A sphere theorem for Bach-flat manifolds with positive constant scalar curvature

Differential Geometry and its Applications 64 (2019) 80–91

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Differential Geometry and its Applications www.elsevier.com/locate/difgeo

A sphere theorem for Bach-flat manifolds with positive constant scalar curvature ✩ Yi Fang a , Wei Yuan b,∗ a b

Department of Applied Mathematics, Anhui University of Technology, Ma’anshan, Anhui 243002, China Department of Mathematics, Sun Yat-sen University, Guangzhou, Guangdong 510275, China

a r t i c l e

i n f o

Article history: Received 9 November 2017 Received in revised form 18 November 2018 Available online 12 February 2019 Communicated by F. Fang MSC: primary 53C24, 53C25 secondary 53C20, 53C21 Keywords: Bach-flat Sphere theorem Positive constant scalar curvature Gap theorem

a b s t r a c t A classic gap theorem for complete non-compact Ricci flat manifolds by M. Anderson suggests that one can get the rigidity of certain spaces simply by passing local curvature estimates on geodesic balls to global. Using similar ideas, Kim showed that a 4-dimensional complete non-compact Bach-flat manifold with vanishing scalar curvature and small L2 -curvature tensor has to be flat. Unfortunately, this method does not generalize to compact manifolds without boundary. Applying a different approach, we show that a closed Bach-flat Riemannian manifold with positive constant scalar curvature has to be locally spherical if its Weyl and traceless n Ricci tensors are small in the sense of either L∞ or L 2 -norm. These results generalize a rigidity theorem of positive Einstein manifolds due to M.-A. Singer. As an application, we can partially recover the well-known Chang–Gursky–Yang’s 4-dimensional conformal sphere theorem. © 2019 Elsevier B.V. All rights reserved.

1. Introduction The notion of Bach tensor was first introduced by Rudolf Bach in 1921 (see [2]) when studying the so-called conformal gravity. That is, instead of using the Hilbert–Einstein functional, one considers the functional  W(g) =

|W (g)|2 dvg M4

✩ Yi Fang was supported by NSFC (Grant No. 11801006) and The Young Teachers’ Science Research Funds of Anhui University of Technology (Grant No. RD16100248). Wei Yuan was supported by NSFC (Grant No. 11601531, No. 11521101) and The Fundamental Research Funds for the Central Universities (Grant No. 2016-34000-31610258). * Corresponding author. E-mail addresses: fl[email protected] (Y. Fang), [email protected] (W. Yuan).

https://doi.org/10.1016/j.difgeo.2019.01.004 0926-2245/© 2019 Elsevier B.V. All rights reserved.

Y. Fang, W. Yuan / Differential Geometry and its Applications 64 (2019) 80–91

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for 4-dimensional manifolds, where W is the Weyl tensor. Critical points of this functional are characterized by the vanishing of certain symmetric 2-tensor Bg . The tensor Bg is usually referred as Bach tensor and the metric is called Bach-flat, if Bg vanishes. Let (M n , g) be an n-dimensional Riemannian manifold (n ≥ 4). The Bach tensor is defined to be Bjk =

1 ∇i ∇l Wijkl + Wijkl S il , n−3

(1.1)

where Sjk =

1 n−2

 Rjk −

1 Rgjk 2(n − 1)

 (1.2)

is the Schouten tensor. Using the Cotton tensor Cijk = ∇i Sjk − ∇j Sik

(1.3)

∇l Wijkl = (n − 3)Cijk ,

(1.4)

and the relation

we can extend the definition of Bach tensor such that it can be defined for 3-dimensional manifolds: Definition 1.1. For manifolds with dimension n ≥ 3, the Bach tensor is defined to be Bjk = ∇i Cijk + Wijkl S il .

(1.5)

We say a metric is Bach-flat, if its Bach tensor vanishes. Typical examples of Bach flat metrics are Einstein metrics and locally conformally flat metrics. Due to the conformal invariance of Bach-flatness on 4-manifolds, metrics conformal to Einstein metrics are also Bach-flat. For 4-dimensional manifolds, it also includes half-locally conformally flat metrics. In general, Tian and Viaclovsky studied the module space of 4-dimensional Bach-flat manifolds ([10,11]). Besides these known “trivial” examples, there are not many examples known about generic Bach-flat manifolds so far. In fact, in some particular situations, one would expect rigidity phenomena occur. In [8], Kim shows that on a complete non-compact 4-dimensional Bach-flat manifold (M, g) with zero scalar curvature and positive Yamabe constant has to be flat, if the L2 -norm of its Riemann curvature tensor is sufficiently small. This result can be easily extended to any dimension n ≥ 3. Kim’s proof is based on a classic idea that one can get global rigidity from local estimates: applying the ellipticity of Bach-flat metric, the Sobolev’s inequality and together the smallness of ||Rm||L2 (M,g)(M,g) , one can get the estimate ||Rm||L4 (Br (p),g) ≤

C ||Rm||L2 (M,g)(M,g) r

for any fixed p ∈ M and r > 0. Now the conclusion follows by letting r → ∞. This method has many successful applications. For example, Anderson used it to get his well-known gap theorem for Ricci flat manifolds ([1]). For other settings, please see, for example, [4,6] etc. However, note that the assumption of complete non-compactness is essential here. One can not get the rigidity by simply letting r → ∞, when the manifold is compact without boundary for instance.

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Y. Fang, W. Yuan / Differential Geometry and its Applications 64 (2019) 80–91

Is it possible for us to have a result similar to Kim’s but on closed manifolds? Here by closed manifolds, we mean compact manifolds without boundary. In fact, Singer proved that even dimensional closed positive Einstein manifolds with non-vanishing Euler characteristic have to be locally spherical, provided the n L 2 -norm of its Weyl tensor is small ([9]). It suggests that this phenomenon might occur in larger classes of metrics, since Einstein metrics are in particular Bach-flat. Applying a global estimate for symmetric 2-tensors (see Proposition 2.3), we can prove the following result: Theorem A. Suppose (M n , g) is a closed Bach-flat Riemannian manifold with constant scalar curvature Rg = n(n − 1). If ||W ||L∞ (M,g) +

2n n−1 ||E||L∞ (M,g) < ε0 (n) := , 3(n − 2) 4

(1.6)

where E is the traceless part of Ricci curvature tensor, then (M n , g) is isometric to a quotient of the round sphere Sn . Remark 1.2. Note that in Theorem A we do not assume the Yamabe constant is uniformly positively lower bounded. This assumption will be needed in Theorem B. It is equivalent to the existence of a uniform Sobolev’s inequality (see section 4), which was applied frequently in the proof of Theorem B. Another one by assuming integral conditions: Theorem B. Suppose (M n , g) is a closed Bach-flat Riemannian manifold with constant scalar curvature Rg = n(n − 1). Assume that there is a constant α0 such that its Yamabe constant satisfies that Y (M, [g]) ≥ α0 > 0.

(1.7)

Then (M n , g) is isometric to a quotient of the round sphere Sn , if ||W ||L n2 (M,g) +

n (n − 2)α0 ||E||L n2 (M,g) < τ0 (n, α0 ) := . 2(n − 2) 4 max{8(n − 1), n(n − 2)}

(1.8)

Remark 1.3. Bach-flat metrics are typical examples of the so-called critical metrics (cf. [10]). By replacing the presumption Bach-flatness with harmonic curvature, which refers to the vanishing of Cotton tensor when the scalar curvature is a constant, the corresponding version of Theorem A and B are still valid without any essential difficulty. In particular, applying Theorem B for 4-dimensional manifolds, we can partially recover the well-known 4-dimensional conformal sphere theorem by Chang–Gursky–Yang (cf. [5]; for a generalization see [7]): Theorem C. Suppose (M 4 , g) is a closed Bach-flat Riemannian manifold. Assume that there is a constant α0 such that its Yamabe constant satisfies that Y (M, [g]) ≥ α0 > 0.

(1.9)

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Then (M 4 , g) is conformal to the round sphere S4 or its canonical quotient RP 4 , if  |Wg |2 dvg <

32 2 α2 π (χ(M 4 ) − 2) + 0 . 3 6912

(1.10)

M4

Remark 1.4. From a perspective of fourth order flow, Bour achieved a similar result. Please see [3] for more details. Remark 1.5. It was shown in [5] that (M 4 , g) is conformal to (CP 2 , gF S ) or a manifold covered isometrically by S1 × S3 endowed with the canonical product metric, if we assume  |Wg |2 dvg = 16π 2 χ(M 4 ) (1.11) M4

instead. 2. θ-Codazzi tensor and related inequality We define the following concept which generalizes the classic Codazzi tensor: Definition 2.1. For any θ ∈ R, we say a symmetric 2-tensor h ∈ S2 (M ) is a θ-Codazzi tensor if Cθ (h)ijk := ∇i hjk − θ∇j hik = 0.

(2.1)

In particular, h is referred to be a Codazzi tensor or anti-Codazzi tensor if θ = 1 or θ = −1 respectively. The motivation for us to define this notion is the following identity associated to it: Lemma 2.2. Let (M n , g) be a closed Riemannian manifold, then for any h ∈ S2 (M ) and θ ∈ R,  

 1 2 |∇h| − |Cθ (h)| dvg 1 + θ2 2

M

2θ = 1 + θ2

  ◦ ◦ |δh|2 + W (h, h) + M

(2.2)

 n 2 Rg ◦ 2 2 (trh)E · h − tr(E × h ) − |h| dvg , n−2 n−2 n−1

◦

◦ ◦

◦

◦

where h := h − n1 (trh)g is the traceless part of the tensor h and W (h, h) := Wijkl hil hjk . Proof. We have  ∇i hjk ∇j hik dvg M

 ∇j ∇i hkj hik dvg

=− M



j l (∇i ∇j hkj + Rjil hkl − Rjik hl j )hik dvg

=− M



=−

(−∇i (δh)k + Ril hkl − Rjikl hjl )hik dvg M

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     1 2 l l ik |δh| − Eil hk + Rg gil hk h dvg = n M

 

 2 Rg (Ejl gik − Ejk gil ) + (gjl gik − gjk gil ) hjl hik dvg n−2 n(n − 1)

Wjikl +

+ M

  |δh|2 + W (h, h) + = M

  ◦ ◦ |δh|2 + W (h, h) +

=

M

 n Rg 2 ((trh)2 − n|h|2 ) + (trh)E · h − tr(E × h2 ) dvg n(n − 1) n−2 n−2  n 2 Rg ◦ 2 (trh)E · h − tr(E × h2 ) − |h| dvg . n−2 n−2 n−1

Thus for any θ ∈ R,  |Cθ (h)|2 dvg M



|∇i hjk − θ∇j hik |2 dvg

= M



=



 (1 + θ2 )|∇h|2 − 2θ∇i hjk ∇j hik dvg

M

   ◦ ◦ 2 2 (1 + θ )|∇h| − 2θ |δh|2 + W (h, h) + = M

n 2 Rg ◦ 2 (trh)E · h − tr(E × h2 ) − |h| n−2 n−2 n−1

 dvg .

That is,   |∇h|2 − M

2θ = 1 + θ2

 1 2 dvg |C (h)| θ 1 + θ2

  ◦ ◦ |δh|2 + W (h, h) + M

 n 2 Rg ◦ 2 2 (trh)E · h − tr(E × h ) − |h| dvg . n−2 n−2 n−1

2

From this, we get the following inequality: Proposition 2.3. Let (M n , g) be a closed Riemannian manifold, then for any h ∈ S2 (M ) and θ ∈ R, 

2θ |∇h| dvg ≥ 1 + θ2 2

M

  ◦ ◦ |δh|2 + W (h, h) + M

 n 2 Rg ◦ 2 2 (trh)E · h − tr(E × h ) − |h| dvg , n−2 n−2 n−1

(2.3)

where equality holds if and only if h is a θ-Codazzi tensor. Remark 2.4. As a special case, when the metric g is Einstein, the above inequality was given by the second named author in [12]. In particular, we have Corollary 2.5. Suppose (M n , g) is a closed Riemannian manifold with constant scalar curvature Rg = n(n − 1)λ.

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Then the traceless part of Ricci tensor satisfies  |∇E|2 dvg ≥

2θ 1 + θ2

M

  W (E, E) − M

 n trE 3 − nλ|E|2 dvg , n−2

(2.4)

In particular when θ = 1, the equality holds if and only if g is of harmonic curvature. Proof. By the second Bianchi identity, we can easily see that δE = −

n−2 dRg = 0. 2n

Note that trE = 0, thus the conclusion follows from Proposition 2.3. When θ = 1, E is a Codazzi tensor if and only if the Cotton tensor vanishes: Cijk =

  1 1 Alt ∇i Rjk − gjk ∇i R = 0. n − 2 i,j 2(n − 1)

2

3. L∞ -sphere theorem We can rewrite the Bach tensor in terms of traceless Ricci tensor: Lemma 3.1. The Bach tensor can be expressed as follows Bg =

  1 1 1 2 ◦ Δg E − ∇2g R − gΔg R + W ·E n−2 2(n − 1) n n−2   1 n R 2 E × E − |E| g − E, − (n − 2)2 n (n − 1)(n − 2)

◦

where (W · E)jk := Wijkl E il . Proof. By definition, ∇i Cijk = ∇i (∇i Sjk − ∇j Sik ) p i Skp − Rijk Spi ) = Δg Sjk − (∇j ∇i Ski + Rijp ◦

= Δg Sjk − ∇j ∇k trS − (Ric × S)jk + (Rm · S)jk , where we used the fact ∇i Ski = ∇k trS by the contracted second Bianchi identity. Since S=

R 1 E+ g n−2 2n(n − 1)

and Rm = W +

R 1 E?g+ g ? g, n−2 2n(n − 1)

(3.1)

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where ? is the Kulkarni–Nomizu product for symmetric 2-tensors, the conclusion follows by substituting them into the expression Bjk = ∇i Cijk + Wijkl S il .

2

As the first step, we show the metric has to be Einstein under given presumptions: Proposition 3.2. Suppose (M n , g) is a closed Bach flat Riemannian manifold with constant scalar curvature Rg = n(n − 1) and ||Wg ||L∞ (M,g) +

2n 2n ||Eg ||L∞ (M,g) < Λn := , 3(n − 2) 3

then the metric g has to be Einstein. Proof. Since the scalar curvature Rg is a constant, by Lemma 3.1, 1 2 ◦ n Δg E + Bg = W ·E− n−2 n−2 (n − 2)2



 1 n 2 E × E − |E| g − E = 0. n n−2

That is, ◦

n Δ g E + 2W · E − n−2



 1 2 E × E − |E| g − nE = 0. n

Thus, −EΔg E = 2W (E, E) −

n tr(E 3 ) − n|E|2 n−2

and hence  



 |∇E| dvg = − 2

M

EΔg Edvg = M

M

 n 3 2 2W (E, E) − tr(E ) − n|E| dvg . n−2

(3.2)

On the other hand, from Corollary 2.5,  |∇E|2 dvg ≥ M

2θ 1 + θ2

  W (E, E) − M

 n trE 3 − n|E|2 dvg , n−2

for any θ ∈ R. Therefore, 2(1 − θ + θ2 ) (1 − θ)2

 W (E, E)dvg ≥ M

n n−2





trE 3 + (n − 2)|E|2 dvg .

M

Since 

 W (E, E)dvg ≤ ||W ||L∞ (M,g)

M

|E|2 dvg , M

(3.3)

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by taking θ = −1, we get n n−2





3 trE 3 + (n − 2)|E|2 dvg ≤ ||W ||L∞ (M,g) 2

M

 |E|2 dvg . M

That is, n n−2



 trE dvg ≤ 3

 3 ||W ||L∞ (M,g) − n |E|2 dvg . 2

M

M

From the inequality 

 trE 3 dvg ≥ − M

 |E|3 dvg ≥ −||E||L∞ (M,g)

M

|E|2 dvg , M

we have 

 n 3 ||W ||L∞ (M,g) + ||E||L∞ (M,g) − n |E|2 dvg ≥ 0. 2 n−2 M

Since ||W ||L∞ (M,g) + we have E = 0.

2n 2n ||E||L∞ (M,g) < Λn = , 3(n − 2) 3

2

It is well-known that the Weyl tensor satisfies an elliptic equation on Einstein manifolds (cf. [9]): Lemma 3.3. Let (M n , g) be an Einstein manifold with scalar curvature Rg = n(n − 1)λ, then its Weyl tensor satisfies Δg W − 2(n − 1)λW − 2Q(W ) = 0,

(3.4)

where Q(W ) := Bijkl − Bjikl + Bikjl − Bjkil is a quadratic combination of Weyl tensors with Bijkl := g pq g rs Wpijr Wqkls . Now we finish this section by proving one of our main theorem: Proof of Theorem A. We take ε0 := min{Λn ,

n−1 n−1 }= . 4 4

From Proposition 3.2, we conclude that g is an Einstein metric. Applying Lemma 3.3, we have  −

 Δg W − 2(n − 1)W, W dvg = −2

M

 Q(W ), W dvg ≤ 8

M

|W |3 dvg . M

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That is, 



|∇W |2 + 2(n − 1)|W |2 dvg ≤ 8

M

 |W |3 dvg .

(3.5)

M

Now we have  2(n − 1)

 |W |2 dvg ≤ 8

M

 |W |3 dvg ≤ 8||W ||L∞ (M,g)

M

|W |2 dvg . M

Thus the Weyl tensor vanishes, since ||W ||L∞ (M,g) < ε0 = Therefore, the metric g is locally spherical.

n−1 . 4

2

n

4. L 2 -sphere theorem Let (M n , g) be a Riemannian manifold. Suppose the Yamabe constant associated to it satisfies that

 Y (M n , [g]) :=

inf

M

0≡u∈C ∞ (M )

+ Rg u2 dvg ≥ α0 > 0. 

n−2 2n n n−2 dv u g M

4(n−1) 2 n−2 |∇u|

Assume the metric g has constant scalar curvature Rg = n(n − 1), then ⎛ ⎞ n−2 n  2n ⎝ u n−2 dvg ⎠ ≤

1 Y (M, [g])

M

  M

 4(n − 1) 2 2 |∇u| + Rg u dvg n−2

4(n − 1) = (n − 2)Y (M, [g]) 4(n − 1) ≤ (n − 2)α0 Denote CS :=

4(n−1) (n−2)α0

 

  |∇u|2 + M

|∇u|2 +

 n(n − 2) 2 u dvg 4

 n(n − 2) 2 u dvg . 4

M

> 0, we get the Sobolev’s inequality ||u||2

2n L n−2

(M,g)

  n(n − 2) ≤ CS ||∇u||2L2 (M,g) + ||u||2L2 (M,g) . 4

(4.1)

Note that, the constant CS > 0 only depends on n and α0 and is independent of the metric g. Lemma 4.1. Let (M n , g) be a Bach flat Riemannian manifold with constant scalar curvature Rg = n(n − 1). Suppose there is a constant α0 such that its Yamabe constant satisfies that Y (M, [g]) ≥ α0 > 0.

(4.2)

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Then (M n , g) is Einstein, if ||W ||L n2 (M,g) +

n−2 n α0 ||E||L n2 (M,g) < δ0 (n, α0 ) := min{1, }. 2(n − 2) 2(n − 1) 4

(4.3)

Proof. From equation (3.2) and Hölder’s inequality,  

 |∇E| dvg = 2

M

M

 n 3 2 2W (E, E) − tr(E ) − n|E| dvg n−2



≤

2||W ||L n2 (M,g) +

≤ 2δ0 ||E||2

2n

L n−2 (M,g)

 n ||E||L n2 (M,g) ||E||2 2n − n||E||2L2 (M,g) L n−2 (M,g) n−2

− n||E||2L2 (M,g) .

Note that the equality in the last line holds if and only if E = 0, since the inequality in (4.3) is strict. By Sobolev’s inequality (4.1) and the Kato’s inequality, ||E||2

2n

L n−2 (M,g)

  n(n − 2) ||E||2L2 (M,g) ≤CS ∇|E| 2L2 (M,g) + 4   n(n − 2) ||E||2L2 (M,g) . ≤CS ||∇E||2L2 (M,g) + 4

Thus, we have   n−2 δ0 CS ||E||2L2 (M,g) ≤ 0. (1 − 2δ0 CS ) ||∇E||2L2 (M,g) + n 1 − 2 Now by taking δ0 := min{

n−2 1 2 α0 min{ , 1}, , }= 2CS (n − 2)CS 2(n − 1) 4

we conclude that E vanishes identically on M and hence (M, g) is Einstein.

2

Now we can show Proof of Theorem B. It is easy to check that τ0 (n, α0 ) < δ0 (n, α0 ) for any n ≥ 3 and hence (M n , g) has to be Einstein by Lemma 4.1. Now from Sobolev’s inequality (4.1), Kato’s inequality and inequality (3.5), we have   ||W ||2

2n L n−2

(M,g)

|∇W |2 +

≤CS

 n(n − 2) |W |2 dvg 4

M

≤CS max{1,

n(n − 2) } 8(n − 1)





|∇W |2 + 2(n − 1)|W |2 dvg

M

4 ≤ max{8(n − 1), n(n − 2)} (n − 2)α0 Applying Hölder’s inequality,

 |W |3 dvg . M

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 |W |3 dvg ≤ ||W ||L n2 (M,g) ||W ||2

2n

L n−2 (M,g)

M

and hence 

 4 n 1− max{8(n − 1), n(n − 2)}||W ||L 2 (M,g) ||W ||2 2n ≤ 0, L n−2 (M,g) (n − 2)α0

which implies that W vanishes identically on M since ||W ||L n2 (M,g) < τ0 (n, α0 ) = Therefore, (M n , g) is isometric to a quotient of Sn .

(n − 2)α0 . 4 max{8(n − 1), n(n − 2)} 2

As for n = 4, we have Proof of Theorem C. Let gˆ ∈ [g] be the Yamabe metric, which means

1 Rgˆ V ol(M 4 , gˆ) 2 = Y (M 4 , [g]). We can also normalize it such that Rgˆ = 12. According to the solution of Yamabe problem,  Y (M , [g]) ≤ Y (S , gS4 ) = 12 · 4

4

8 2 π 3

 12

√ = 8 6π

and hence V ol(M 4 , gˆ) ≤ V ol(S4 , gS4 ) =

8 2 π . 3

From the Gauss–Bonnet–Chern formula,  

 1 2 Qgˆ + |Wgˆ | dvgˆ = 8π 2 χ(M 4 ), 4

(4.4)

M4

where 1 1 1 Qgˆ := − Δgˆ Rgˆ − |Egˆ |2 + Rg2ˆ 6 2 24

(4.5)

is the Q-curvature for metric gˆ. Thus, ||Egˆ ||2L2 (M,ˆg) = and hence

1 1 ||Wgˆ ||2L2 (M,ˆg) + 12V ol(M 4 , gˆ) − 16π 2 χ(M 4 ) ≤ ||Wgˆ ||2L2 (M,ˆg) + 16π 2 (2 − χ(M 4 )) 2 2

Y. Fang, W. Yuan / Differential Geometry and its Applications 64 (2019) 80–91



||Wgˆ ||L2 (M,ˆg) + ||Egˆ ||L2 (M,ˆg)

2

91



≤ 2 ||Wgˆ ||2L2 (M,ˆg) + ||Egˆ ||2L2 (M,ˆg) ≤ 3||Wgˆ ||2L2 (M,ˆg) + 32π 2 (2 − χ(M 4 )) = 3||Wg ||2L2 (M,g) + 32π 2 (2 − χ(M 4 )) <

α02 , 482

where we used the fact that ||Wg ||L2 (M,g) is conformally invariant for 4-dimensional manifolds. On the other hand, the metric gˆ is also Bach-flat, since Bach-flatness is conformally invariant for 4-dimensional manifolds. Applying Theorem B to the Yamabe metric gˆ, we conclude that (M 4 , gˆ) is isometric to a quotient of the round sphere S4 . For the quotient of an even dimensional sphere, only identity and Z2 -actions make it a smooth manifold. Therefore, (M 4 , g) is conformal to S4 or RP 4 with their canonical metrics. 2 Acknowledgements The authors would like to express their appreciations to Professor Chen Bing-Long, Professor Gilles Carron, Professor Huang Xian-Tao, Professor Zhang Hui-Chun for their interests in this problem and inspiring discussions. Especially, we would like to thank Professor Gilles Carron for bringing the work [3] to our attention and valuable comments. References [1] M. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 (2) (1990) 429–445. [2] R. Bach, Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs, Math. Z. 9 (1921) 110–135. [3] V. Bour, Fourth order curvature flows and geometric applications, arXiv:1012.0342, 2010. [4] G. Carron, Some old and new results about rigidity of critical metrics, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 13 (4) (2014) 1091–1113. [5] S.-Y.A. Chang, M. Gursky, P. Yang, A conformally invariant sphere theorem in four dimensions, Publ. Math. IHÉS 98 (2003) 105–143. [6] B.-L. Chen, On stationary solutions to the vacuum Einstein field equations, arXiv:1606.00543, 2016. [7] B.-L. Chen, X.-P. Zhu, A conformally invariant classification theorem in four dimensions, Commun. Anal. Geom. 22 (2014) 811–831. [8] S. Kim, Rigidity of noncompact complete Bach-flat manifolds, J. Geom. Phys. 60 (2010) 637–642. n [9] M.-A. Singer, Positive Einstein metrics with small L 2 -norm of the Weyl tensor, Differ. Geom. Appl. 2 (1992) 269–274. [10] G. Tian, J. Viaclovsky, Bach-flat asymptotically locally Euclidean metrics, Invent. Math. 160 (2) (2005) 357–415. [11] G. Tian, J. Viaclovsky, Moduli spaces of critical Riemannian metrics in dimension four, Adv. Math. 196 (2) (2005) 346–372. [12] W. Yuan, Volume comparison with respect to scalar curvature, arXiv:1609.08849, 2016.