬁnite quotient and appropriate scaling,

(i) f is a non-zero constant if and only if M is Ricci ﬂat;

(ii) f is non-constant if and only if M is isometric to

• Sn;

• Hn;

• the warped product cases.

In the warped product cases, we can divide again into compact and non-compact

1 2 2 ones. For the compact ones S ×r E with metric g = ds + (r(s)) gE, r(s) appears

to be one of the following:

• r(s) is a constant and E is an arbitrary compact Einstein manifold of positive

scalar curvature without boundary (cf. Example 2 in [36]);

• r(s) is non-constant and periodic and E is an arbitrary compact Einstein

manifold of positive scalar curvature without boundary (cf. Example 4 in

[36]).

2 2 For the non-compact ones R ×r E with metric g = ds + (r(s)) gE, r(s) appears to be one of the following:

38 • r(s) is a constant and E is an arbitrary compact Einstein manifold without

boundary (cf. Example 1 in [36]);

• r(s) is non-constant and peroidic and E is an arbitrary compact Einstein

manifold of positive scalar curvature without boundary (cf. Example 3 in

[36]);

• r(s) is given in Proposition 2.5 in [36] and E is an arbitrary compact Ein-

stein manifold without boundary (cf. Example 5 in [36]) .

Remark 2.2.8. We would like to mention again, since we only assume Bach ﬂatness,

our list includes the warped metric where the level sets are only Einstein instead

of constant curvature as in [36, 38] .

On the other hand, as a consequence of Theorem 2.2.6, a Bach ﬂat CPE metric

turns out to be of harmonic Riemann curvature. Namely,

Lemma 2.2.9. Suppose the metric g is a CPE metric satisfying assumptions in

Theorem 2.2.6. Then the Cotton tensor C of g vanishes identically and therefore

g is of harmonic Riemann curvature.

Proof. We simply choose a local coordinate system {∂1, ∂2, ··· , ∂n−1, ∂n = ∂s} and calculate directly. It is easily seen that

Cαβγ = Cαβn = Cnβn = 0.

39 The only term that needs some eﬀort is Cnβγ, which in fact is seen to be zero from

(2.1.14) and the fact that both Bach tensor and the augmented Cotton tensor

D are identically zero. Notice that Wnjkn is known to be identically zero from

the proof of Proposition 2.2.5. To see the harmonicity of Riemann curvature we

calculate as follows:

1 R l = W l + (S g − S g − S g + S g ) l ijkl, ijkl, n − 2 ik jl il jk jk il jl ik , n − 3 1 = − C + (S − S ) n − 2 ijk n − 2 ik,j jk,i

= −Cijk = 0 using the fact that the Ricci scalar curvature R is constant.

Then, using the result in [14] (also [38] for n = 3), we can verify Conjecture

1.3.1 for Bach ﬂat manifolds.

Theorem 2.2.10. Suppose that (M n, g) (n ≥ 3) is Bach ﬂat CPE manifold

admitting a non-constant solution to (1.3.1). Then (M n, g) is isometric to a

round sphere.

2.3 3-dimensional cases

In dimension 3 we recall that the Bach tensor is given as the divergence of the

Cotton tensor in (2.1.5). What we will do in this section is to establish another

40 integral identity on compact manifold with a static metric or a CPE metric. Then

ij we will be able to conclude that the full divergence Bij, of the Bach tensor (the

ijk full divergence Cijk, of the Cotton tensor) vanishes if and only if the Cotton tensor vanishes in dimension 3 for a static metric as well as a CPE metric on a compact manifold.

Proposition 2.3.1. Suppose that (M n, g) (n ≥ 3) is a compact Riemannian manifold with no boundary admitting a non-constant smooth solution to (2.1.9). Then, for any p ≥ 2, we have the following integral identity:

Z Z p ij p(n − 4) p−2 f Bij, = − f D · C. (2.3.1) M 2(n − 1)(n − 2) M

Proof. First, applying integration by parts twice, we get

Z Z Z p i j 1 p+2 ij 1 p+1 i,j f Bijf f = f Bij, − f Bijf . (2.3.2) M (p + 1)(p + 2) M p + 1 M

Then we use Proposition 2.1.3 to calculate the second term in the right hand side of the above equation. Namely,

Z Z Z p+1 i,j p+1 k Dkij i,j n − 3 p k i,j (n − 2) f Bijf = − f ∇ ( 2 )f + f Ckijf f M M f n − 2 M Z p−1 k l i,j + f Wikljf f f . M

Now we deal with each term separately. For the ﬁrst term, we perform once again integrating by part and get:

Z Z Z p+1 k Dkij i,j (n − 2)(p + 2) p−2 2 1 p f ∇ ( 2 )f = f |D| − f D · C. M f 2(n − 1) M 2 M

41 For the second term we simply use Proposition 2.1.1: Z Z p i,j k n − 2 p f Ckijf f = − f D · C. M 2(n − 1) M

And for the last term, we use the deﬁnition of Bach tensor and again perform

more integrating by part: Z Z Z p−1 k l i,j p i j n − 2 p f Wkijlf f f = (n − 2) f Bijf f − f D · C. M M 2(n − 1) M

Combining all the three terms together, we get Z Z Z p+1 i,j n − 4 p p i j f Bijf = − f D · C − (p + 1) f Bijf f , (2.3.3) M 2(n − 1)(n − 2) M M

where we have applied Proposition 2.1.4. Going back and rewriting (2.3.2) as

follows: Z Z Z 1 p+2 ij p i j p+1 i,j f Bij, = (p + 1) f Bijf f + f Bijf , p + 2 M M M which implies, from (2.3.3), Z Z 1 p+2 ij n − 4 p f Bij, = − f D · C. p + 2 M 2(n − 1)(n − 2) M

So the proof is complete.

In particular, when n = 3, we obtain

Corollary 2.3.2. Suppose that (M 3, g) is a compact Riemannian manifold with

no boundary admitting a non-constant smooth solution to (2.1.9). Then, for any

p ≥ 2, Z Z p ijk p p 2 f Cijk, = − f |C| . (2.3.4) M 4 M

42 Hence we have improved Theorem 2.2.6 in dimension 3.

Theorem 2.3.3. Suppose that (M 3, g) is a compact Riemannian manifold with no boundary with a static metric or CPE metric and non-constant function f.

ijk If Cijk, vanishes identically, then the Cotton tensor vanishes identically and therefore Theorem 2.2.6 holds.

More interestingly we have the improved version of Theorem 2.2.7, which gives a partial answer to the Fischer-Marsden’s problem (cf. [29]).

Theorem 2.3.4. Suppose that (M 3, g) is a compact vacuum static space with

ijk Cijk, vanishing identically. Then the vacuum static space must be one of the following up to a ﬁnite quotient and appropriate scaling,

(i) Flat space;

(ii) Sn;

(iii) S1 × S2;

1 2 2 2 (iv) S ×r S with warped metric g = ds + r (s)gS2 , where r(s) is a periodic function given in Example 4 in [36].

Similarly we have the improved version of Theorem 2.2.10 as follows:

Theorem 2.3.5. Conjecture 1.3.1 holds for compact 3-manifold (M 3, g) with no

ijk boundary satisfying Cijk, = 0.

43 Chapter 3

Local rigidity phenomena

3.1 Local scalar curvature rigidity in general

In this section we will investigate the local scalar curvature rigidity phe-

nomenon for general vacuum static spaces. For convenience of readers, we will

present the calculations in [29] and [9, 10] (cf. see also [21, 42]) for general vacuum

static spaces. We ﬁrst recall the deformations of scalar curvature. In this paper

we use the conventions that Greek indices run through 1, 2, ··· , n while Latin

indices run through 1, 2, ··· , n − 1.

Lemma 3.1.1. ([29, Lemma 2.2 and Lemma 7.2]) For the deformation of metrics gt = g + th, we have

d Rgt = DRg(h) = −∆(Trh) + δδh − Ric · h (3.1.1) dt t=0

44 and

2 d 2 2 2 1 2 1 2 2 Rgt =D Rg(h, h) = −2DRg(h ) − ∆(|h| ) − |∇h| − |d(Trh)| dt t=0 2 2 (3.1.2)

2 β αγ + 2h · ∇ (Trh) − 2δh · d(Trh) + ∇αhβγ ∇ h ,

β where (δh)α = −∇ hαβ.

Note that the operator ∆ diﬀers from that in [29] by a sign. Let (M n, g,¯ f)

(n ≥ 3) be a vacuum static space and Ω ia subdomain in M n. As in [29, 9, 10, 42], we consider the functional

Z Fg = fRgdvolg¯, (3.1.3) Ω where dvolg¯ is the volume element with respect to the static metric g¯ instead of g.

It is well known that such geometric problems need to appropriately ﬁx a gauge in order to derive rigidity results. In [29] they relied on the slice theorem in [25] for closed manifolds. The following lemma from [10] is a version of the slice theorem that is applicable to domains instead of closed manifolds without boundary .

Lemma 3.1.2. ([10, Proposition 11]) Suppose that Ω is a domain in a Rieman- nian manifold (M n, g¯). Fix a real number p > n, there exists an ε > 0, such that

45 for a metric g on Ω with

||g − g¯||W 2,p(Ω,g¯) < ε,

∗ there exists a diﬀeomorphism ϕ :Ω → Ω such that ϕ|∂Ω = id and h = ϕ (g) − g¯

is divergence-free in Ω with respect to g¯. Moreover,

||h||W 2,p(Ω,g¯) ≤ C||g − g¯||W 2,p(Ω,g¯), for some constant C > 0 that only depends on Ω.

For the convenience of calculations we are using a Fermi coordinate of the boundary ∂Ω with respect to the vacuum static metric g¯ such that ∂n = ∂ν on the boundary ∂Ω. Let A be the second fundamental form and H be mean curvatures of ∂Ω. In the following calculations from now on in this section everything is with respect to the vacuum static metric g¯ unless it will be indicated otherwise. But,

ﬁrst, in the light of Lemma 3.1.2, we may assume that

δh = 0 in Ω and hij = 0 on ∂Ω. (3.1.4)

We would like to mention that it is not necessarily true that h vanishes on the boundary after requiring δh = 0 in Ω. For the convenience of readers, we present

46 the calculations:

hij,k = Ajkhin + Aikhjn

i ∂Ω i hin, = (∇ ) hin + Hhnn

j (3.1.5) hnn,i = ∂ihnn − 2Aijhn

j j hj ,i = 2Aijhn

n i hαn, = −hαi, .

n Lemma 3.1.3. Suppose that (M , g,¯ f) is a vacuum static space and that gt =

g¯ + th is a deformation. Then Z d Fgt (h) = ((Trh)∂νf − f∂ν(Trh) − h(ν, ∇f) − fδh · ν) dσg¯ (3.1.6) dt t=0 ∂Ω and, if in addition one assumes δh = 0, 2 Z d 1 2 2 2 Fgt (h, h) = − (|∇h| + |d(Trh)| − 2R(h, h))fdvolg¯ dt t=0 2 Ω Z 2 2 + (f(∂ν(|h| ) + δ(h ) · ν + 2h(∇(Trh), ν)))dσg¯ (3.1.7) ∂Ω Z 2 2 + (h (ν, ∇f) − |h| ∂νf − 2(Trh)h(ν, ∇f))dσg¯, ∂Ω αγ βδ 2R 2 where R(h, h) = Rαβγδh h + 2(Trh)Ric · h − n−1 (Trh) .

Proof. One only needs to apply Lemma 3.1.1 and perform integrating by parts.

One calculation is worth to present here. Z β αγ ∇αhβγ · ∇ h fdvolg¯ Ω Z Z 2 2 αγ βδ 2 2 = (∇ f − fRicg¯) · h + fRαβγδh h dvolg¯ − h (ν, ∇f) + fδ(h ) · ν dσg¯ Ω ∂Ω Z Z 2 αγ βδ 2 2 = |h| ∆f + fRαβγδh h dvolg¯ − h (ν, ∇f) + fδ(h ) · ν dσg¯, Ω ∂Ω 47 where the static equation (1.1.6) and the Ricci identity in Riemannian geometry

R 2 are used. One may also use (3.1.6) to handle the term Ω DRg¯(h )dvolg¯.

The following expansion of the mean curvature from [10] gives us the ﬁrst and second deformation of the mean curvature.

Lemma 3.1.4. ([10, Proposition 5]) Suppose that g =g ¯+ h be another metric on

n a domain Ω in a Riemannian manifold (M , g¯). Assume that h|T ∂Ω = 0. Then

1 i 1 i Hg − Hg¯ = ( Hg¯hnn − hin, + hi ,n) 2 2 (3.1.8) 1 1 1 + ((− h2 + h hi )H + h (h i − h i )) + O(|h|2(|∇h| + |h|)), 2 4 nn in n g¯ nn in, 2 i ,n where

|O(|h|2(|∇h| + |h|))| ≤ C(|h|2(|∇h| + |h|)) for some constant C that only depends on n.

The vacuum metric g¯ is not a critical point for the functional F according to

(3.1.6), instead it follows from (3.1.8) that

Z Z d ( fRgt dvolg¯ + 2 fHgdσg¯) = 0 (3.1.9) dt t=0 Ω ∂Ω for gt =g ¯ + th, where h|T ∂Ω vanishes, as observed in [21]. An immediate consequence of (3.1.8) is

i i (2 − hnn)(Hg − Hg¯) = −(1 − hnn)(2hni, − hi ,n) (3.1.10) 3 +(h − h2 + h hi )H + O(|h|2(|∇h| + |h|)). nn 4 nn in n g¯ 48 For the convenience we denote

Z 1 2 2 IΩ = |∇h| + |d(Trh)| − 2R(h, h) fdvolg¯ (3.1.11) 4 Ω

and

Z 1 2 1 2 BΩ = −(Trh)∂νf + h(ν, ∇f) − h (ν, ∇f) + |h| ∂νf + (Trh)h(ν, ∇f) dσg¯ ∂Ω 2 2 (3.1.12) Z 1 2 1 2 + ∂ν(Trh) − ∂ν(|h| ) − δ(h ) · ν − h(∇(Trh), ν) fdσg¯. ∂Ω 2 2

Therefore we may write

Z 0 1 00 Fg − Fg¯ − Fg¯(h) − Fg¯ (h, h) = (Rg − Rg¯) fdvolg¯ + IΩ + BΩ (3.1.13) 2 Ω

when h = g − g¯ satisﬁes (3.1.4).

Proposition 3.1.5. Assume that

1 |h| < , δh = 0, and h = 0. 2 T ∂Ω

Then

Z 1 ij 1 2 BΩ = (2 − hnn)(Hg − Hg¯) + A hinhjn + |h| Hg¯ fdσg¯ (3.1.14) ∂Ω 2 4 Z Z 1 2 2 2 + hnnh(ν, ∇f) + |h| − hnn ∂νf dσg¯ + O |h| (|∇h| + |h|)dσg¯ , ∂Ω 4 ∂Ω

where Z Z 2 2 |O( |h| (|∇h| + |h|)dvolg¯)| ≤ C |h| (|∇h| + |h|)dvolg¯ Ω Ω

for some constant C that only depends on n.

49 Proof. First we calculate

Z 1 1 2 1 2 BΩ : = −(Trh)∂νf + h(ν, ∇f) + |h| ∂νf − h (ν, ∇f) + (Trh)h(ν, ∇f) dσg¯ ∂Ω 2 2 Z 2 1 i 1 i = hnn + hnih n ∂νf + (1 + hnn)hin · ∂ f dσg¯. ∂Ω 2 2

To get the second part of BΩ we calculate

n i ν(Trh) = hn ,n + hi ,n 1 − ν(|h|2) = −h h n − 2h ih 2 nn n ,n n ni,n 1 1 1 1 − δ(h2) · ν = h h n + h ih + h ih 2 2 nn n ,n 2 n nn,i 2 n ni,n

i i j −h(∇(Trh), ν) = −hnn(hnn,n + hi ,n) − hn (hnn,i + hj ,i).

Then, using (3.1.5), we obtain

Z 2 1 2 1 2 BΩ := ∂ν(Trh) − ∂ν(|h| ) − δ(h ) · ν − h(∇(Trh), ν) fdσg¯ ∂Ω 2 2 Z ∂Ω i 1 i i = (∇ ) (1 − hnn)hin − (1 − hnn) 2hn ,i − hi ,n fdσg¯ ∂Ω 2 Z 1 1 ij 3 i + (1 − hnn)hnnH + A hinhjn + Hhnih n fdσg¯ ∂Ω 2 2 2 Z 1 i i i = − 1 − hnn hin∂ f − (1 − hnn) 2hn ,i − hi ,n f dσg¯ ∂Ω 2 Z 1 1 ij 3 i + (1 − hnn)hnnH + A hinhjn + Hhnih n fdσg¯. ∂Ω 2 2 2

50 1 2 Therefore, adding BΩ and BΩ, we arrive at

Z 1 1 ij 3 i BΩ = (1 − hnn)hnnH + A hinhjn + Hhnih n fdσg¯ ∂Ω 2 2 2 Z i i − (1 − hnn) 2hn ,i − hi ,n fdσg¯ ∂Ω Z i 2 1 i + hnnhin∂ f + hnn + hnih n ∂νf dσg¯. ∂Ω 2

Finally, using (3.1.10), we may ﬁnish the calculation and establish (3.1.14).

Now we are ready to show the boundary integral is non-negative for small geodesic balls in a vacuum static spaces.

Proposition 3.1.6. Suppose (M n, g,¯ f) is a vacuum static space and that g =g ¯+h where 1 |h| + |∇h| < , δh = 0, and h = 0. 2 T ∂Ω

n Then, for a p0 ∈ M where f(p0) > 0, there exists r0 > 0 and C > 0 such that

Z 2 BBr(p0) ≥ −C |h| (|∇h| + |h|)dσg¯ ∂Br(p0) for any geodesic ball Br(p0) with r < r0, provided that

Hg ≥ Hg¯ and on ∂Br(p0).

n Proof. First of all, for p0 ∈ M with f(p0) > 0, one may have r1 such that

f(p) ≥ 1 and f(p) + |∇f|(p) ≤ β1

51 for p ∈ Br1 (p0) and positive constants 1 and β1. Hence, from (3.1.14), we obtain Z Z 1 ij 1 2 2 BBr(p0) ≥ A hinhjn + |h| Hg¯ fdσg¯ − C |h| dvolg¯ ∂Br(p0) 2 4 ∂Br(p0) Z 2 − C |h| (|∇h| + |h|)dσg¯ ∂Br(p0) for some constant C > 0, here we have used the assumptions that (2 − hnn)(Hg −

Hg¯) ≥ 0. Therefore it is easy from here to ﬁnish the proof based on the geometry of small geodesic balls. Namely,

1 n − 1 A = g¯ + O(r) and H = + O(r). ij r ij g¯ r

Before we give the estimate on the interior term, we study the following eigenvalue problem of the Laplace on symmetric 2-tensors. Namely we consider

R 1 |∇h|2dvol µ(Ω) = inf{ Ω 2 g¯ : h 6≡ 0 and h = 0} (3.1.15) R 2 T ∂Ω Ω |h| dvolg¯

The following is an easy but very useful fact to us (please see a similar result in

[34]).

n Lemma 3.1.7. Suppose (M , g,¯ f) is a vacuum static space and that Br(p) is a

n geodesic ball of radius r in M . Then, there are constants r0 and c0 such that

c µ(B (p)) ≥ 0 (3.1.16) r r2

n for all point p ∈ M and r < r0.

52 Proof. We ﬁrst observe that there are constant r0 and c1 such that

0 µ(Br(p)) ≥ c1µ(Br (0))

n 0 for all p ∈ M and r < r0, where µ(Br (0)) is the ﬁrst eigenvalue for the Euclidean

0 ball Br (0) with respect to the Euclidean metric. Therefore it suﬃces to show

(3.1.16) for Euclidean balls with respect to the Euclidean metric. In fact, by scaling property, we simply need to show

0 µ(B1 ) > 0.

For this purpose, we consider the functional

R 1 2 B0 2 |∇h| dx J(h) = 1 , ∀h ∈ R 2 W 0 |h| dx B1 where

1,2 0 = {h ∈ W (B ): h 6≡ 0, h| 0 = 0}. W 1 T ∂B1

Then the Euler-Lagrange equation for the minimizers of J is   0 0 ∆h + µ(B1 )h = 0 in B1

 0 hij = 0 and ∂νhin = 0 on ∂B1 ,

0 1 n−1 where µ(B1 ) = infh∈W J(h) ≥ 0, if we use the spherical coordinate {θ , ··· , θ , ν}

0 on the unit sphere. What we need to show is that µ(B1 ) is in fact positive. As-

0 sume otherwise µ(B1 ) = 0. Then we may easily see that, by integral by parts, the

0 eigen-tensor h has to be parallel (constant) in B1 , which forces h ≡ 0 since all the

53 0 n tangent vectors at the boundary ∂B1 together span the full space R (One may take the advantage to ignore the base point for a vector in the Euclidean space) .

This ﬁnishes the proof.

We remark that, in case the domain Ω is a square in the plane R2, the ﬁrst eigenvalue µ is zero. Consequently, we have

Proposition 3.1.8. Suppose (M n, g,¯ f) is a vacuum static space and that g =g ¯+h where

δh = 0, and hT ∂Ω = 0.

n Then, for a p0 ∈ M where f(p0) > 0, there exists r0 > 0 such that Z 1 2 2 IBr(p0) ≥ (|∇h| + |h| )dvolg¯ 8 Br(p0) for a geodesic ball Br(p0) with radius r < r0.

Proof. Recall (3.1.11), Z 1 2 2 IBr(p0) = |∇h| + |d(trh)| − 2R(h, h) fdvolg¯. 4 Br(p0)

Clearly there is a constant C (C depends on (M n, g¯)) such that

2R(h, h) ≤ C |h|2.

Hence Z 1 2 2 IBr(p0) ≥ |∇h| − C|h| fdvolg¯ 4 Br(p0) Z Z 1 2 2 1 2 2 = |∇h| + |h| fdvolg¯ + |∇h| − (2C + 1)|h| fdvolg¯. 8 Br(p0) 8 Br(p0)

54 The rest of proof easily follows from Lemma 3.1.7.

Now we are ready to prove the main theorem following the approach from

[9, 10, 21] .

Proof of Theorem 1.3.4. First, due to the assumption that ||g − g¯|| 2 is C (Br(p0)) suﬃciently small, in the light of Lemma 3.1.2, we may assume g =g ¯ + h, where h satisﬁes

δh = 0 in Br(p0) and h|T ∂Br(p0) = 0.

Following the approach in [9, 10, 21], we have

1 F (g) − F (¯g) − F 0(¯g) · h − F 00(¯g) · (h, h) 2 Z 2 2 ≤Ckhk 2 (|∇h| + |h| )dvol . C (Br(p0)) g¯ Br(p0)

On the other hand, by Proposition 3.1.6 and Proposition 3.1.8, one arrives at

Z 2 2 (|∇h| + |h| )dvolg¯ Br(p0) Z Z 2 2 2 ≤C||h|| 2 ( (|∇h| + |h| )dvol + |h| dσ ) C (Br(p0)) g¯ g¯ Br(p0) ∂Br(p0) Z 2 2 ≤C ||h|| 2 (|∇h| + |h| )dvol , 1 C (Br(p0)) g¯ Br(p0)

by Trace Theorem of Sobolev spaces, which implies that h ≡ 0, when khk 2 C (Br(p0)) is small enough. Thus the proof is complete.

55 3.2 Local scalar curvature rigidity of space forms

In the previous section, we investigated the local scalar curvature rigidity of domains of suﬃciently small size in general vacuum static spaces. In this section we consider the local scalar curvature rigidity of space forms.

3.2.1 Euclidean spaces

In this subsection, we consider local scalar curvature rigidity of domains in the Euclidean space Rn. In Euclidean cases it turns out one has the local scalar curvature rigidity of bounded domain of any size, which may be compared with the rigidity result of closed ﬂat spaces in [29] but a much a weaker version of the rigidity in [41] (cf. also [50]), where the positive mass theorem is employed. In

Euclidean cases the lapse function f may be taken to be 1. We calculate from

(3.1.11) and (3.1.14) that

Z 1 2 2 IΩ = |∇h| + |d(Trh)| dx 4 Ω and Z 1 ij 1 2 BΩ = (2 − hnn)(Hg − Hg n ) + A hinhjn + |h| Hg n dθ R R ∂Ω 2 4 Z + O |h|2(|∇h| + |h|)dθ . ∂Ω

Theorem 3.2.1. Let Ω be a bounded smooth domain in the Euclidean space Rn.

56 Assume that 1 A + Hg n ≥ 0 on ∂Ω. (3.2.1) 2 R

Then there is > 0 such that, for any Riemnnian metric g on Ω satisfying

• g = gRn on ∂Ω,

• Rg ≥ 0 in Ω,

• Hg ≥ Hg n on ∂Ω, and R

• kg − gRn kC2(Ω) ≤ ,

∗ there is a diﬀeomorphism ϕ :Ω → Ω such that ϕ g = gRn in Ω and ϕ = id on ∂Ω.

Proof. Again, in the light of Lemma 3.1.2, we may assume that g = gRn + h and

δh = 0 in Ω and h|T ∂Ω = 0 on ∂Ω.

Then, using the smoothness of the boundary ∂Ω and the fact that h|T ∂Ω = 0 on

∂Ω, one derives that

Z Z 2 2 |∇h| dvolg¯ ≥ µ(Ω) |h| dvolg¯ Ω Ω for some positive number µ(Ω), based on the argument similar to the one in the proof of Lemma 3.1.7. Therefore one can show that h has to vanish in Ω, whenever khkC2(Ω) is suﬃciently small. Thus the proof is complete.

It is easily seen that, for example, (3.2.1) holds on convex domains including round balls in Rn.

57 3.2.2 Hyperbolic spaces

The natural way to describe the hyperbolic space Hn is to identify it as the hyperboloid

n n+1 2 2 H = {(t, x) ∈ R : −t + |x| = −1 and t > 0} in the Minkowski space-time (Rn+1, −(dt)2 + |dx|2). In this coordinate

2 (d|x|) 2 g n = + |x| g n−1 . H 1 + |x|2 S

p 2 n With the lapse function f = t = 1 + |x| , the hyperbolic space (H , gHn ) is a vacuum static space of negative cosmological constant (cf. [47]). We will consider

H the geodesic balls Br center from the vertex (1, 0). We again calculate from

(3.1.11) and (3.1.14) that

Z 1 2 2 2 2 IBH = |∇h| + |d(Trh)| − 2|h| − 2|Trh| tdvolg n r 4 H BrH and Z n−1 ! 1 2 n X 2 BBH ≥ (2 − hnn)(Hg − Hg n ) + ( h + h )Hg n cosh rdσg n r H 4 nn 2(n − 1) in H H ∂BrH i=1 Z Z 3 2 1 2 2 + ( h + |h| ) sinh rdσg n + O |h| (|∇h| + |h|)dσg n , 4 nn 4 H H ∂BrH ∂BrH where t = cosh r and ∂νt = sinh r are positive.

H Theorem 3.2.2. For n ≥ 3, let Br be the geodesic ball centered at the vertex

(1, 0) with radius r > 0 on the hyperboloid. There exists an ε0 > 0, such that, for

H any metric g on Br satisfying

58 H • g = gHn on ∂Br ;

H • Rg ≥ −n(n − 1) in Br ;

H • Hg ≥ Hg n on ∂B ; H r

• ||g − g n || 2 < ε0, H C (BrH)

H H ∗ H there is a diﬀeomorphism ϕ : Br → Br such that ϕ g = gHn in Br and ϕ = id

H on ∂Br .

Proof. Again, in the light of Lemma 3.1.2, we may assume that g = gHn + h and

δh = 0 in Ω and h|T ∂Ω = 0 on ∂Ω.

From the assumptions we have

I + B BrH BrH Z Z 1 2 2 2 2 1 2 ≥ |∇h| + |d(Trh)| − 2|h| − 2|Trh| tdvolg n + |h| cosh rdσg n 4 H 4 H BrH ∂BrH Z Z 3 2 1 2 2 + ( h + |h| )∂νtdσg n − C |h| (|∇h| + |h|)dσg n 4 nn 4 H H ∂BrH ∂BrH (3.2.2) for some constant C > 0. Similar to the idea used in [21], we want to use the positive boundary terms to help to cancel the negative interior terms. For that,

59 we perform integral by parts and estimate

Z 2 2 (|Trh| + |h| )∂νtdvolg n H ∂BrH Z 2 2 = div((|Trh| + |h| )∇t)dvolg n H BrH Z Z 2 2 = (|Trh| + |h| )∆tdvolg n + 2 ((Trh)∇(Trh) + h∇h) · ∇tdvolg n H H BrH BrH Z Z 2 2 ≥n (|Trh| + |h| )tdvolg n − 2 (|Trh||∇Trh| + |h||∇h|)|∇t|dvolg n H H BrH BrH Z Z 2 2 2 2 1 2 2 ≥n (|Trh| + |h| )tdvolg n − (a(|Trh| + |h| ) + (|∇Trh| + |∇h| ))tdvolg n . H a H BrH BrH Here we use the fact that ∆t = nt and |∇t| < t. Going back to (3.2.2) we get, for

3 11 the choices b = 4 and a = 6 ,

I + B BrH BrH Z 1 (1 + b) 2 2 ≥ ( − )(|∇h| + |d(Trh)| ) tdvolg n 4 4a H BrH Z Z (1 + b) 1 2 2 1 − b 2 + ( (n − a) − )(|h| + |Trh| ) tdvolg n + |h| cosh rdσg n 4 2 H 4 H BrH ∂BrH Z Z 2 − b 2 2 + h sinh rdσg n − C |h| (|∇h| + |h|)dσg n 4 nn H H ∂BrH ∂BrH Z 1 2 2 7 1 2 2 = (|∇h| + |d(Trh)| ) + (n − 3 + )(|h| + |Trh| ) tdvolg n 88 16 42 H BrH Z Z Z 1 2 5 2 2 + |h| cosh rdσg n + h sinh rdσg n − C |h| (|∇h| + |h|)dσg n . 16 H 16 nn H H ∂BrH ∂BrH ∂BrH

Now one may conclude that h = 0, when ||h|| 2 is suﬃciently small and C (BrH)

n ≥ 3.

60 3.2.3 Hemispheres

n The upper hemisphere S+ with the standard round metric is a vacuum static

space of positive cosmological constant, where the lapse function f = xn+1 is the high function when the sphere is the unit round sphere Sn centered at the origin

n+1 S in the Euclidean space R . One considers the geodesic balls Br centered at the north pole on the hemisphere. One then calculates from (3.1.11) and (3.1.14) that

Z 1 2 2 2 2 IBS = |∇h| + |d(Trh)| + 2|h| + 2|Trh| xn+1dvolg n r 4 S BrS

and Z n−1 ! ! 1 2 n X 2 BBS = (2 − hnn)(Hg − Hg n ) + h + h Hg n cos rdσg n r R 4 nn 2(n − 1) in R S ∂BrS i=1 Z n−1 ! Z 2 1 X 2 2 − h + h sin rdσg n + O |h| (|∇h| + |h|)dσg n nn 2 in S S ∂BrS i=1 ∂BrS Z 2 2 n−1 ! n − 1 cos r 2 1 cos r X 2 ≥ − sin r h + n − sin r h dσg n 4 sin r nn 2 sin r in S ∂BrS i=1 Z 2 − C |h| (|∇h| + |h|)dσg n , S ∂BrS

S where f = f(r) = cos r and Hg n = (n − 1) cot r for ∂B in the hemisphere. S r

S √ 2 Theorem 3.2.3. ([10]) Consider the geodesic ball Br with cos r ≥ n+3 . Let g

S be a Riemannian metric on Br with the following properties:

S • Rg ≥ n(n − 1) in Br ;

S • Hg ≥ (n − 1) cot r on ∂Br ;

61 S • g and gSn induced the same metric on ∂Br ,

2 ∗ If g − gSn is suﬃciently small in the C -norm, then ϕ (g) = gSn for some diﬀeo-

morphism ϕ : BS → BS with ϕ| = id. r r ∂BrS

Remark 3.2.4. In [21], the interior integral

Z 1 2 2 2 2 IBS = |∇h| + |d(Trh)| + 2|h| + 2|Trh| xn+1dvolg n r 4 S BrS

S is used cleverly to improve the size of the geodesic ball Br that is bigger than

√ 2 cos r ≥ n+3 of Theorem 3.2.3 of [10].

3.3 Conformal rigidity of vacuum static space

In this section we consider the scalar curvature rigidity among conformal de-

formations. This is inspired by the work in [30], where the scalar curvature rigidity

among conformal deformations of the hemispheres is established. For n ≥ 2, let

n (M , g,¯ f) be a static space with positive scalar curvature Rg¯ > 0. We denote

Ω+ = {x ∈ M : f(x) > 0}.

Theorem 3.3.1. Let (M n, g,¯ f) be a complete n-dimensional static space with

+ Rg¯ > 0 (n ≥ 2). Assume Ω is a pre-compact subset in M. Then, if a metric g ∈ [¯g] on M satisﬁes that

+ • Rg ≥ Rg¯ in Ω ,

62 • g and g¯ induced the same metric on ∂Ω+, and

+ • Hg = Hg¯ on ∂Ω ,

then g =g ¯.

Proof. Since g ∈ [¯g] we may write as usual   2u  e g¯ when n = 2 g =  4  u n−2 g¯ when n ≥ 3.

Hence   −2u  e (Rg¯ − 2∆u) when n = 2 Rg =  − n+2 4(n − 1)  u n−2 R u − ∆u when n ≥ 3  g¯ n − 2 and    Hg¯ + 2∂νu when n = 2 Hg =  2(n − 1)  Hg¯ + ∂νu when n ≥ 3. n − 2 If we let  2u(x)  Rg¯(e −1) 2ξ  2u(x) = R[¯g]e when n = 2 Λ(x) = 4 n−2 Rg¯u(x) u(x) n−2 −1  4(n−1) R[¯g] 4 u  n−2  u(x)−1 = n−1 ξ ξ when n ≥ 3,

where ξ is between 0 and u(x) when n = 2; ξ is between 1 and u(x) when n ≥ 3,

and   u(x) when n = 2 v(x) =  u(x) − 1 when n ≥ 3,

63 then we may rewrite the assumptions Rg ≥ R[¯g] and Hg ≥ H[¯g] as follows:   +  −∆v − Λ(x)v ≥ 0 in Ω   v = 0 on ∂Ω+ (3.3.1)    +  ∂νv = 0 on ∂Ω .

R[¯g] On the other hand, if denote Λ = n−1 , we deduce from the static equation that

−∆f − Λf = 0 and f > 0 in Ω+. (3.3.2)

In the following we want to ﬁrst use the positive lapse function f in Ω+ to show

v(x) + v ≥ 0. To do so we consider the quotient ϕ(x) = f(x) in Ω and calculate

1 −∆ϕ = (−∆v + ϕ∆f + 2∇ϕ · ∇f) f (3.3.3) ∇f ≥ −ϕ (Λ − Λ(x)) + 2∇ϕ · . f

Assume otherwise that there exists x¯ ∈ Ω+ such that ϕ(¯x) < 0. In order to apply

the maximum principle we would like to use L0hospital0s rule to see that ϕ = 0 on

∂Ω+. Here we need to use that fact that ∇f 6= 0 at ∂Ω+ = {x ∈ M n : f(x) = 0}

according [29, Theorem 1]. Therefore we may assume that x¯ be a minimum point

for ϕ. Notice that Λ(¯x) < Λ when ϕ(¯x) < 0. Thus, from (3.3.3), we arrive at

0 ≥ −∆ϕ(¯x) ≥ −ϕ(¯x) (Λ − Λ(¯x)) > 0,

which is a contradiction. Therefore we have shown that v(x) ≥ 0 in Ω+.

64 Finally, applying the Hopf maximum principle, for instance, [23, Theorem

7.3.3], to (4.1), we conclude that v ≡ 0 in Ω+, that is, g ≡ g¯ in Ω+. So the proof is compete.

Next we want to show that the domain Ω+ is the biggest of which the scalar

curvature rigidity holds. This generalizes the work in [30]. Our construction is

diﬀerent from [30] and is based on the idea in [9], particularly [9, Theorem 5 and

Lemma 21].

Theorem 3.3.2. Let (M n, g,¯ f) be a complete n-dimensional static space with

+ Rg¯ > 0 (n ≥ 2). Assume the level set Ω is a pre-compact subset in M. For any

open domain Ω in M n that contains Ω+ , there is a smooth metric g ∈ [¯g] such

that

• Rg > Rg¯ at some point in Ω and

• supp(g − g¯) ⊂ Ω

Proof. By [29, Theorem 1] , we know ∇f 6= 0 on ∂Ω+. Hence we may assume the

level set Ω− = {x ∈ M n : f(x) > −} ⊂ Ω, at least for suﬃciently small , is

pre-compact in M n.

First, we need a smooth metric in the neighborhood of ∂Ω− that smoothly

extends to outside Ω−. Based the same idea used in [9, Lemma 21], we consider

4 2 g˜ = w n−2 g¯ when n ≥ 3 and g˜ = w g¯ when n = 2,

65 1 − f+ − n for w = 1 − e in Ω0 = {x ∈ M : − < f(x) < 0}. Similar calculations as in the proof of [9, Lemma 21] show that

− • R[˜g] > R[¯g] in Ω0 and

• supp(˜g − g¯) ⊂ Ω.

− 2 Next, in Ω , we consider ut = 1 − t(f + 2 ) > 0 on M for t ∈ [0, δ] and suﬃciently small δ > 0, and the family of conformal deformations

4 n−2 2 gt = ut g¯ when n ≥ 3 and gt = ut g¯ when n = 2, for t ∈ [0, δ).

− Then, when n ≥ 3, we calculate the expansion of scalar curvature, in Ω 2 ,

− n+2 4(n − 1) R[g ] = u n−2 (R u − ∆u ) t t g¯ t n − 2 t 4(n − 1) R = R + ∆f + g¯ (f + ) t + O(t2) g¯ n − 2 n − 1 2 2 = R + tR + O(t2) g¯ n − 2 g¯ 1 > R + tR . g¯ n g¯

2 1 Similarly, when n = 2, we have R[gt] = R[¯g] + tR[¯g] + O(t ) > R[¯g] + 2 tR[¯g].

Let

4 n−2 2 gˆt = w gt when n ≥ 3 and gˆt = w− gt when n = 2, − 2 2

− 2 where w := w(− ) = 1 − e ∈ (0, 1). Thus, − 2 2

1 R > R[¯g] + tR . gˆt n g¯ 66 As for mean curvature, we have when n ≥ 3,

2 n 2 1 − n−2 2(n − 1) − n−2 − Hgˆt = w− Hg¯ + w− 1 − e t|∇f| 2 2 n − 2 2 and when n = 2,

1 2 −1 −2 − Hgˆt = w− Hg¯ + w− 1 − e t|∇f| 2 2 2

− on ∂Ω 2 .

On the other hand, when n ≥ 3,

2 n 2 1 − n−2 2(n − 1) − n−2 4 − H[˜g] = w− Hg¯ + w− e |∇f| 2 2 n − 2 2 2 and when n = 2,

1 2 4 −1 −2 − H[˜g] = w− Hg¯ + w− e |∇f| 2 2 2 2

− 2 By choosing suﬃciently small, we have Hgˆt > H[˜g] on ∂Ω .

− Hence we constructed a conformal metric gˆt in Ω 2 for some suﬃciently small t and such that

1 − 2 • Rgˆt > Rg¯ + n tRg¯ in Ω ;

− • gˆt =g ˜ on ∂Ω 2 ;

− 2 • Hgt > H[¯g] on ∂Ω .

− Now, applying [9, Theorem 5] to glue gˆt and g˜ on Ω 2 , we get a smooth metric g that ﬁnishes the proof, so long as one realizes that g stays in [¯g] if both g˜ and gt are conformal to g¯ due to the construction of the gluing in [9].

67 Chapter 4

Brown-York mass and compactly conformal deformations

4.1 Brown-York mass and conformal rigidity

In this section, we will prove results associated to the Brown-York mass and investigate rigidity phenomena of conformally compact deformations.

Proof of Theorem 1.3.9. For g ∈ [¯g], there exists a smooth function u such that   2u  e g¯ when n = 2 g =  4  u n−2 g¯ when n ≥ 3.

68 It is well known that for u = 1 on ∂M,   −2u  e (Rg¯ − 2∆g¯u) when n = 2 Rg =  − n+2 4(n − 1)  u n−2 R u − ∆ u when n ≥ 3  g¯ n − 2 g¯ and    Hg¯ + 2∂νu when n = 2 Hg =  2(n − 1)  Hg¯ + ∂νu when n ≥ 3, n − 2 where ν is the outward normal of ∂M with respect to g¯.

Let  2u(x)  Rg¯(e −1) 2ξ  2u(x) = Rg¯e when n = 2 w(x) = 4 n−2 n−2 4(n−1) Rg¯u(x) u(x) −1 4  Rg¯ u  n−2  u(x)−1 = n−1 ξ ξ when n ≥ 3,

where ξ(x) is between 0 and u(x) when n = 2; ξ(x) is between 1 and u(x) when n ≥ 3.

Assume that for (I),

mBY (∂M, g¯; g+) ≤ 0; and for (II),

mBY (∂M, g¯; g−) ≥ 0.

We are going to show that in both cases g+ = g− =g ¯ on M and thus the theorem follows.

Let u+ be the smooth function for the conformal factor of g+ and u− for g− respectively.

69 We take for (I),   u+(x) when n = 2 v(x) =  u+(x) − 1 when n ≥ 3 and for (II)   −u−(x) when n = 2 v(x) =  1 − u−(x) when n ≥ 3.

Thus we have   −∆ v − w(x)v ≥ 0 in M  g¯   v = 0 on ∂M   Z   ∂νv dσg¯ ≥ 0. ∂M We claim that

v(x) ≥ 0

on M.

Suppose not, there exists a point x¯ ∈ M such that

v(¯x) < 0.

For any ε > 0, let Mε be an ε-extension of (M, g¯) along the boundary ∂M.

That is, we extend g¯ smoothly on a smooth manifold Mε with boundary ∂Mε

which contains the interior of M as an open submanifold and satisﬁes that

distg¯(∂M, ∂Mε) = ε.

70 Since Λ1(M, g¯) > 0, for any 0 < δ < 1, there exists an ε > 0 small such that

Λ1(Mε, g¯) > δΛ1(M, g¯) > 0.

Let φ be a ﬁrst eigenfunction associated to Λ1(Mε, g¯) and from the standard

v(x) theory we can choose φ to be positive on Mε. Take ϕ(x) = φ(x) , then

ϕ ≡ 0 on ∂M and

ϕ(¯x) < 0.

Let x0 ∈ M be the point where ϕ attains its minimum on M. Then we have

−∆g¯ϕ(x0) ≤ 0, ∇ϕ(x0) = 0.

Clearly, ϕ(x0) < 0, which would imply v(x0) < 0 and thus

R R w(x ) < g¯ < g¯ + δΛ (M, g¯). 0 n − 1 n − 1 1

This is trivial for (I) and for (II) with maxM Rg¯ ≤ 0; as for the case of

max Rg¯ > 0, M it can be deduced from the presumption that

4 n+2 √ (n − 1)δΛ1(M, g¯) ||g−||C0(M,g¯) < α = n 1 + . maxM Rg¯

71 In fact, for n ≥ 3, v(x0) < 0 implies that 1 < ξ(x0) < u−(x0) and thus

n+2 n+2 n+2 0 4 Rg¯ n−2 Rg¯ n−2 Rg¯ ||g−||C (M,g¯) w(x0) ≤ u− (x0) ≤ max u− = √ n − 1 n − 1 M n − 1 n R ≤ g¯ + δΛ (M, g¯), n − 1 1

√ 4 n−2 where ||g−||C0(M,g¯) = n (maxM u−) .

The estimate for n = 2 follows with similar calculations.

Now at x0, we have

0 ≥ −∆g¯ϕ(x0) 1 v(x0) 2 = −∆g¯v(x0) + ∆g¯φ(x0) + ∇ϕ(x0) · ∇φ(x0) φ(x0) φ(x0) φ(x0) 1 Rg¯ = −∆g¯v(x0) − + Λ1(Mε, g¯) v(x0) φ(x0) n − 1 1 Rg¯ ≥ w(x0) − − Λ1(Mε, g¯) v(x0) φ(x0) n − 1 1 ≥ − (Λ1(Mε, g¯) − δΛ1(M, g¯)) v(x0) φ(x0)

> 0.

Contradiction! Therefore,

v(x) ≥ 0, ∀x ∈ M.

On the other hand, the non-negativity of v implies that

∂νv(x) < 0

72 for any x ∈ ∂M by Generalized Hopf’s lemma (cf. Theorem 7.3.3 in [23]), unless

v is identically vanishing. Thus

Z ∂νv dσg¯ < 0, ∂M

which contradicts to the fact

Z ∂νv dσg¯ ≥ 0, ∂M hence v ≡ 0 on M. i.e.

g ≡ g.¯

In order to prove Corollary 1.3.11, we recall the following well-known fact on

the estimate of the ﬁrst eigenvalue of Laplacian.

Lemma 4.1.1. (Karp-Pinsky, [34]) For any p ∈ M, there exist positive constants

r0 and c0 such that the ﬁrst eigenvalue of Laplacian satisﬁes that

c λ (B (p), g¯) ≥ 0 , 1 r r2

provided r < r0.

Proof of corollary 1.3.11. Take

R (x) Λ := max g¯ . x∈B1(p) n − 1

73 We can choose 0 < r0 < 1 such that

c0 λ1(Br0 (p), g¯) ≥ 2 > Λ, r0

due to Lemma 4.1.1. Clearly,

Λ1(Br0 (p), g¯) ≥ λ1(Br0 (p), g¯) − Λ > 0.

Now the conclusion follows by applying Theorem 1.3.9 on the geodesic ball

Br0 (p).

We ﬁnish this section by showing Corollary 1.3.12 is true.

Proof of corollary 1.3.12. Note that if Rg¯ ≤ 0, then

Λ1(Ω, g¯) > 0

for any compactly contained domain Ω ⊂ M. The conclusion follows automatically from Theorem 1.3.9.

4.2 Non-rigidity phenomena

In this section, we will construct compactly conformal deformations which

suggests that the vanishing of Brown-York mass does not imply the conformal

rigidity for large domains, assuming the positivity of scalar curvature.

Let φ be a ﬁrst Dirichlet eigenfunction of Laplacian on Ω. In particular, we can choose φ > 0 on Ω. Note that, in fact ∂νφ < 0 on ∂Ω by Generalized Hopf’s lemma

74 (cf. Theorem 7.3.3 in [23]). Following the idea of Lemma 20 and Proposition 21

in [10], we can construct metrics such that they satisfy the desired properties near

∂Ω:

Proposition 4.2.1. For any ε > 0 suﬃciently small, there are smooth metrics g˜+ and g˜− on M such that   R > R on Ω − Ω  g˜+ g¯ 2ε (I):  g˜+ =g ¯ on M − Ω and   R < R on Ω − Ω  g˜− g¯ 2ε (II):  g˜− =g ¯ on M − Ω, where Ω2ε := {x ∈ Ω: φ(x) > 2ε}.

Proof. Since ∂νφ 6= 0 on ∂Ω, we can ﬁnd an ε > 0 such that

2 3 2 − 1 − 1 |∇φ| − λ1(Ω, g¯)φ − 2|∇φ| φ ∆ e φ = e φ ≥ 0, g¯ φ4

− 1 i.e. e φ is subharmonic on Ω − Ω2ε.

− 1 − 1 For n ≥ 3, we take w+ = 1 − e φ , w− = 1 + e φ on Ω and extend them to be

constantly 1 outside domain Ω. Clearly, w+ and w− are smooth on M. Now let

4 4 n−2 n−2 g˜+ = w+ g¯ and g˜− = w− g¯ respectively. Then we have

− n+2 4(n − 1) − 4 R = w n−2 R w − ∆ w ≥ w n−2 R > R g˜+ + g¯ + n − 2 g¯ + + g¯ g¯

75 and

− n+2 4(n − 1) − 4 R = w n−2 R w − ∆ w ≤ w n−2 R < R g˜− − g¯ − n − 2 g¯ − − g¯ g¯

on Ω − Ω2ε.

2 2 Similar constructions work for n = 2, if we take g˜+ = w+g¯, g˜− = w−g¯ respectively.

Now let

Ωε := {x ∈ Ω: φ(x) > ε} and choose ε suﬃciently small such that

Λ1(Ωε, g¯) < 0

and let ψ be a positive ﬁrst eigenfunction of Lg¯ on Ωε which vanishes on ∂Ωε. We will produce perturbed metrics as follow:

+ − Proposition 4.2.2. There are metrics gt , gt ∈ [¯g] with   ¯ R + ≥ Rg¯ on Ωε  gt   (I): g+ =g ¯ on ∂Ω  t ε    H + > Hg¯ on ∂Ωε  gt

76 and   ¯ R − ≤ Rg¯ on Ωε  gt   (II): g− =g ¯ on ∂Ω  t ε    H − < Hg¯ on ∂Ωε.  gt

4 n−2 ∞ Proof. For n ≥ 3, let gt = ut g¯, where ut = 1 + tϕ, where ϕ ∈ C (M) and supported in Ω. Then

− n+2 4(n − 1) R =u n−2 R u − ∆ u gt t g¯ t n − 2 g¯ t 4(n − 1) R =R − ∆ ϕ + g¯ ϕ t + O(t2) g¯ n − 2 g¯ n − 1

and

2(n − 1) H = H + t ∂ ϕ. gt g¯ n − 2 ν

We take

ϕ+ := −ψ

for (I) and

ϕ− := ψ

for (II).

Then we have

Rg¯ ∆g¯ϕ+ + ϕ+ = Lg¯ψ = Λ1(Ωε, g¯)ψ < 0 n − 1

77 on Ω and ϕ+ = 0, ∂νϕ+ > 0 on ∂Ωε.

Similarly, R ∆ ϕ + g¯ ϕ > 0 g¯ − n − 1 − on Ω and ϕ− = 0, ∂νϕ− < 0 on ∂Ωε.

Let

4 + + n−2 gt = (ut ) g¯ and

4 − − n−2 gt = (ut ) g,¯

+ − where ut = 1 + tϕ+ and ut = 1 + tϕ− respectively. Then they are our desired

metrics, if we choose t suﬃciently small.

+ + 2 − − 2 For n = 2, we take gt = (ut ) g¯ and gt = (ut ) g¯. By similar calculation, we

can see they satisfy (I) and (II) respectively.

In order to glue metrics we derived previously, we need to match them at zero’s

order ﬁrst.

+ − Proposition 4.2.3. There are metrics gˆt , gˆt ∈ [¯g] with   ¯ Rgˆ+ > Rg¯ on Ωε  t  0  (I ): gˆ+ =g ˜ on ∂Ω  t + ε    H + > Hg¯ on ∂Ωε  gˆt

78 and   ¯ Rgˆ− < Rg¯ on Ωε  t  0  (II ): gˆ− =g ˜ on ∂Ω  t − ε    H − < Hg¯ on ∂Ωε.  gˆt

1 4 1 4 + − n−2 + − − n−2 − + Proof. Let gˆt := (1−e ε ) gt and gˆt := (1+e ε ) gt . Then clearly, gˆt =g ˜+

− and gˆt =g ˜− on ∂Ωε with

− 1 − 4 R + = (1 − e ε ) n−2 R + > R + ≥ Rg¯ gˆt gt gt and

− 1 − 4 R − = (1 + e ε ) n−2 R − < R − ≤ Rg¯ gˆt gt gt

¯ on Ωε.

As for mean curvatures,

− 1 − 2 2(n − 1) H + = (1 − e ε ) n−2 Hg¯ + t(−∂νψ) gˆt n − 2 − 1 ! − 1 − 2 2(n − 1) e ε > (1 − e ε ) n−2 Hg¯ + · (−∂νφ) − 1 n − 2 ε2(1 − e ε )

= Hg˜+

79 and

− 1 − 2 2(n − 1) H − = (1 + e ε ) n−2 Hg¯ − t(−∂νψ) gˆt n − 2 − 1 ! − 1 − 2 2(n − 1) e ε < (1 + e ε ) n−2 Hg¯ − · (−∂νφ) − 1 n − 2 ε2(1 + e ε )

= Hg˜−

on ∂Ωε, if we choose ε > 0 suﬃciently small.

The following crucial gluing theorem (part I) was originally proved in [9]. We observed that it holds within conformal classes and also a similar statement (part

II) holds by minor modiﬁcations on the original proof.

Theorem 4.2.4 (Brendle-Marques-Neves [9]). Let (M, g) be a Riemannian manifold with boundary ∂M. Suppose g˜ ∈ [g] is another metric on M with the same induced metric on ∂M.

Then for any δ > 0 and any neighborhood K of ∂M the following two statements hold.

(I): If Hg ≥ Hg˜, then there exists a metric gˆ ∈ [g] such that   gˆ = g on ∂M − K    gˆ =g ˜ in a neighborhood of ∂M     Rgˆ ≥ minx∈M {Rg(x),Rg˜(x)} − δ on M

80 and

(II): If Hg ≤ Hg˜, then there exists a metric gˆ ∈ [g] such that   gˆ = g on ∂M − K    gˆ =g ˜ in a neighborhood of ∂M     Rgˆ ≤ maxx∈M {Rg(x),Rg˜(x)} + δ on M.

Now we can prove the main theorem in this section.

Proof of Theorem 1.3.13. Applying Theorem 4.2.4, for any δ > 0, we can glue

+ − metrics g˜+, g˜− from Proposition 4.2.1 and gˆt , gˆt from Proposition 4.2.3 along

+ − ∂Ωε to get metrics gδ and gδ respectively, such that

+ + Rg ≥ min{Rgˆ (x),Rg˜+ (x)} − δ δ x∈Ωε t and

− − Rg ≤ max{Rgˆ (x),Rg˜− (x)} + δ δ x∈Ωε t on Ωε. In particular, we get

R + > Rg¯ gδ and

R − < Rg¯ gδ inside Ωε, if we choose δ suﬃciently small.

81 Now we take   + gδ on Ωε g+ =  g˜+ on M − Ωε

and   − gδ on Ωε g− =  g˜− on M − Ωε.

Clearly, g+ and g− are smooth metrics which satisfy all requirement in the statement of Theorem 1.3.13.

Corollary 1.3.14 holds automatically, if we can justify the existence of a com-

pact domain Ω with

Λ1(Ω, g¯) < 0.

In fact, for closed manifolds this can be achieved with the aid of the following

lemma:

Lemma 4.2.5. Let (M n, g¯) be a closed Riemannian manifold with scalar curvature

Rg¯ > 0. Then for any ε > 0, there exists a smooth domain Ω such that Ω 6= M

and its ﬁrst Dirichlet eigenvalue of Laplacian satisﬁes

λ1(Ω, g¯) < ε.

82 In particular, we can take Ω such that

λ1(Ω, g¯) < Λ,

Rg¯(x) where Λ := minx∈M n−1 > 0.

Proof. For any domain Ω & M, we have

R |∇ϕ|2dv λ (Ω, g¯) = inf{ Ω g¯ : ϕ| ≡ 0}. 1 R 2 ∂Ω Ω ϕ dvg¯

For any p ∈ M and any r > 0, let Br(p) and B2r(p) be geodesic balls around

p with radii r and 2r respectively.

We take Ωr := M −Br(p) and 0 ≤ ϕ ≤ 1 a smooth test function which satisﬁes

that   ϕ = 1 on M − B (p)  2r   ϕ = 0 on Br(p)     2 |∇ϕ| ≤ r on B2r(p) − Br(p).

Then ϕ is supported in Ωr and we have

R 2 n−2 |∇ϕ| dvg¯ 4 V ol (B (p)) c r Ωr ≤ · g¯ 2r ≤ 0 R 2 2 ϕ dv r V olg¯(M − B2r(p)) V olg¯(M − B2r(p)) Ωr g¯

for r suﬃciently small, where c0 is a constant depends only on n.

Thus for n ≥ 3,

R 2 n−2 |∇ϕ| dvg¯ c r λ (Ω , g¯) ≤ Ωr ≤ 0 → 0, 1 r R 2 ϕ dv V olg¯(M − B2r(p)) Ωr g¯

83 as r → 0 and hence we can ﬁnd some r0 > 0 such that for any ε > 0 we have

λ1(Ωr0 , g¯) < ε.

As for n = 2, if M is orientable, then M is diﬀeomorphic to the standard

2 2u sphere S since Rg¯ > 0. And there is a smooth function u such that e g¯ = gS2 ,

which is the canonical spherical metric on S2.

2 Let q ∈ S be the south pole and Br(q) a geodesic ball with respect to gS2

2 centered at q with radius 0 < r < π. Take Ωr := S − Br(q), then for any smooth

∞ function ϕ ∈ C0 (Ωr),

R 2 R 2 R 2 |∇ϕ| dvolg¯ |∇ϕ| dvolg |∇ϕ| dvolg λ (Ω , g¯) ≤ Ωr = Ωr S2 ≤ c · Ωr S2 , 1 r R ϕ2dvol R ϕ2e2udvol 1 R ϕ2dvol Ωr g¯ Ωr gS2 Ωr gS2

−2 minM u where c1 := e is independent of r. Therefore,

λ1(Ωr, g¯) ≤ c1λ1(Ωr, gS2 ) → 0,

as r → 0 (cf. Theorem 6, P. 50 in [22]). Hence for any ε > 0, we can ﬁnd an

r0 > 0 such that

λ1(Ωr0 , g¯) < ε.

Suppose M is not orientable, then M is diﬀeomorphic to RP 2, the real pro-

jective plane whose double covering S2. Let p, q be the north and south pole of

2 ˜ 2 π 2 S and consider the domain Ωr = S − (Br(p) ∪ Br(q)), 0 < r < 2 on S . For

2 any ε > 0, take r0 > 0 suﬃciently small such that its quotient Ωr0 ⊂ RP is a

84 domain. With similar calculations, we can ﬁnd an r0 such that

˜ 2 λ1(Ωr0 , g¯) ≤ c1λ1(Ωr0 , gS ) < ε.

Proof of Corollary 1.3.14. Let Ω ⊂ M be the domain in Lemma 4.2.5 with

λ1(Ω, g¯) < Λ,

Rg¯(x) where Λ := minx∈M n−1 > 0. Then we have

Λ1(Ω, g¯) ≤ λ1(Ω, g¯) − Λ < 0.

The conclusion follows from Theorem 1.3.13.

And for noncompact case:

Proof of corollary 1.3.15. Since (M, g¯) has quadratic volume growth, we have the

ﬁrst Dirichlet eigenvalue of Laplacian satisﬁes that

λ1(M, g¯) = 0

(see Proposition 9 in [15]) and it implies that for any ε > 0 there is a compactly contained domain Ω ⊂ M with

λ1(Ω, g¯) < ε.

85 Q In particular, if we take ε < 2 , we have

Q Λ (Ω, g¯) ≤ λ (Ω, g¯) − Q < − < 0. 1 1 2

Now by Theorem 1.3.13, the corollary follows.

Finally, we would like to discuss the subtle issue of domains which has critical eigenvalue

Λ1(Ω, g¯) = 0 brieﬂy. According to [4], in order to get rigidity we only need to assume g =g ¯ on

∂Ω and Rg ≥ Rg¯ on Ω, which means the Brown-York mass can only be vanished in this case and we won’t get its positivity. But it is not clear right now whether the same phenomena appear when we assume scalar curvature nonincreasing instead and we hope this issue can be addressed in the future.

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94 Appendix A

Variations of curvature

In this part, we will give the calculations of the 1st and 2nd variations formulae

of curvature. Note that, some of the conventions we adopted here may be diﬀerent

from the one in the main part of the dissertation.

Let (M, g) be an n-dimensional manifold and g(t) be a one-parameter family

of metrics, where g(t) = g + th and h is a symmetric 2-tensor.

n For a any p ∈ M ﬁxed, let {∂i}i=0 be a normal coordinates around p with

k respect to the metric g. We have gij = δij and ∂kgij = 0 and hence Γij = 0.

95 We adopt the convention for Laplacian as

ij ∆ = g ∇i∇j, (A.0.1) and formal adjoint of ∇ as

j (δh)i = −(divh)i = −∇ hij, (A.0.2)

where hij is a symmetric 2-tensor.

A.1 Metrics

Proposition A.1.1 (Variations of metrics).

g˙ ij = −hij (A.1.1) and

ij j ik g¨ = 2hkh . (A.1.2)

kj j Proof. Since glkg = δl ,

kj kj g˙lkg + glkg˙ = 0.

Thus

ij kj li ij g˙ = −g˙lkg g = −h .

96 Similarly,

kj kj kj g¨lkg + 2g ˙lkg˙ + glkg¨ = 0.

Since g¨ij = 0,

ij kj il kj il i kj g¨ = −2g ˙lkg˙ g = 2hlkh g = 2hkh .

A.2 Christoﬀel symbols

Proposition A.2.1 (Variations of Christoﬀel symbols).

1 Γ˙ k = gkl(∇ h + ∇ h − ∇ h ) (A.2.1) ij 2 i jl j il l ij and

¨k kl Γij = −h (∇ihjl + ∇jhil − ∇lhij) (A.2.2)

Proof. We have

1 1 Γ˙ k = g˙ kl(∂ g + ∂ g − ∂ g ) + gkj(∂ g˙ + ∂ g˙ − ∂ g˙ ) ij 2 i jl j il l ij 2 i jl j il l ij 1 = gkl(∇ h + ∇ h − ∇ h ) 2 i jl j il l ij

97 and

1 Γ¨k = g¨kl(∂ g + ∂ g − ∂ g ) +g ˙ kl(∂ g˙ + ∂ g˙ − ∂ g˙ ) ij 2 i jl j il l ij i jl j il l ij 1 + gkl(∂ g¨ + ∂ g¨ − ∂ g¨ ) 2 i jl j il l ij

kl = − h (∇ihjl + ∇jhil − ∇lhij).

A.3 Riemannian curvature tensor

For a tensor T , we denote

Alt{Tij} := Tij − Tji i,j to be the anti-symmetrization of T with respect to indices i and j.

Proposition A.3.1 (Variations of Riemannian curvature tensor of type (3,1)).

˙ l 1 l l l Rijk = Alt{(∇i∇jhk + ∇i∇khj − ∇i∇ hjk)} (A.3.1) 2 i,j

and

¨l pl Rijk = − h Alt{(∇i∇jhkp + ∇i∇khjp − ∇i∇phjk} (A.3.2) i,j

1 p p p l l l − Alt{(∇jh + ∇khj − ∇ hjk)(∇ihp − ∇phi + ∇ hip)} 2 i,j k

Proof. We have

l l l p l p l l p l Rijk = ∂iΓjk − ∂jΓik + Γ Γip − Γ Γjp = Alt{∂iΓjk + Γ Γip}. jk ik i,j jk

98 Thus,

˙ l ˙ l ˙ p l p ˙ l Rijk =Alt{∂iΓjk + Γ Γip + Γ Γip} i,j jk jk 1 pl =Alt ∂i g (∂jhkp + ∂khjp − ∂phjk) i,j 2

1 pl = Alt g ∇i(∇jhkp + ∇khjp − ∇phjk) 2 i,j

1 l l l = Alt{(∇i∇jhk + ∇i∇khj − ∇i∇ hjk)}. 2 i,j

Similarly,

n o ¨l ¨l ˙ p ˙ l Rijk =Alt ∂iΓjk + 2Γ Γip i,j jk

pl = − Alt ∂i(h (∂jhkp + ∂khjp − ∂phjk)) i,j

1 pq rl + Alt g (∂jhkq + ∂khjq − ∂qhjk)g (∂ihpr + ∂phir − ∂rhip) 2 i,j

pl = − Alt ∇i(h (∇jhkp + ∇khjp − ∇phjk)) i,j

1 pq rl + Alt g (∇jhkq + ∇khjq − ∇qhjk)g (∇ihpr + ∇phir − ∇rhip) 2 i,j

pl pl = − Alt (∇ih )(∇jhkp + ∇khjp − ∇phjk) + h ∇i(∇jhkp + ∇khjp − ∇phjk) i,j

1 p p p l l l + Alt (∇jh + ∇khj − ∇ hjk)(∇ihp + ∇phi − ∇ hip) 2 i,j k

pl = − h Alt {∇i∇jhkp + ∇i∇khjp − ∇i∇phjk)} i,j

1 p p p l l l + Alt (∇jh + ∇khj − ∇ hjk)(−∇ihp + ∇phi − ∇ hip) 2 i,j k

99 pl = − h Alt {∇i∇jhkp + ∇i∇khjp − ∇i∇phjk} i,j

1 p p p l l l − Alt (∇jh + ∇khj − ∇ hjk)(∇ihp − ∇phi + ∇ hip) . 2 i,j k

Proposition A.3.2 (Variations of Riemannian curvature tensor of type (4,0)).

˙ p 1 Rijkl =hplR + Alt {∇i∇jhkl + ∇i∇khjl − ∇i∇lhjk} (A.3.3) ijk 2 i,j and

¨ 1 p p p Rijkl = − Alt (∇jh + ∇khj − ∇ hjk)(∇ihpl − ∇phil + ∇lhip) . (A.3.4) 2 i,j k

Proof. By Proposition A.3.1, we have

˙ p ˙ p Rijkl =g ˙plRijk + gplRijk

p 1 p p p =hplR + gplAlt ∇i∇jh + ∇i∇khj − ∇i∇ hjk ijk 2 i,j k

p 1 =hplR + Alt {∇i∇jhkl + ∇i∇khjl − ∇i∇lhjk} ijk 2 i,j

and

¨ p ˙ p ¨p Rijkl =¨glpRijk + 2g ˙lpRijk + glpRijk

˙ p ¨p =2g ˙lpRijk + glpRijk

100 p p p =hlpAlt ∇i∇jh + ∇i∇khj − ∇i∇ hjk i,j k

qp − glpAlt {h (∇i∇jhkq + ∇i∇khjq − ∇i∇qhjk)} i,j

1 q q q p p p − glpAlt (∇jh + ∇khj − ∇ hjk)(∇ihq − ∇qhi + ∇ hiq) 2 i,j k

p p p =hlpAlt ∇i∇jh + ∇i∇khj − ∇i∇ hjk i,j k

q − h Alt {∇i∇jhkq + ∇i∇khjq − ∇i∇qhjk} l i,j

1 q q q − Alt (∇jh + ∇khj − ∇ hjk)(∇ihql − ∇qhil + ∇lhiq) 2 i,j k

1 p p p = − Alt (∇jh + ∇khj − ∇ hjk)(∇ihpl − ∇phil + ∇lhip) . 2 i,j k

A.4 Ricci tensor

Proposition A.4.1 (Variations of Ricci curvature tensor).

1 R˙ = − (∆ h + ∇ ∇ (trh) + ∇ (δh) + ∇ (δh) ) (A.4.1) jk 2 L jk j k j k k j 1 = − (∆Lhjk + (LX g)jk), (A.4.2) 2

1 ] where X := 2 d(trh) + δh and

101 ¨ pi l l Rjk =h (Rijklhp + Rijplhk − ∇i∇khjp + ∇i∇phjk + ∇j∇khip − ∇j∇phik) 1 + (∇ hp + ∇ hp − ∇ph )(2(δh) + ∇ (trh)) 2 j k k j jk p p 1 + (∇ hp + ∇ hp − ∇ph )(∇ hi − ∇ hi + ∇ih ), (A.4.3) 2 i k k i ik j p p j jp where ∆L is the Lichnerowicz Laplacian, which is deﬁned by

il i i ∆Lhjk = ∆hjk + 2Rijklh − Rjihk − Rkihj. (A.4.4)

Proof. By Proposition A.3.1,

˙ i ˙ l Rjk =δl Rijk

1 i l l l = δl Alt ∇i∇jhk + ∇i∇khj − ∇i∇ hjk 2 i,j 1 = [(∇ ∇ hi − ∇ ∇ hi ) + (∇ ∇ hi − ∇ ∇ hi) + (∇ ∇ih − ∇ ∇ih )] 2 i j k j i k i k j j k i j ik i jk 1 = [(∇ ∇ hi + ∇ (δh) ) + (∇ ∇ hi − ∇ ∇ (trh)) + (−∇ (δh) − ∆h )] 2 i j k j k i k j j k j k jk 1 = [(∇ ∇ hi + ∇ ∇ hi − ∇ ∇ (trh) − ∆h )] 2 i j k i k j j k jk 1 = [(∇ ∇ hi − Rp hi + Ri hp) + (∇ ∇ hi − Rp hi + Ri hp)] 2 j i k ijk p ijp k k i j ikj p ikp j 1 − [∇ ∇ (trh) + ∆h )] 2 j k jk 1 = (−∇ (δh) − ∇ (δh) − R hpi − R hpi + R hp + R hp) 2 j k k j ijkp ikjp jp k kp j 1 − (∇ ∇ (trh) + ∆h ) 2 j k jk 1 = [−∆h − 2R hil + R hp + R hp − ∇ ∇ (trh) − ∇ (δh) − ∇ (δh) ] 2 jk ijkl jp k kp j j k j k k j 1 = − (∆ h + ∇ ∇ (trh) + ∇ (δh) + ∇ (δh) ). 2 L jk j k j k k j

102 Since

∇j∇k(trh) + ∇j(δh)k + ∇k(δh)j 1 1 =∇ ∇ (trh) − (δh) + ∇ ∇ (trh) − (δh) j 2 k k k 2 j j

=(LX g)jk, we have

˙ 1 Rjk = − (∆Lhjk + (LX g)jk). 2

Similarly,

¨ i ¨l Rjk =δl Rijk

pi = − h Alt {∇i∇jhkp + ∇i∇khjp − ∇i∇phjk} i,j 1 − (∇ hp + ∇ hp − ∇ph )(∇ hi − ∇ hi + ∇ih ) 2 j k k j jk i p p i ip 1 + (∇ hp + ∇ hp − ∇ph )(∇ hi − ∇ hi + ∇ih ) 2 i k k i ik j p p j jp

pi l l = − h (−Rijkhlp − Rijphlk + ∇i∇khjp − ∇i∇phjk − ∇j∇khip + ∇j∇phik) 1 − (∇ hp + ∇ hp − ∇ph )(−(δh) − ∇ (trh) − (δh) ) 2 j k k j jk p p p 1 + (∇ hp + ∇ hp − ∇ph )(∇ hi − ∇ hi + ∇ih ) 2 i k k i ik j p p j jp

pi l l =h (Rijklhp + Rijplhk − ∇i∇khjp + ∇i∇phjk + ∇j∇khip − ∇j∇phik) 1 + (∇ hp + ∇ hp − ∇ph )(2(δh) + ∇ (trh)) 2 j k k j jk p p 1 + (∇ hp + ∇ hp − ∇ph )(∇ hi − ∇ hi + ∇ih ). 2 i k k i ik j p p j jp

103 A.5 Scalar curvature

Proposition A.5.1 (The variation of scalar curvature).

˙ R = γgh := −∆(trh) + δδh − h · Ric (A.5.1) and

1 1 R¨ = − 2γ (h × h) − ∆(|h|2) − |∇h|2 − |d(trh)|2 (A.5.2) g 2 2

2 jk i + 2h · ∇ (trh) − 2(δh) · d(trh) + ∇ih ∇jhk,

k where (h × h)ij := hikhj .

Proof. By Proposition C.6.1,

˙ jk jk ˙ R =g ˙ Rjk + g Rjk 1 = −hjkR − gjk (∆ h + ∇ ∇ (trh) + ∇ (δh) + ∇ (δh) ) jk 2 L jk j k j k k j 1 = −hjkR − (2∆(trh) − 2δδh) jk 2 = −∆(trh) + δδh − h · Ric.

And we have

¨ il jk il jk il jk ¨ il jk il jk ˙ il jk ˙ R =¨g g Rijkl + g g¨ Rijkl + g g Rijkl + 2g ˙ g˙ Rijkl + 2g ˙ g Rijkl + g g˙ Rijkl

il il jk ¨ il jk il jk ˙ =2¨g Ril + g g Rijkl + 2h h Rijkl − 4h g Rijkl

i kl il jk il jk p ˙ p il jk ¨ =4hkh Ril + 2h h Rijkl − 4h g (hlpRijk + glpRijk) + g g Rijkl

104 i kl il jk il p i jk ˙ p il jk ˙ p ¨p =4hkh Ril + 2h h Rijkl − 4h hl Rip − 4hpg Rijk + g g (2hlpRijk + glpRijk)

il jk i jk ˙ p jk i ˙ p jk i ¨p =2h h Rijkl − 4hpg Rijk + 2g hpRijk + g δpRijk

il jk i jk ˙ p jk ¨ =2h h Rijkl − 2hpg Rijk + g Rjk.

For the second second term,

i jk ˙ p i jk p p p 2hpg R =hpg Alt ∇i∇jh + ∇i∇khj − ∇i∇ hjk ijk i,j k

i p p p =hp(−2∇i(δh) − ∆hi − ∇i∇ (trh))

= − 2h · ∇(δh) − h · ∆h − h · ∇2(trh).

And for the third term,

jk ¨ jk pi l l g Rjk =g h (Rijklhp + Rijplhk − ∇i∇khjp + ∇i∇phjk + ∇j∇khip − ∇j∇phik) 1 + gjk(∇ hp + ∇ hp − ∇ph )(2(δh) + ∇ (trh)) 2 j k k j jk p p 1 + gjk(∇ hp + ∇ hp − ∇ph )(∇ hi − ∇ hi + ∇ih ) 2 i k k i ik j p p j jp

pi l lj j =h [Rilhp + Rijplh + ∇i(δh)p + ∇i∇p(trh) + ∆hip − ∇j∇phi ] 1 + (−2(δh)p − ∇p(trh))(2(δh) + ∇ (trh)) 2 p p 1 + (∇ hjp + ∇jhp − ∇phj)(∇ hi − ∇ hi + ∇ih ) 2 i i i j p p j jp

105 pi l lj =h [Rilhp + Rijplh + ∇i(δh)p + ∇i∇p(trh)]

pi j q j j q + h [∆hip − ∇p∇jhi + Rjpihq − Rjpqhi ] 1 3 − [4|δh|2 + 4(δh) · d(trh) + |d(trh)|2] + |∇h|2 − ∇ihjp∇ h 2 2 j ip

pi lj lj =h [Rijplh + 2∇i(δh)p + ∇i∇p(trh) + ∆hip + Rjpilh ] 1 3 − [4|δh|2 + 4(δh) · d(trh) + |d(trh)|2] + |∇h|2 − ∇ihjp∇ h 2 2 j ip

On the other hand,

2 i jk i jk 2 ∆(|h| ) = ∇ ∇i(h hjk) = ∇ (2hjk∇ih ) = 2h∆h + 2|∇h| . and

i jk jk i i jk δδ(h × h) =∇i∇j(hkh ) = ∇i(h ∇jhk + hk∇jh )

jk i 2 jk i =∇ih ∇jhk + |δh| − h · ∇δh + h ∇i∇jhk

jk i 2 jk i l i i l =∇ih ∇jhk + |δh| − h · ∇δh + h (∇j∇ihk − Rijkhl + Rijlhk)

jk i 2 il jk =∇ih ∇jhk + |δh| − 2h · ∇δh − Rijklh h + Ric · (h × h).

106 i.e.

il jk jk i 2 Rijklh h = −δδ(h × h) + ∇ih ∇jhk + |δh| − 2h · ∇δh + Ric · (h × h).

Now we have

¨ il jk i jk ˙ p jk ¨ R =2h h Rijkl − 2hpg Rijk + g Rjk

il jk 2 =2h h Rijkl + 4h · ∇δh + 2h · ∆h + 2h · ∇ (trh) 1 3 − 2|δh|2 − 2|δh| · d(trh) − |d(trh)|2 + |∇h|2 − ∇ihjp∇ h 2 2 j ip

jk i 2 =2[−δδ(h × h) + ∇ih ∇jhk + |δh| − 2h · ∇δh + Ric · (h × h)]

+ 4h · ∇δh + ∆(|h|2) − 2|∇h|2 + 2h · ∇2(trh) 1 3 − 2|δh|2 − 2(δh) · d(trh) − (d(trh))2 + |∇h|2 − ∇ihjp∇ h 2 2 j ip = − 2[−∆(tr(h × h)) + δδ(h × h) − 2Ric · (h × h)] − ∆(|h|2) 1 1 − |∇h|2 − |d(trh)|2 + 2h · ∇2(trh) − 2(δh) · d(trh) + ∇ hjk∇ hi 2 2 i j k 1 1 = − 2γ (h × h) − ∆(|h|2) − |∇h|2 − |d(trh)|2 + 2h · ∇2(trh) g 2 2

jk i − 2(δh) · d(trh) + ∇ih ∇jhk.

107 Appendix B

Conformal transformations of

Riemannian metric

In this part, we give the computations of various of curvature under conformal transformation of Riemannian metric. Note that, some of the conventions we adopted here may be diﬀerent from the one in the main part of the dissertation.

Let (M, g) be a Riemannian manifold. For any p ∈ M, take a local coordinates

n 2f {∂i}i=1 round p. Consider g˜ = e g a metric conformally related to g.

108 B.1 Metric

Proposition B.1.1.

2f g˜ij = e gij, (B.1.1)

g˜ij = e−2f gij. (B.1.2)

B.2 Christoﬀel symbols

Proposition B.2.1.

˜k k k Γij = Γij + Aij, (B.2.1)

where the tensor

k k k k Aij = δj ∇if + δi ∇jf − gij∇ f.

Proof.

1 Γ˜k = g˜kl (∂ g˜ + ∂ g˜ − ∂ g˜ ) ij 2 i jl j il l ij 1 = e−2f gkl (∂ e2f )g + (∂ e2f )g − (∂ e2f )g + e2f (∂ g + ∂ g − ∂ g ) 2 i jl j il l ij i jl j il l ij 1 = gkl (∂ g + ∂ g − ∂ g ) + gkl [(∂ f)g + (∂ f)g − (∂ f)g ] 2 i jl j il l ij i jl j il l ij

k k k k = Γij + (δj ∇if + δi ∇jf − gij∇ f).

109 B.3 Hessian and Laplacian

Proposition B.3.1. Let u be a smooth function, then

˜ ˜ k ∇i∇ju = ∇i∇ju − Aij∇ku (B.3.1)

and

∆˜ u = e−2f (∆u + (n − 2) h∇f, ∇ui) . (B.3.2)

Proof. We have

˜ ˜ ˜k k k ∇i∇ju − ∇i∇ju = − Γij − Γij ∂ku = −Aij∇ku

and

˜ ij ˜ ˜ −2f ij k −2f ∆u =g ˜ ∇i∇ju = e g ∇i∇ju − Aij∇ku = e (∆u + (n − 2) h∇f, ∇ui) .

B.4 Second fundamental form and mean curva-

ture

Proposition B.4.1. Let Σn−1 ,→ M n be a hypersurface and choose local coordinates at any p ∈ Σ such that ∂n = ν is the normal to Σ. We have

˜ f Aij = e (Aij + gij∂νf) (B.4.1)

110 and

˜ −f H = e (H + (n − 1)∂νf) . (B.4.2)

Thus the traceless fundamental form is

1 1 A˜ − H˜ g˜ = ef A − Hg . (B.4.3) ij n − 1 ij ij n − 1 ij

Proof. We have

g˜(˜ν, ν˜) = e2f g(˜ν, ν˜) = g(ef ν,˜ ef ν˜) = 1. i.e. ν˜ = e−f ν.

Now the second fundamental form is

1 1 1 A˜ = ∂ g˜ = e−f ∂ e2f g = ef ∂ (g ) + ef g ∂ f = ef (A + g ∂ f) ij 2 ν˜ ij 2 ν ij 2 ν ij ij ν ij ij ν

and the mean curvature is

˜ ij ˜ −2f ij f −f H =g ˜ Aij = e g · e (Aij + gij∂νf) = e (H + (n − 1)∂νf) .

B.5 Riemann curvature tensor

For a tensor T , we denote

Alt{Tij} := Tij − Tji i,j

111 to be the anti-symmetrization of T with respect to indices i and j.

Recall that the Kulkarni-Nomizu product of symmetric 2-tensors αij and βij is deﬁned to be

(α β)ijkl := αilβjk + αjkβil − αikβjl − αjlβik = Alt{αilβjk + αjkβil}. (B.5.1) ? i,j

Proposition B.5.1.

˜l l l l Rijk = Rijk + Alt{gikaj + δjaik} (B.5.2) i,j and

˜ 2f Rijkl = e Rijkl − (a g)ijkl , (B.5.3) ? where 1 a = ∇ ∇ f − ∇ f∇ f + |∇f|2g . ij i j i j 2 ij

n Proof. Choose {∂i}i=1 to be a normal coordinates around p with respect to the

k metric g. Then we have ∂igjk(p) = 0 and thus Γij(p) = 0. Now we have

l l Rijk = Alt{∂iΓjk} i,j

˜l ˜l ˜m ˜l Rijk = Alt{∂iΓjk + ΓjkΓim}. i,j

Therefore,

n o ˜l l ˜l l ˜m ˜l l m l Rijk − Rijk = Alt ∂i Γjk − Γjk + ΓjkΓim = Alt ∇iAjk + AjkAim . i,j i,j

112 While

l l l Alt ∇iAjk = Alt δj∇i∇kf + gik∇j∇ f i,j i,j and

m l l l 2 l Alt AjkAim = Alt δi∇jf∇kf − δigjk|∇f| − gik∇jf∇ f , i,j i,j thus

˜l l Rijk − Rijk

l l l 2 l =Alt δj∇i∇kf + gik∇j∇ f − δj ∇if∇kf − |∇f| gik − gik∇jf∇ f i,j l 1 2 =Alt δj ∇i∇kf − ∇if∇kf + |∇f| gik i,j 2 l l 1 2 l + Alt gik ∇j∇ f − ∇jf∇ f + |∇f| δj i,j 2

l l =Alt δjaik + gikaj . i,j

Hence

˜ ˜m Rijkl =g ˜lmRijk 2f m m m = e glm Rijk + Alt δj aik + gikaj i,j 2f = e Rijkl + Alt {gjlaik + gikajl} i,j 2f = e Rijkl − Alt {gilajk + gjkail} i,j

2f = e Rijkl − (a g)ijkl . ?

113 B.6 Ricci curvature tensor

Note that, we adopt the convention for Ricci tensor that

i il Rjk = Rijk = g Rijkl.

Proposition B.6.1.

˜ 2 Rjk = Rjk − (n − 2)∇j∇kf − gjk∆f + (n − 2)∇jf∇kf − (n − 2)|∇f| gjk.

(B.6.1)

Proof. By Proposition C.5.1, we have

˜ ˜i i Rjk − Rjk = Rijk − Rijk

i i i i = gikaj − gjkai + δjaik − δiajk

i = −(n − 2)ajk − gjkai

2 = −(n − 2)∇j∇kf − gjk∆f + (n − 2)∇jf∇kf − (n − 2)|∇f| gjk.

B.7 Scalar curvature

Proposition B.7.1.

R˜ = e−2f R − 2(n − 1)∆f − (n − 1)(n − 2)|∇f|2 . (B.7.1)

114 For n = 2, let u2 = e2f , i.e. f = log u, then

K˜ = u−2 (K − ∆ log u) , (B.7.2)

R where K = 2 is the Gaussian curvature.

4 n−2 2f 2 For n ≥ 3, let u = e , i.e. f = n−2 log u, then − n+2 4(n − 1) R˜ = u n−2 Ru − ∆u . (B.7.3) n − 2

Proof. By Proposition C.6.1,

˜ jk ˜ R =g ˜ Rjk

−2f i = e R − 2(n − 1)ai

= e−2f R − 2(n − 1)∆f − (n − 1)(n − 2)|∇f|2 .

Let n = 2 and f = log u, we get equation (B.7.2) immediately.

2 For n ≥ 3, let f = n−2 log u, we get − 4 4(n − 1) 4(n − 1) 2 R˜ = u n−2 R − ∆ log u − |∇ log u| n − 2 n − 2 − n+2 4(n − 1) = u n−2 Ru − ∆u . n − 2

Remark B.7.2. For n ≥ 3, let

Lg := −∆g + aRg

115 n−2 be the conformal Laplacian of g, where a := 4(n−1) . Then we have

˜ p−1 Lgu = aRu , (B.7.4)

2n where p = n−2 .

B.8 Traceless Ricci tensor

Let 1 E = R − Rg jk jk n jk be the traceless Ricci tensor.

Proposition B.8.1.

1 E˜ = E + (n − 2)v−1 ∇ ∇ v − ∆vg , (B.8.1) jk jk j k n jk

where v = e−f i.e. f = − log v.

Proof. By Propositions C.6.1 and C.7.1,

1 E˜ = R˜ − R˜g˜ jk jk n jk 1 = R − (n − 2)a − g ai − R − 2(n − 1)ai g jk jk jk i n i jk 1 = E − (n − 2) a − aig jk jk n i jk 1 1 = E − (n − 2) ∇ ∇ f − ∇ f∇ f − ∆fg + |∇f|2g jk j k j k n jk n jk 1 = E + (n − 2)v−1 ∇ ∇ v − ∆vg . jk j k n jk

116 B.9 Schouten tensor

Let

1 1 S = R − Rg (B.9.1) jk n − 2 jk 2(n − 1) jk be the Schouten tensor.

Proposition B.9.1.

˜ Sjk = Sjk − ajk. (B.9.2)

Proof. By Propositions C.6.1 and C.7.1,

1 1 S˜ = R˜ − R˜g˜ jk n − 2 jk 2(n − 1) jk 1 1 = R − (n − 2)a − g ai − R − 2(n − 1)ai g n − 2 jk jk jk i 2(n − 1) i jk 1 = S − a − ai − ai g jk jk n − 2 i i jk

= Sjk − ajk.

B.10 Weyl tensor

Recall that Riemann curvature tensor can be expressed by

Rijkl = Wijkl + (S g)ijkl . (B.10.1) ?

117 Proposition B.10.1.

˜ 2f Wijkl = e Wijkl. (B.10.2)

Proof. By Propositions C.5.1 and C.9.1,

˜ ˜ ˜ Wijkl = Rijkl − S g˜ ? ijkl 2f ˜ = e Rijkl − (a g)ijkl − S g˜ ? ? ijkl 2f 2f ˜ = e Wijkl + e (S g)ijkl − (a g)ijkl − S g˜ ? ? ? ijkl 2f 2f ˜ = e Wijkl + e ((S − a) g)ijkl − S g˜ ? ? ijkl

2f = e Wijkl.

B.11 Cotten Tensor

Let

Cijk = ∇iSjk − ∇jSik = Alt {∇iSjk} (B.11.1) i,j be the Cotten tensor.

Proposition B.11.1.

˜ l Cijk = Cijk + Wijkl∇ f. (B.11.2)

118 Proof. By Proposition C.9.1,

˜ n ˜ ˜ o Cijk = Alt ∇iSjk i,j

n ˜ ˜p ˜ ˜p ˜ o = Alt ∂iSjk − ΓijSpk − Γ Sjp i,j ik

p p p p = Alt ∂i(Sjk − ajk) − (Γij + Aij)(Spk − apk) − (Γ + A )(Sjp − ajp) i,j ik ik

p = Alt {∇i(Sjk − ajk) − A (Sjp − ajp)} i,j ik

p p p = Cijk − Alt {∇iajk + (δi fk + δ fi − gikf )(Sjp − ajp)} i,j k

l l = Cijk − f Alt {Sjkgil − Sjlgik)} + f Alt {ajkgil − ajlgik)} − Alt {∇iajk} i,j i,j i,j

l l = Cijk − (S g)ijklf + (a g)ijklf − Alt {∇iajk} . ? ? i,j

By Ricci identity,

1 2 −Alt {∇iajk} = −Alt ∇ifjk − fijfk − fjfik + ∇i|∇f| gjk i,j i,j 2 l 1 2 = Rijklf + Alt fjfik − ∇i|∇f| gjk i,j 2

l l = Rijklf + f Alt {gjlfik − gjkfil} i,j

l 2 l = Rijklf − (∇ f g)ijklf . ?

Therefore,

˜ l l l 2 l Cijk = Cijk − (S g)ijklf + (a g)ijklf + Rijklf − (∇ f g)ijklf ? ? ? l 2 l = Cijk + (Rijkl − (S g)ijkl)f + ((a − ∇ f) g)ijklf ? ? l = Cijk + Wijklf ,

119 where

2 l 1 2 l l ((a − ∇ f) g)ijklf = |∇f| (g g)ijklf − ((df ⊗ df) g)ijklf = 0. ? 2 ? ?

B.12 Bach tensor

For n ≥ 4, the Bach tensor is deﬁned to be

1 1 B = ∇i∇lW + SilW . (B.12.1) jk n − 3 ijkl n − 2 ijkl

Note that with our convention of Schouten tensor,

l ∇ Wijkl = (n − 3)Cijk, (B.12.2) we can extend the deﬁnition of Bach tensor to any n ≥ 3:

i il Bjk = ∇ Cijk + S Wijkl. (B.12.3)

For a tensor T , we denote

Sym{Tij} := Tij + Tji i,j to be the symmetrization of T with respect to indices i and j.

Proposition B.12.1.

˜ −2f i i l Bjk = e Bjk + (n − 4) f Sym {Cijk} + Wijklf f . (B.12.4) j,k

120 Proof. By Proposition C.11.1,

˜ i ˜ il ˜ ˜ ∇ Cijk =g ˜ ∇lCijk −2f il ˜ ˜p ˜ ˜p ˜ ˜p ˜ = e g ∂lCijk − ΓliCpjk − ΓljCipk − ΓlkCijp −2f il q p ˜ p ˜ p ˜ = e g ∇l(Cijk + Wijkqf ) − AliCpjk − AljCipk − AlkCijp −2f i q il p ˜ p ˜ p ˜ = e ∇ (Cijk + Wijkqf ) − g AliCpjk + AljCipk + AlkCijp −2f i i l il il p ˜ p ˜ p ˜ = e ∇ Cijk + ∇ Wijklf + Wijklf − g AliCpjk + AljCipk + AlkCijp .

While

˜il ˜ ip lq ˜ ˜ S Wijkl =g ˜ g˜ SpqWijkl

−2f ip lq = e g g (Spq − apq)Wijkl

−2f il il = e S Wijkl − a Wijkl

−2f il il i l = e S Wijkl − Wijklf + Wijklf f ,

thus we have

˜ ˜ i ˜ ˜il ˜ −2f i il −2f Bjk = ∇ Cijk + S Wijkl = e ∇ Cijk + S Wijkl + Djk = e (Bjk + Djk) ,

where

i l i l il p ˜ p ˜ p ˜ Djk := Wijklf f + ∇ Wijklf − g AliCpjk + AljCipk + AlkCijp .

121 Since

il p ˜ p ˜ p ˜ − g AliCpjk + AljCipk + AlkCijp il p p p ˜ p p p ˜ = − g (δi fl + δl fi − gilf )Cpjk + (δj fl + δl fj − gjlf )Cipk il p p p ˜ − g δkfl + δl fk − gklf )Cijp p ˜ p i ip i p ˜ p i ip i p ˜ = − (2 − n)f Cpjk + (δj f + g fj − δjf )Cipk + (δkf + g fk − δkf )Cijp i ˜ i ˜ ˜ = − (4 − n)f Cijk + f (Cijk + Cjki) i ˜ ˜ =f (n − 4)Cijk + Ckij , we have

i l i l i ˜ ˜ Djk = Wijklf f + ∇ Wijklf + f (n − 4)Cijk + Ckij

i l l i l l = Wijklf f + (n − 3)Clkjf + f (n − 4)(Cijk + Wijklf ) + (Ckij + Wkijlf )

i i l i l = (n − 4)f (Cijk + Cikj) + (n − 4)Wijklf f + (Wijkl + Wkijl)f f i i l i l = (n − 4) f Sym {Cijk} + Wijklf f + (Wijkl − Wljki)f f j,k i i l = (n − 4) f Sym {Cijk} + Wijklf f . j,k

Therefore,

˜ −2f i i l Bjk = e Bjk + (n − 4) f Sym {Cijk} + Wijklf f . j,k

122 Appendix C

Curvature in warped product

metrics

In this part, we give the computations of various of curvature in warped prod-

uct metric. Note that, some of the conventions we adopted here may be diﬀerent

from the one in the main part of the dissertation.

Let (M n, g) be a Riemannian manifold. Suppose M is diﬀeomorphic to R×M

2 2 n and g = dt + σ(t) g¯. For any p ∈ M, let {∂α}α=1 be a local coordinates around

n−1 p, where ∂n = ∂t and {∂i}i=1 is a local coordinates of p ∈ M.

123 C.1 Metric

Proposition C.1.1.

2 gtt = 1, gti = 0, gij = σ g¯ij (C.1.1) and

gtt = 1, gti = 0, gij = σ−2g¯ij. (C.1.2)

C.2 Christoﬀel symbols

Proposition C.2.1.

t i t Γtt = Γtt = Γti = 0, (C.2.1)

t 0 Γij = −σσ g¯ij, (C.2.2)

σ0 Γk = δk (C.2.3) tj σ j and

k ¯k Γij = Γij. (C.2.4)

124 Proof. By deﬁnition,

1 Γγ = gγδ (∂ g + ∂ g − ∂ g ) . αβ 2 α βδ β αδ δ αβ

Clearly,

t i t Γtt = Γtt = Γti = 0.

And we have

1 1 Γt = gtt (∂ g + ∂ g − ∂ g ) = − ∂ g = −σσ0g¯ , ij 2 i jt j it t ij 2 t ij ij

1 1 σ0 Γk = gkl (∂ g + ∂ g − ∂ g ) = gkl∂ g = δk (C.2.5) tj 2 t jl j tl l tj 2 t jl σ j and

1 1 Γk = gkl (∂ g + ∂ g − ∂ g ) = g¯kl (∂ g¯ + ∂ g¯ − ∂ g¯ ) = Γ¯k . (C.2.6) ij 2 i jl j il l ij 2 i jl j il l ij ij

C.3 Hessian and Laplacian

Proposition C.3.1. Let u be a smooth function, then

2 ∇t∇tu = ∂t u, (C.3.1)

125 σ0 ∇ ∇ u = ∂ ∂ u − ∂ u, (C.3.2) t i t i σ i

¯ ¯ 0 ∇i∇ju = ∇i∇ju + σσ g¯ij∂tu (C.3.3)

and

σ0 1 ∆u = ∂2u + (n − 1) ∂ u + ∆¯ u. (C.3.4) t σ t σ2

Proof. We have

2 α 2 ∇t∇tu = ∂t u − Γtt∂αu = ∂t u,

σ0 σ0 ∇ ∇ u = ∂ ∂ u − Γα∂ u = ∂ ∂ u − Γk ∂ u = ∂ ∂ u − δk∂ u = ∂ ∂ u − ∂ u t i t i ti α t i ti k t i σ i k t i σ i

and

α t k ∇i∇ju = ∂i∂ju − Γij∂αu = ∂i∂ju − Γij∂tu − Γij∂ku

¯k 0 ¯ ¯ 0 = ∂i∂ju − Γij∂ku + σσ g¯ij∂tu = ∇i∇ju + σσ g¯ij∂tu.

Now

σ0 1 ∆u = gαβ∇ ∇ u = gtt∇ ∇ u + gij∇ ∇ u = ∂2u + (n − 1) ∂ u + ∆¯ u. α β t t i j t σ t σ2

126 C.4 Second fundamental form and mean curva-

ture

n−1 Proposition C.4.1. Consider M ,→ M n as a hypersurface, then

0 Aij = σσ g¯ij (C.4.1) and

σ0 H = (n − 1) . (C.4.2) σ

Thus the traceless fundamental form is

1 A − Hg = 0. (C.4.3) ij n − 1 ij

Proof. We have

1 1 A = ∂ g = ∂ σ2g¯ = σσ0g¯ ij 2 ν ij 2 t ij ij and

σ0 H = gijA = σ−2g¯ijσσ0g¯ = (n − 1) . ij ij σ

127 C.5 Riemann curvature tensor

For a tensor T , we denote

Alt{Tij} := Tij − Tji i,j to be the anti-symmetrization of T with respect to indices i and j.

Proposition C.5.1.

t 00 Rtjk = −σσ g¯jk, (C.5.1)

t t t l Rijk = Rttk = Rtjt = Rttt = 0 (C.5.2) and

l ¯l 0 2 l l Rijk = Rijk − (σ ) (¯gjkδi − g¯ikδj). (C.5.3)

Also,

00 Rtjkt = −σσ g¯jk, (C.5.4)

Rijkt = Rittt = 0 (C.5.5)

and

2 ¯ 0 2 Rijkl = σ Rijkl − (σ ) (¯gjkg¯il − g¯ikg¯jl) . (C.5.6)

128 n−1 Proof. Choose {∂i}i=1 to be a normal coordinates around p with respect to the

¯k metric g¯. Then we have ∂ig¯jk(p) = 0 and thus Γij(p) = 0.

By deﬁnition,

δ δ µ δ Rαβγ = Alt{∂αΓβγ + ΓβγΓαµ}. α,β

By the symmetry of Riemann curvature tensor,

t t l Rttk = Rtjt = Rttt = 0. (C.5.7)

And we have

t t t α t α t t t t i t Rtjk = ∂tΓjk − ∂jΓtk + ΓjkΓtα − ΓtkΓjα = ∂tΓjk − ΓtkΓjt + ΓtkΓji

0 2 00 0 2 00 = −((σ ) + σσ )¯gjk + (σ ) g¯jk = −σσ g¯jk,

t t α t 0 t t l t Rijk = Alt{∂iΓjk + ΓjkΓiα} = Alt{∂i(−σσ g¯jk) + ΓjkΓit + ΓjkΓil} i,j i,j

0 = Alt{−σσ ∂i(¯gjk)} = 0 i,j

and

l l α l l t l p l Rijk = Alt{∂iΓjk + ΓjkΓiα} = Alt{∂iΓjk + ΓjkΓit + Γ Γip} i,j i,j jk 0 ¯l 0 σ l ¯l 0 2 l l = Rijk + Alt{(−σσ g¯jk)( δi)} = Rijk − (σ ) (¯gjkδi − g¯ikδj). i,j σ

129 On the other hand,

t 00 Rtjkt = gttRtjk = −σσ g¯jk, (C.5.8)

t Rijkt = gttRijk = 0 (C.5.9) and

p 2 ¯ 0 2 Rijkl = glpRijk = σ Rijkl − (σ ) (¯gjkg¯il − g¯ikg¯jl) . (C.5.10)

C.6 Ricci curvature tensor

Note that, we adopt the convention for Ricci tensor that

α αδ Rβγ = Rαβγ = g Rαβγδ.

Proposition C.6.1. " ! # σ00 σ00 σ0 2 Ric = −(n − 1) dt2 + Ric − + (n − 2) σ2g¯ . (C.6.1) σ σ σ

Proof. By Proposition C.5.1, we have

σ00 R = gαβR = gttR + gjkR = −(n − 1) , tt tαβt tttt tjkt σ

130 αβ tt jk Rit = g Riαβt = g Rittt + g Rijkt = 0 and

αβ tt il Rjk = g Rαjkβt = g Rtjkt + g Rijkl

00 il ¯ 0 2 = −σσ g¯jk +g ¯ Rijkl − (σ ) (¯gjkg¯il − g¯ikg¯jl) ! σ00 σ0 2 = R¯ − + (n − 2) σ2g¯ . jk σ σ jk

C.7 Scalar curvature

Proposition C.7.1.

σ00 σ0 2 1 R = −2(n − 1) − (n − 1)(n − 2) + R.¯ (C.7.1) σ σ σ2

Proof. By Proposition C.6.1,

αβ tt jk R = g Rαβ = g Rtt + g Rjk " ! # σ00 1 σ00 σ0 2 = −(n − 1) + g¯jk R¯ − + (n − 2) σ2g¯ σ σ2 jk σ σ jk σ00 σ0 2 1 = −2(n − 1) − (n − 1)(n − 2) + R.¯ σ σ σ2

131 C.8 Traceless Ricci tensor

Let 1 E = Ric − Rg n be the traceless Ricci tensor.

Proposition C.8.1.

" # (n − 1)(n − 2) σ00 σ0 2 R¯ E = − − + dt2 (C.8.1) n σ σ (n − 1)(n − 2)σ2 " ! # 1 (n − 2) σ00 σ0 2 + Ric − R¯g¯ + − σ2g¯ . (C.8.2) n n σ σ

Proof. By Propositions C.6.1 and C.7.1,

1 E =Ric − Rg n " ! # σ00 σ00 σ0 2 = − (n − 1) dt2 + Ric − + (n − 2) σ2g¯ σ σ σ " # 1 σ00 σ0 2 1 − −2(n − 1) − (n − 1)(n − 2) + R¯ (dt2 + σ2g¯) n σ σ σ2 " # (n − 1)(n − 2) σ00 σ0 2 R¯ = − − + dt2 n σ σ (n − 1)(n − 2)σ2 " ! # (n − 2) σ00 σ0 2 R¯ + Ric + − − σ2g¯ n σ σ (n − 2)σ2 " # (n − 1)(n − 2) σ00 σ0 2 R¯ = − − + dt2 n σ σ (n − 1)(n − 2)σ2 " ! # 1 (n − 2) σ00 σ0 2 + Ric − R¯g¯ + − σ2g¯ . n n σ σ

132 C.9 Schouten tensor

Let

1 1 S = Ric − Rg (C.9.1) n − 2 2(n − 1) be the Schouten tensor.

Proposition C.9.1. ! 1 σ00 σ0 2 R¯ S = − 2 − + dt2 (C.9.2) 2 σ σ (n − 1)(n − 2)σ2 1 1 1 + Ric − R¯g¯ − σ02g¯ . (C.9.3) n − 2 2(n − 1) 2

Proof. By Propositions C.6.1 and C.7.1,

1 1 S = Ric − Rg n − 2 2(n − 1) " ! #! 1 σ00 σ00 σ0 2 = −(n − 1) dt2 + Ric − + (n − 2) σ2g¯ n − 2 σ σ σ " # 1 σ00 σ0 2 1 − −2(n − 1) − (n − 1)(n − 2) + R¯ (dt2 + σ2g¯) 2(n − 1)(n − 2) σ σ σ2 ! 1 σ00 σ0 2 R¯ = − 2 − + dt2 2 σ σ (n − 1)(n − 2)σ2 " ! # 1 1 σ0 2 R¯ + Ric − + σ2g¯ n − 2 2 σ 2(n − 1)(n − 2)σ2 ! 1 σ00 σ0 2 R¯ = − 2 − + dt2 2 σ σ (n − 1)(n − 2)σ2 1 1 1 + Ric − R¯g¯ − σ02g¯ . n − 2 2(n − 1) 2

133 C.10 Weyl tensor

Recall that the Kulkarni-Nomizu product of symmetric 2-tensors αij and βij

is deﬁned to be

(α β)ijkl := αilβjk + αjkβil − αikβjl − αjlβik = Alt{αilβjk + αjkβil}. (C.10.1) ? i,j

And Riemann curvature tensor can be expressed by

Rαβγδ = Wαβγδ + (S g)αβγδ . (C.10.2) ?

Proposition C.10.1.

1 W = − E¯ , (C.10.3) tjkt n − 2 jk

Wijkt = Wittt = 0 (C.10.4)

and

2 1 ¯ Wijkl = σ W ijkl + E g¯ , (C.10.5) (n − 2)(n − 3) ? ijkl where E¯ is the traceless Ricci tensor of g¯.

134 Proof. By Propositions C.5.1 and C.9.1,

Wtjkt =Rtjkt − (S g)tjkt ?

=Rtjkt − (Sttgjk + Sjkgtt) ! 1 σ00 σ0 2 R¯ = − σσ00g¯ + 2 − + σ2g¯ jk 2 σ σ (n − 1)(n − 2)σ2 jk 1 1 1 − R¯ − R¯g¯ − σ02g¯ n − 2 jk 2(n − 1) jk 2 jk 1 1 = − R¯ − R¯g¯ n − 2 jk n − 1 jk 1 = − E¯ , n − 2 jk

Wijkt =Rijkt − (S g)ijkt = Rijkt − (Sitgjk + Sjkgit − Sikgjt − Sjtgik) = 0 ? and

Wijkl =Rijkl − (S g)ijkl ? 2 ¯ 02 =σ Rijkl − σ (¯gjkg¯il − g¯ikg¯jl) 1 1 1 − σ2 Ric − R¯g¯ − σ02g¯ g¯ n − 2 2(n − 1) 2 ? ijkl 2 ¯ 1 2 02 1 2 1 ¯ =σ Rijkl − σ σ (¯g g¯)ijkl − σ Ric − Rg¯ g¯ 2 ? n − 2 2(n − 1) ? ijkl

1 2 02 + σ σ (¯g g¯)ijkl 2 ? 1 1 =σ2 R¯ − Ric − R¯g¯ g¯ n − 2 2(n − 1) ? ijkl

135 1 1 =σ2 W + (S¯ g¯) − Ric − R¯g¯ g¯ ? n − 2 2(n − 1) ? ijkl 1 1 =σ2 W + S¯ − Ric − R¯g¯ g¯ n − 2 2(n − 1) ? ijkl 2 1 ¯ =σ W ijkl + E g¯ ijkl . (n − 2)(n − 3) ? ijkl

C.11 Cotten Tensor

Let

Cαβγ = ∇αSβγ − ∇βSαγ = Alt {∇αSβγ} (C.11.1) α,β be the Cotten tensor.

Proposition C.11.1.

Cttt = Cijt = 0, (C.11.2)

1 σ0 1 C = − E¯ + ∂ R σ2g¯ , (C.11.3) tjk n − 2 σ jk n − 1 t jk

∂ R¯ C = − i , (C.11.4) itt 2(n − 1)(n − 2)σ2 and

1 ¯ ¯ 1 ¯ ¯ Cijk = Alt ∇iRjk − ∇iRg¯jk . (C.11.5) n − 2 i,j 2(n − 1)

136 Proof. By Proposition C.9.1,

Ctjk = Alt {∇tSjk} t,j

= ∇tSjk − ∇jStk

α α α α = (∂tSjk − ΓtjSαk − ΓtkSjα) − (∂jStk − ΓjtSαk − ΓjkStα)

i i i t = ∂tSjk − ΓtjSik − ΓtkSij + ΓjtSik + ΓjkStt σ0 = ∂ S − S − σσ0S t jk σ jk tt 1 σ0 1 = − E¯ + ∂ R σ2g¯ , n − 2 σ jk n − 1 t jk

1 ∇ R ∂ R¯ C = (∇ S − ∇ S ) = − i = − i itt n − 2 i tt t it 2(n − 1)(n − 2) 2(n − 1)(n − 2)σ2 and

1 ¯ ¯ 1 ¯ ¯ Cijk = Alt {∇iSjk} = Alt ∇iRjk − ∇iRg¯jk . i,j n − 2 i,j 2(n − 1)

137