DOI 10.1007/s12220-014-9532-z
Bach-Flat Critical Metrics of the Volume Functional
on 4-Dimensional Manifolds with Boundary
A. Barros · R. Diógenes· E. Ribeiro Jr.
Received: 8 November 2013 / Published online: 4 September 2014 © Mathematica Josephina, Inc. 2014
Abstract The purpose of this article is to investigate Bach-flat critical metrics of the volume functional on a compact manifold M with boundary∂M. Here, we prove that a Bach-flat critical metric of the volume functional on a simply connected 4-dimensional manifold with boundary isometric to a standard sphere must be isometric to a geodesic ball in a simply connected space formR4,H4orS4. Moreover, we show that in dimension three the result even is true replacing the Bach-flat condition by the weaker assumption thatMhas divergence-free Bach tensor.
Keywords Volume functional·Critical point equation·Bach-flat metrics
Mathematics Subject Classification Primary 53C25, 53C20, 53C21·Secondary 53C65
Communicated by Eduardo Garcia-Rio. A. Barros·R. Diógenes·E. Ribeiro Jr. (
B
)Departamento de Matemática, Universidade Federal do Ceará - UFC,
Campus do Pici, Av. Humberto Monte, Bloco 914, Fortaleza, CE 60455-760, Brazil e-mail: ernani@mat.ufc.br
A. Barros
e-mail: abarros@mat.ufc.br R. Diógenes
e-mail: rafaeljpdiogenes@gmail.com
Present Address:
E. Ribeiro Jr.
1 Introduction
A fruitful problem in Riemannian geometry is to study the critical points of the vol-ume functional associated with the space of smooth Riemannian structures. In the last decades very much attention has been given to study the critical points of the volume functional. Here, we shall study the space of smooth Riemannian structures on com-pact manifolds with boundary that satisfies a critical point equation associated with a boundary value problem.
Recently, inspired by a result obtained in [12] as well as by the characterization of the critical points of the scalar curvature functional, Miao and Tam studied variational properties of the volume functional constrained to the space of metrics of constant scalar curvature on a given compact manifold with boundary. For more details, we refer the reader to [15] and [16]. Afterward, in a celebrated article [11] Corvino, Eichmair and Miao studied this problem in a general context. In fact, they studied the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equal to a given constant. To do this, they localized a condition satisfied by such stationary points to smooth bounded domains. We now recall the definition of critical metrics studied by Miao and Tam. Here, for simplicity, these metrics will be called Miao–Tam critical metrics.
Definition 1 A Miao–Tam critical metric is a 3-tuple(Mn,g, f), where(Mn,g), is a compact Riemannian manifold of dimension at least three with a smooth boundary ∂M and f : Mn → Ris a smooth function such that f−1(0)=∂M satisfying the overdetermined-elliptic system
L∗
g(f)=g. (1.1)
Here,L∗gis the formalL2-adjoint of the linearization of the scalar curvature operator
Lg. Such a function f is called a potential function.
We recall thatL∗g(f)= −(f)g+Hess f−fRic;see, for instance, [5]. Therefore, the fundamental equation of a Miao–Tam critical metric (1.1) can be written as
−(f)g+Hess f − f Ric=g. (1.2)
We highlight that some explicit examples of Miao–Tam critical metrics are in the form of warped products. Those examples include the spatial Schwarzschild metrics and AdS-Schwarzschild metrics restricted to certain domains containing their horizon and bounded by two spherically symmetric spheres (cf. Corollaries 3.1 and 3.2 in [16]).
It is not to hard to show that critical metrics have constant scalar curvature [15]. In 2009, Miao and Tam were able to prove that these metrics arise as critical points of the volume functional onMnwhen restricted to the class of metricsgwith prescribed constant scalar curvature such thatg |T∂M=hfor a prescribed Riemannian metrich
on the boundary.
we want to know if there exist non-constant sectional curvature critical metrics on a compact manifold whose boundary is isometric to a standard round sphere. If yes, what can we say about the structure of such metrics?
Indeed, they studied these critical metrics under Einstein and conformally flat assumptions. In particular, they proved that a connected, compact, Einstein manifold (Mn, g)with smooth boundary that satisfies (1.2) must be isometric to a geodesic ball in a simply connected space formRn,HnorSn. Moreover, based on the techniques developed in a work of Kobayashi [13], Kobayashi and Obata [14], Miao and Tam showed that the result even is true replacing the Einstein condition by the assumption that(Mn,g)is locally conformally flat with boundary isometric to a standard sphere. More precisely, they proved the following result.
Theorem 1 (Miao–Tam, [16])Let(Mn,g, f)be a locally conformally flat simply connected, compact Miao–Tam critical metric with boundary isometric to a standard sphereSn−1. Then(Mn, g)is isometric to a geodesic ball in a simply connected space formRn,HnorSn.
It should be emphasized that the hypothesis that the boundary of Mnis isometric to a standard sphereSn−1considered by Miao and Tam is not artificial. To clarify this, we consider that the boundary ofMnis totally geodesic and is isometric to a standard sphereSn−1. Under these conditions, motivated by the positive mass theorem, Min-Oo conjectured that ifMnhas scalar curvature at leastn(n−1), thenMnmust be isometric to the hemisphereSn+with standard metric (cf. [17]). However, an elegant article due to Brendle, Marques and Neves shows counterexamples to Min-Oo’s conjecture in dimensionsn ≥3. For more details, see [6]. We also highlight thatSn
+satisfies (1.2)
for a suitable potential function (cf. [15] p. 153).
We now recall that the Bach tensor on a Riemannian manifold (Mn,g),n ≥ 4, which was introduced in the early 1920s to study conformal relativity, see [3], is defined in terms of the components of the Weyl tensorWi k jlas
Bi j =
1 n−3∇
k∇lW i k jl+
1
n−2RklWi
k
jl, (1.3)
while forn =3 it is given by
Bi j = ∇kCki j. (1.4)
We say that(Mn,g)is Bach-flat whenBi j =0. On 4-dimensional compact
mani-folds, Bach-flat metrics are precisely critical points of the conformally invariant func-tionalW(g)defined on the space of smooth Riemannian structures as
W(g)=
M
|Wg|2d Mg,
whereWgdenotes the Weyl tensor ofg. It is not difficult to check that locally
classification for gradient Ricci solitons under the Bach-flat assumption. For more details, we refer the reader to [7] and [8].
It is well known that 4-dimensional compact Riemannian manifolds have special behavior; for more details, see, for instance, [1,2,5] and [18]. Here, we shall investigate Bach-flat critical metrics of the volume functional on 4-dimensional manifolds with boundary. More precisely, we replace the assumption of locally conformally flat in the Miao–Tam result by the Bach-flat condition, which is weaker that the former. We now state our first result.
Theorem 2 Let(M4,g, f)be a simply connected, compact Miao–Tam critical metric with boundary isometric to a standard sphere S3. Then(M4,g) is isometric to a geodesic ball in a simply connected space formR4,H4orS4provided
M
f2B(∇f,∇f)d Mg≥0,
where B is the Bach tensor.
The proof of Theorem2was inspired by the trend developed by Cao and Chen in [7]. In the sequel, as an immediate consequence of Theorem2we deduce the following corollary.
Corollary 1 Let (M4,g, f)be a Bach-flat simply connected, compact Miao–Tam critical metric with boundary isometric to a standard sphere S3. Then (M4,g)is isometric to a geodesic ball in a simply connected space formR4,H4orS4.
Based on the previous result, it is natural to ask what occurs in lower dimension. To do so, inspired by the ideas developed in [9] (see also [8] and [7]) we shall prove a rigidity result for a 3-dimensional Miao–Tam critical metric with divergence-free Bach tensor, i.e., divB=0, and boundary isometric to a standard sphereS2. Clearly, the assumption of divergence-free Bach tensor is weaker than the Bach-flat condition considered in Theorem2. More precisely, we have the following result.
Theorem 3 Let(M3,g, f)be a simply connected, compact Miao–Tam critical met-ric with boundary isometmet-ric to a standard sphereS2. IfdivB(∇f)=0in M, where B is the Bach tensor, then(M3, g)is isometric to a geodesic ball in a simply connected space formR3,H3orS3.
Finally, we get the following rigidity result.
2 Preliminaries and Key Lemmas
In this section we shall present a couple of lemmas that will be useful in the proof of our main results. We begin by recalling that
L∗g(f)= −(f)g+Hess f − f Ric.
So, as was previously mentioned, the fundamental equation of a Miao–Tam critical metric (1.1) becomes
−(f)g+Hess f − f Ric=g. (2.1) Tracing (2.1) we arrive at
(n−1)f +R f = −n. (2.2) Moreover, by using (2.2) it is not difficult to check that
fRic˚ =Hess˚ f, (2.3) where ˚T stands for the traceless ofT.
For simplicity, we now rewrite Eq. (2.1) in the tensorial language as
−(f)gi j+ ∇i∇jf − f Ri j =gi j. (2.4)
Next, since a Miao–Tam critical metric has constant scalar curvature (cf. [15]), we use the last identity in order to obtain the following lemma.
Lemma 1 LetMn, g, f)be a Miao–Tam critical metric. Then
f∇iRj k−∇jRi k
=Ri j ks∇sf+
R n−1
∇i f gj k−∇jf gi k
−∇if Rj k − ∇jf Ri k
.
Proof Computing∇i(f Rj k)with the aid of (2.4) we infer
(∇if)Rj k + f∇iRj k = ∇i∇j∇kf −(∇if)gj k. (2.5)
Since Ris constant (2.2) yields∇if = −n−R1∇if. Hence we use this data in
(2.5) to deduce
f∇iRj k = −(∇i f)Rj k+ ∇i∇j∇kf +
R
To fix notation we recall three special tensors in the study of curvature for a Rie-mannian manifold (Mn,g),n ≥ 3. The first one is the Weyl tensor W, which is defined by the decomposition formula
Ri j kl =Wi j kl+
1 n−2
Ri kgjl+Rjlgi k−Rilgj k−Rj kgil
− R
(n−1)(n−2)
gjlgi k−gilgj k, (2.7)
whereRi j klstands for the Riemann curvature operator. The second tensor is the Cotton
tensorC, given by
Ci j k = ∇iRj k− ∇jRi k−
1 2(n−1)
∇iRgj k− ∇jRgi k). (2.8)
These two tensors are related by
Ci j k = −
(n−2)
(n−3)∇lWi j kl, (2.9)
providedn≥4.
Finally, the Schouten tensorAis defined by
Ai j =
1 n−2
Ri j −
R 2(n−1)gi j
. (2.10)
Combining Eqs. (2.7) and (2.10) we have the splitting
Ri j kl=
1
n−2(A⊙g)i j kl+Wi j kl, (2.11) where⊙is the Kulkarni–Nomizu product. For more details about these tensors, we refer to [5].
From now on we introduce the covariant 3-tensorTi j kby
Ti j k =
n−1 n−2
Ri k∇jf −Rj k∇if−
R n−2
gi k∇jf −gj k∇i f
+ 1
n−2
gi kRj s∇sf −gj kRi s∇sf
. (2.12)
It is important to highlight thatTi j k was defined similarly toDi j k in [7]. Now, we
may announce our second lemma.
Lemma 2 Let(Mn,g, f)be a Miao–Tam critical metric. Then the following identity
holds:
Proof First of all, we compare (2.8) with Lemma1to arrive at
f Ci j k =Ri j ks∇sf +
R n−1
∇i f gj k − ∇jf gi k−∇if Rj k− ∇jf Ri k. (2.13)
On the other hand, from (2.7) we obtain
Ri j ks∇s f =Wi j ks∇sf +
1 n−2
Ri kgj s+Rj sgi k−Ri sgj k−Rj kgi s∇sf
− R
(n−1)(n−2)
gj sgi k−gi sgj k
∇sf.
From this it follows that f Ci j k =Wi j ks∇sf +
(n−1) (n−2)
Ri k∇jf −Rj k∇i f−
R (n−2)
gi k∇jf −gj k∇i f
+ 1
n−2
gi kRj s∇s f −gj kRi s∇sf
=Ti j k+Wi j ks∇sf,
which concludes the proof of the lemma. ⊓⊔
To simplify some computations we shall define a functionρonMnby
ρ = |∇f|2+ 2
n−1 f + R n−1 f
2. (2.14)
We claim that
1
2∇ρ = fRic(∇f). (2.15) Indeed, since Ris constant we have 12∇ρ = Hess f(∇f)+ n−11∇f +n−R1f∇f. Next, we use (2.1) and (2.2) to obtain
1
2∇ρ =Hess f(∇f)−(f +1)∇f = f Ric(∇f), which settles our claim.
Proceeding, we recall that, at regular points of a smooth function f, the vector fieldν = |∇∇ff| is normal toc = {p ∈ M : f(p) = c}. In particular, the second
fundamental form ofcis given by
hi j = −∇eiν,ej, (2.16)
where{e1, . . . ,en−1}is an orthonormal frame onc. Then the mean curvature
com-puted at these points, denoted byH, is given as
H = − 1
| ∇f |
n−1
i=1
We now follow the trend of Cao and Chen (cf. [7] and [8]) to study the level sets of the potential function of Miao–Tam critical metrics. To this end, first, we deduce a similar result concerning the tensorT defined by (2.12) in the next lemma.
Lemma 3 Let(Mn,g, f)be a Miao–Tam critical metric. Let = {f = f(p)}be a level set of f . If gab denotes the induced metric on , then, at any point where
∇f =0, we have
|f T|2=2(n−1)
2 (n−2)2 |∇f|
4
n
a,b=2
|hab−
H n−1gab|
2+ n−1 2(n−2)|∇
ρ|2,
whereρis given by(2.14), haband H are the second fundamental form and the mean
curvature of, respectively, while∇is the Riemannian connection of.
Proof We consider an orthonormal frame {e1,e2, . . . ,en} with e1 = |∇∇ff| and
e2, . . . ,entangent to. A straightforward computation allows us to deduce
|T|2=2(n−1)
2 (n−2)2 (|Ric|
2|∇f|2− |Ric(∇f)|2)+2(n−1)R2 (n−2)2 |∇f|
2
+2(n−1)
(n−2)2|Ric(∇f)|
2−4(n−1)R (n−2)2 (R|∇f|
2−Ric(∇f,∇f))
+4(n−1)
(n−2)2(RRic(∇f,∇f)− |Ric(∇f)|
2)−4(n−1)R
(n−2)2 Ric(∇f,∇f). Proceeding, we can use (2.15) to obtain
|f T|2=2(n−1)
2 (n−2)2 f
2|∇f|2|Ric|2− n(n−1) 2(n−2)2|∇ρ|
2
−2(n−1)R
2 (n−2)2 f
2|∇f|2+2(n−1)R
(n−2)2 f∇ρ ,∇f. (2.18) On the other hand, the second fundamental formhabof the level set, as well as
its mean curvatureH, are given, respectively, by
hab= −
∇ea
∇f
|∇f|
,eb = −
1
|∇f|∇a∇bf
= − 1 |∇f|
f Rab−
1 n−1+
f R n−1
gab
and
H = − 1
|∇f|(f R− f R11− f R−1)=
1
From which we deduce
|h|2= 1 |∇f|2
f2|Ric|2−2f2
n
a=2
R12a−f2R112 −2f(R−R11) 1
n−1+ f R n−1
+ 1 |∇f|2
(f R+1)2 n−1 and
H2= 1 |∇f|2
f2R112 +2f R11+1
.
After some computations we obtain
n
a,b=2
|hab−
H n−1gab|
2= 1
|∇f|2
f2|Ric|2−n R
2
11f2+ f2R2−2f2R R11 n−1
− 2f
2
|∇f|2
n
a=2
R12a. (2.19)
On the other hand, by using (2.15) once more we get
f R11= 1
|∇f|2 f Ric(∇f,∇f)= 1
2|∇f|2∇ρ ,∇f and
f R1a=
1
|∇f|f Ric(∇f,ea)=
1
2|∇f|∇ρ ,ea =
1 2|∇f|∇aρ .
Substituting the last two identities into (2.19) we obtain
n
a,b=2
|hab−
H n−1gab|
2= 1
|∇f|2
f2|Ric|2− 1
2|∇f|2|∇
ρ|2− R2f2
n−1
− n
4(n−1)|∇f|4∇ρ ,∇f
2+ R f
(n−1)|∇f|2∇ρ ,∇f
,
which can be rewritten as
f2|Ric|2= |∇f|2
n
a,b=2
|hab−
H n−1gab|
2+ n
4(n−1)|∇f|4∇ρ ,∇f
2
+ 1
2|∇f|2|∇
ρ|2+ R2f2
n−1 −
R f
Finally, comparing (2.18) with (2.20), we deduce
|f T|2=2(n−1)
2 (n−2)2 |∇f|
4
n
a,b=2
|hab−
H
n−1gab|
2+ n−1 2(n−2)|∇
ρ|2,
which completes the proof of the lemma. ⊓⊔
We point out that some of these calculations above were also done in [4] and [19] in a different context to study the CPE conjecture (cf. Besse [5], p. 128).
Next, as a consequence of Lemma3we derive the following properties concerning a level set of the quoted metrics.
Proposition 1 Let(Mn,g, f)be a Miao–Tam critical metric with T ≡0. Let c be a regular value of f and = {p ∈ M; f(p)=c}be a level set of f . We consider e1= |∇∇ff| and choose an orthonormal frame{e2, . . . ,en}tangent to. Under these
conditions the following assertions hold.
(1) The second fundamental form habofis hab= n−H1gab.
(2) |∇f|is constant on.
(3) R1a=0for any a≥2and e1is an eigenvector ofRic. (4) The mean curvature ofis constant.
(5) On, the Ricci tensor either has a unique eigenvalue or two distinct eigenvalues with multiplicity1and n−1. Moreover, the eigenvalue with multiplicity1is in the direction of∇f .
(6) R1abc=0, for a,b,c∈ {2, . . . ,n}.
Proof The first two items follow directly from Lemma3 jointly with (2.14). Since T ≡0 we may use (2.12) to deduce
0=T(ei,∇f,∇f)
=Ric(ei,∇f)|∇f|2−Ric(∇f,∇f)∇f,ei,
in other words,
Ric(ei,∇f)|∇f|2=Ric(∇f,∇f)∇f,ei.
So, fori =a ≥ 2, we obtainR1a =0. Furthermore, Ric(e1)=gi jR1jei = R11e1. Therefore,e1=|∇∇ff| is an eigenvector of Ric, which establishes the third assertion.
Proceeding, we consider the Codazzi equation
R1abc= ∇bhca− ∇chba, a,b,c=2, . . . ,n. (2.21)
By contracting (2.21) with respect to indicesaandc, and using item (1), we get
R1b= ∇bH−gac∇chba=
n−2 n−1∇
Now, we use R1b = 0 to conclude that H is constant on, which gives the fourth
item. Next, sincee1= |∇∇ff| is an eigenvector of Ric, we may choose the frame
e1= ∇f
|∇f|,e2, . . . ,en
diagonalizing Ric such that Ric(ek)=λkekfork=1,2, . . . ,n. Using once more that
T ≡0, we have for alla,b≥2
0=Ta1b=
n−1
n−2(Rab∇1f −Rb1∇af)− R
n−2(gab∇1f −g1b∇af)
+ 1
n−2(gabR1s∇
sf −g
1bRas∇sf)
= n−1
n−2Rab|∇f| − R
n−2gab|∇f| + 1
n−2gabλ1|∇f|.
From here it follows thatRab= R−n−λ11gaband thenλ2= · · · =λn= R−n−λ11, which
gives the fifth assertion. Finally, we use (2.21) as well as items (1) and (4) of the proposition to obtain the last one. So, we complete the proof of the proposition. ⊓⊔
Proceeding with such a metric withT ≡0 we obtain the following lemma. Lemma 4 Let(Mn,g, f)be a Miao–Tam critical metric with T ≡0. Then C ≡0, namely,(Mn,g)has harmonic Weyl tensor.
Proof The first part of the proof is standard, and it follows the proof of Lemma 4.2 of [7]. Here we present its proof for the sake of completeness.
First, since T ≡ 0 we invoke Lemma2 to deduce f Ci j k = Wi j kl∇lf, which
implies
f Ci j k∇kf =0. (2.22)
We now consider a regular point p ∈Mn, with associated level set. We choose any local coordinates(θ2, . . . , θn)onand split the metric in the local coordinates (f, θ2, . . . , θn)as
g= 1
|∇f|2d f 2+g
ab(f, θ )dθadθb.
Denoting∂f =∂1= |∇∇ff|2 we get
∇1f =1 and ∇af =0, for a≥2.
From (2.22) we have f Ci j1=0 for alli,j=1, . . . ,n. Moreover, fora,b,c≥2, by using the Codazzi equation jointly with the first and fourth items of Proposition1we have
In particular, usingR1a=0 we get
W1abc=R1abc=0.
From which we obtain fora,b,c≥2
f Cabc=Wabcs∇sf =Wabc1∇1f =0.
We now claim that f C1ab=0 for alla,b≥2. To do this, first, we notice that
f C1ab=W1abs∇s f =W1abigi s∇sf =W1ab1|∇f|2= − 1
|∇f|2W(∇f, ∂a,∇f, ∂b).
On the other hand, from (2.7) we have 1
|∇f|2W(∇f, ∂a,∇f, ∂b)= 1
|∇f|2R(∇f, ∂a,∇f, ∂b)+
R
(n−1)(n−2)gab
− 1
(n−2)
1
|∇f|2Ric(∇f,∇f)gab+Rab
.
(2.23) We now analyze the second fundamental form in the local coordinate(f, θ2, . . . , θn). It is easy to see that
hab=
1
|∇f|∇f,∇a∂b =
1
|∇f|∇f, Ŵ
1
ab∂f =
Ŵab1
|∇f|.
Moreover, a standard computation allows us to obtain
Ŵab1 = 1
2g 1j
∂agbj+∂bgj a−∂jgab
= 1
2g 11
∂agb1+∂bg1a−∂fgab
= −1
2∇f(gab). From here it follows that
hab= −
∇f
2|∇f|(gab). (2.24)
By Proposition1,|∇f|is constant on, which implies that
Since∇f |∇f|, ∂a
= 0, we conclude∇∇f
|∇f|
∇f
|∇f| = 0. Hence, we can use (2.25) to
arrive at
1
|∇f|2R(∇f, ∂a,∇f, ∂b)= 1
|∇f|∇|∇∇ff|
∇a∂b− ∇a∇∇f
|∇f| ∂b,∇f
= 1 |∇f|2∇∇f
∇a∂b+ ∇a⊥∂b,∇f
− 1
|∇f|∇a∇|∇∇ff| ∂b,∇f
= ∇f
|∇f|(hab)+hach
c b.
From this, we deduce
1
|∇f|2R(∇f, ∂a,∇f, ∂b)=
∇f
(n−1)|∇f|H gab−
H2
(n−1)2gab. (2.26)
In particular, taking the trace in (2.26) with respect toaandbwe have
1
|∇f|2Ric(∇f,∇f)= ∇f
|∇f|H−
H2 (n−1),
and then (2.26) can be written as
R(∇f, ∂a,∇f, ∂b)=
Ric(∇f,∇f)
(n−1) gab. (2.27)
By using Proposition1we have
1
|∇f|2Ric(∇f,∇f)=λ
and
Ric(∂a, ∂b)=μgab,
Therefore, substituting (2.27) into (2.23) we get
f C1ab= −
1
|∇f|2W(∇f, ∂a,∇f, ∂b) = − 1
|∇f|2
Ric(∇f,∇f) (n−1) gab+
1
|∇f|2
Ric(∇f,∇f) (n−2) gab+
Rab
(n−2)
− R
(n−1)(n−2)gab
= − λ
(n−1)gab+ λ
(n−2)gab+ μ
(n−2)gab−
λ+(n−1)μ (n−1)(n−2)gab
=0,
which completes our claim.
Finally, we have f Ci j k = 0 at a point p where∇f(p) = 0. Therefore, we use
Lemma2to conclude that f Ci j k ≡0 inMn. Using this, we obtainCi j k ≡0 inM\∂M
and then the proof of the lemma follows from the continuity of the Cotton tensor. ⊓⊔
To finish this section, we shall present a fundamental integral formula. Lemma 5 Let(Mn,g,f)be a Miao–Tam critical metric. Then
M
f2B(∇f,∇f)d Mg= −
1 2(n−1)
M
f2|T|2d Mg.
Proof From (2.9) we can write the Bach tensor as
Bi j =
1 n−2
∇kCki j+RklWi k jl.
Under this notation we get
f2Bi j =
1 n−2
f2∇kCki j+ f2RklWi k jl
= 1
n−2
∇k
f2Cki j
−2f Cki j∇kf + f2RklWi k jl.
We now use Lemma2jointly with (2.9) to obtain
f2Bi j =
1 n−2
∇kf(Wki jl∇lf +Tki j)−2f Cki j∇kf + f2RklWi k jl
= 1
n−2
∇k
f Tki j
+ f∇kWki jl∇lf + f Wki jl∇k∇lf
+Wki jl∇kf∇lf −2f Cki j∇kf + f2RklWi k jl
= 1
n−2
∇kf Tki j+
n−3
n−2f Cj ki∇kf + f (∇k∇lf − f Rkl)Wki jl
+Wki jl∇kf∇lf +2f Ci k j∇kf
We recall that the Weyl tensor is trace-free on any pair of indices. Next, by using (2.4) we deduce
f2Bi j =
1 n−2
∇k(f Tki j)+
n−3
n−2 f Cj ki∇kf +2f Ci k j∇kf +Wki jl∇kf∇lf
.
From which we obtain
f2B(∇f,∇f)= 1
n−2∇k
f Tki j∇if∇jf. (2.28)
On the other hand, we notice that
∇k
f Tki j
∇if∇jf = ∇k
f Tki j∇i f∇jf
− f Tki j∇k∇if∇jf − f Tki j∇if∇k∇jf.
Now, on integrating (2.28) overMand using Stokes’s formula we arrive at
M
f2B(∇f,∇f)d Mg= −
1 n−2
M
f Tki j∇k∇if∇jf d Mg
+
M
f Tki j∇if∇k∇j f d Mg
= − 1
n−2
M
f2Tki jRki∇j f d Mg
+
M
f2Tki jRk j∇if d Mg
,
where in the last equality we have used once more (2.4). Finally, we changekbyi above and then using the properties ofT defined in (2.12) we achieve
M
f2B(∇f,∇f)d Mg= −
1 2(n−2)
M
f2Tki jRki∇jf d Mg
+
M
f2Tki jRk j∇i f d Mg
− 1
2(n−2)
M
f2Ti k jRi k∇jf d Mg
+
M
f2Ti k jRi j∇kf d Mg
= 1
2(n−2)
M
f2Ti k jRk j∇if −Ri j∇kfd Mg
= − 1
2(n−1)
M
f2|T|2d Mg,
3 Proof of the Results 3.1 Proof of Theorem2
Proof First, since M4 satisfies
M f2B(∇f,∇f)d M ≥ 0 we invoke Lemma 5to
conclude that T ≡ 0. Therefore, from Lemma4 we have C ≡ 0. Hence, we use Lemma2to obtain
Wi j kl∇lf =0.
We now consider a pointp∈ M4such that∇f(p)=0. Choosing an orthonormal frame{e1,e2,e3,e4}withe1=|∇∇ff| at the point pwe arrive at
Wi j k1=0, (3.1)
for all 1≤i, j,k≤4.
We now claim thatWi j kl = 0 whenever ∇f(p) = 0. Indeed, recalling that the
Weyl tensor is trace-free on any pair of indices, we have
W2121+W2222+W2323+W2424=0. By using (3.1) we have
W2323= −W2424. In a similar way we have
W2424= −W3434=W2323.
From here it follows thatW2323=0. Moreover, we also have
W1314+W2324+W3334+W4344=0.
Therefore,W2324=0. This proves thatWabcd =0 unlessa,b,c,dare all distinct.
3.2 Proof of Theorem3
Proof The first part of the proof will follow [9]. To begin with, we consider(M3,g, f) to be a simply connected, compact Miao–Tam critical metric with boundary isometric to a standard sphere S2. Next, we recall that the Cotton tensor can be written as Ci j k = ∇iAj k− ∇jAi k. From here it follows that
∇iBi j = ∇i∇kCki j
=∇i∇k− ∇k∇i∇kAi j.
From which the previous commutator term implies
∇iBi j = −Ril∇lAi j+Rkl∇kAl j+Ri k jl∇kAil
and then we get
∇iBi j =Ri k jl∇kAil. (3.2)
We now remember thatW ≡0 in dimension three. So, a straightforward computa-tion involving (2.11) and (3.2) gives
∇iBi j∇jf = Ai kCk j i∇jf +Ai k∇jAki∇jf +Ajl∇iAil∇j f
−Ail∇jAil∇j f −Aj kgilCkil∇j f −Aj kgil∇iAkl∇jf
= −Ri kCj ki∇j f
= −1
2
Cj kiRi k∇jf +Ck j iRi j∇kf, (3.3)
where we have used thatC is skew-symmetric in the first two indices and thatC is trace-free in any two indices. Now, we combine (3.3) with (2.12) to deduce
(divB)(∇f)= 1
2Ck j i
Ri k∇jf −Ri j∇kf
= 1
4Ck j iTk j i.
Using once more thatW ≡0 jointly with Lemma2we arrive at
(divB)(∇f)= f
4|C| 2.
part of this work was started. Finally, he wishes to thank the Department of Mathematics—Lehigh Univer-sity for the warm hospitality and for the fruitful research environment. A. Barros was partially supported by CNPq/Brazil/. R. Diógenes was partially supported by FUNCAP/Brazil. E. Ribeiro was partially supported by Grants from PJP-FUNCAP/Brazil and CNPq/Brazil
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