Isoperimetric inequality and Weitzenb\" ock type formula for critical ...

arXiv:1802.01390v1 [math.DG] 5 Feb 2018

¨ ISOPERIMETRIC INEQUALITY AND WEITZENBOCK TYPE FORMULA FOR CRITICAL METRICS OF THE VOLUME ´ H. BALTAZAR, R. DIOGENES, AND E. RIBEIRO JR. Abstract. We provide an isoperimetric inequality for critical metrics of the volume functional with nonnegative scalar curvature on compact manifolds with boundary. In addition, we establish a Weitzenb¨ ock type formula for critical metrics of the volume functional on four-dimensional manifolds. As an application, we obtain a classification result for such metrics.

1. Introduction A classical topic in Riemannian geometry is to find canonical metrics on a given manifold M n . A promising way for that purpose is that of critical metrics of the Riemannian functionals, as for instance, the total scalar curvature functional and the volume functional. Einstein and Hilbert proved that the critical points of the total scalar curvature functional restricted to the set of smooth Riemannian structures on M n of unitary volume are Einstein (cf. Theorem 4.21 in [9]), and this result stimulated several interesting works. In this spirit, Miao and Tam [20, 21] studied variational properties of the volume functional constrained to the space of metrics of constant scalar curvature on a given compact manifold with boundary. Indeed, volume is one of the natural geometric quantities used to study gemetrical and topological properties of a Riemannian manifold. In order to make our approach more understanble, we need to recall some terminology. Let (M n , g) be a connected compact Riemannian manifold with dimension n at least three and smooth boundary ∂M. According to [5, 7, 14, 20] and [21], we say that g is, for simplicity, a Miao-Tam critical metric if there is a nonnegative smooth function f on M n such that f |∂M = 0 and satisfies the overdeterminedelliptic system (1.1)

L∗g (f ) = −(∆f )g + Hessg f − f Ricg = g,

Date: December 1, 2017. 2010 Mathematics Subject Classification. Primary 53C25, 53C20, 53C21; Secondary 53C65. Key words and phrases. Volume functional; critical metrics; isoperimetric inequality; Weitzenb¨ ock formula. H. Baltazar was partially supported by CNPq/Brazil. E. Ribeiro was partially supported by CNPq/Brazil, Grant: 303091/2015-0. 1

2

´ H. BALTAZAR, R. DIOGENES, AND E. RIBEIRO JR.

where L∗g is the formal L2 -adjoint of the linearization of the scalar curvature operator Lg , which plays a fundamental role in problems related to prescribing the scalar curvature function. Moreover, Ric, ∆ and Hess stand, respectively, for the Ricci tensor, the Laplacian operator and the Hessian form on M n . It is proved in [20, 21] that these critical metrics arise as critical points of the volume functional on M n when restricted to the class of metrics g with prescribed constant scalar curvature such that g|T ∂M = h for a prescribed Riemannian metric h on the boundary. While Corvino, Eichmair and Miao [14] 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. In particular, they were able to prove a deformation result which suggests that the information of scalar curvature is not sufficient in giving volume comparison. The classification problem for critical metrics of the volume functional is important and relevant in understanding the influence of the scalar curvature in controlling the volume of a given manifold. For more details on such a subject, we refer the reader to [3, 5, 7, 14, 18, 20, 21, 26] and references therein. Isoperimetric problems are classical objects of study in mathematics. Generally speaking, the isoperimetric inequality is a geometric inequality involving the surface area of a set and its volume. The isoperimetric inequality on the plane states that for the length L of a closed curve on R2 and the area A of the planar region that it encloses must to satisfy L2 ≥ 4πA. Moreover, the equality holds if and only if the curve is a circle. In Rn , the classical isoperimetric inequality asserts that if M ⊂ Rn is a compact domain with smooth boundary ∂M, then n−1

(1.2)

Vol(M ) n |∂M | ≥ n−1 , n |∂B | Vol(Bn ) n

where |∂M | denotes the (n − 1)-dimensional volume of ∂M and Vol(M ) is the volume of M. For sufficiently smooth domains, the n-dimensional isoperimetric inequality is equivalent to the Sobolev inequality on Rn (cf. [19]). We refer to [22] for a general discussion on this topic. Inspired by a classical result obtained in [10] and [24], it has been shown by Batista, Ranieri and the last two named authors [7] that the boundary ∂M of a compact three-dimensional oriented Miao-Tam critical metric (M 3 , g) with connected boundary and nonnegative scalar curvature must be a 2-sphere whose area satisfies the inequality 4π |∂M | ≤ , C where C is a positive constant. Moreover, the equality holds if and only if M 3 is isometric to a geodesic ball in R3 or S3 . This result also holds for negative scalar

CRITICAL METRICS OF THE VOLUME FUNCTIONAL

3

curvature, provided that the mean curvature of the boundary satisfies H > 2, as was proven in [4]; see also [6]. Another upper bound estimate for the area of the boundary was obtained by Corvino, Eichmair and Miao (cf. [14], Proposition 2.5). In the spirit of these quoted results and stimulated by the isoperimetric problem, we provide a lower bound estimate for the area of the boundary of a compact manifold satisfying (1.1). More precisely, we have established the following result. Theorem 1. Let (M n , g, f ) be a compact, oriented, Miao-Tam critical metric with connected boundary ∂M. Then the area of the boundary |∂M | must satisfy (1.3)

|∂M | ≥

n+2 H 3 CR , 2n(n − 1)2

where H is the mean curvature of ∂M with respect to the outward unit normal and CR is a positive constant given by Z (R2 f 3 + 3nRf 2 + 2n2 f )dMg . CR = M

Moreover, equality holds if and only if M n is isometric to a geodesic ball in a simply connected space form Rn , Hn or Sn . In order to justify our second main result it is important to recall a classical example of Miao-Tam critical metric built in a Euclidean ball in Rn (cf. Example 1 in [7], see also [20]). Firstly, we consider the triple (Bn , g0 , f ), where (Bn , g0 ) denotes the Euclidean ball in Rn of radius r with standard metric g0 and potential 1 (r2 − |x|2 ). Thus, it is not hard to verify that (Bn , g0 , f ) is a function f = 2(n−1) Miao-Tam critical metric. Furthermore, using the Co-Area formula (cf. [23]) jointly with a suitable change of variable we obtain Z

f dx =

Bn

(1.4)

=

1 rn+2 2 n n r Vol(B ) − |∂B1 | 2(n − 1) n+2 1 rn+2 |∂Bn1 |, n(n − 1)(n + 2)

where ∂Bn1 stands for the boundary of the unit ball. Consequently, a straightforward computation gives

(1.5)

|∂Bn |

Vol(Bn )

n−1 n

=

(n + 2)nn H n+2 (n − 1)n+1 n

Z

Bn

n1 , f dx

where H is the mean curvature of ∂B with respect to the outward unit normal which is, in this case, given by H = n−1 r . One question that naturally arises from Eq. (1.5) is to establish the interplay between volume and area of the boundary for a general critical metric of the volume functional on compact manifold with boundary. In this context, Corvino, Eichmair and Miao [14] were able to show that the area of the boundary ∂M of

4


an n-dimensional Miao-tam critical metric with zero scalar curvature must have an upper bound depending on the volume of M n as follows p Z − 12 (n − 2)(n − 1) 3 |∂M | 2 , Rh Vol(M ) ≥ n ∂M where Rh is the scalar curvature of (∂M, h). In the sequel, motivated by this discussion above as well as the isoperimetric problem, we shall use Theorem 1 to obtain a lower bound estimate (depending on the volume of M n ) for the area of the boundary of a compact manifold M n with nonnegative scalar curvature satisfying (1.1). More precisely, we have the following isoperimetric type inequality for Miao-Tam critical metrics with nonnegative scalar curvature. Theorem 2. Let (M n , g, f ) be a compact, oriented, Miao-Tam critical metric with connected boundary ∂M and nonnegative scalar curvature. Then we have: (1.6)

1

|∂M | ≥ (CR,H ) n Vol(M )

n−1 n

, n−2

n+2 CR , H is the where CR,H is a positive constant given by CR,H = (n+2)n 2(n−1)n+1 H mean curvature of ∂M with respect to the outward unit normal and CR is described in Theorem 1. Moreover, equality holds if and only if M n is isometric to a geodesic ball in Rn .

Recently, mainly motivated by [1], Batista et al. [7] obtained a Böchner type formula for three-dimensional Riemannian manifolds satisfying (1.1) involving the traceless Ricci tensor and the Cotton tensor; see also [2]. As consequence of such a Böchner type formula, they obtained some classification results for critical metrics of the volume functional on three-dimensional compact manifolds with boundary. Before to discuss the four-dimensional case, it is worth mentioning that dimension four display fascinating and peculiar features, for this reason very much attention has been given to this dimension; see, for instance [9, 16], for more details on this specific dimension. In [15], Derdzi´ nski showed that every oriented four-dimensional Einstein mani4 fold (M , g) satisfies the Weitzenb¨ ock formula ∆|W ± |2 = 2|∇W ± |2 + R|W ± | − 36 det W ± , where W + , W − and R stand for the self-dual, anti-self-dual and scalar curvature of M 4 , respectively. The Weitzenb¨ ock formula is a powerful ingredient in the theory of canonical metrics on 4-manifolds. It may be used to obtain classification results as well as rule out some possible new examples. In conjunction with Hitchin’s theory [9], the Weitzenb¨ ock formula provides the classification of four-dimensional Einstein manifolds with positive scalar curvature. It was recently shown by Wu [25] an alternative proof of the classical Weitzenb¨ ock formula as well as some classification results for Einstein and conformally Einstein four-dimensional manifolds.


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Moreover, he obtained a Weitzenb¨ ock type formula for a large class of metrics on four-manifolds which are called generalized m-quasi-Einstein metrics. The proof uses, among other ingredients, an elegant argument of Hamilton [17] and Berger curvature decomposition [8, 9]. In the second part of this article, by using a method analogous to the one of Wu [25], we shall provide a Weitzenb¨ ock type formula for critical metrics of the volume functional on four-dimensional manifolds. More precisely, we have established the following result. Theorem 3. Let (M 4 , g) be a four-dimensional connected, smooth Riemannian manifold and f is a smooth function on M 4 satisfying (1.1). Then we have: 4Rf 2 4f 2 2 ± 2 2 ± 2 + + 8|∇f | |W ± |2 div(f ∇|W | ) = 2f |∇W | + 3 3 +f h∇|W ± |2 , ∇f i − 144f 2 det W ± ±

±

−4f 2 h(Ric ⊙ Ric) , W ± i + 6h(Ric ⊙ df ⊗ df ) , W ± i, where ⊙ stands for the Kulkarni-Nomizu product and ⊗ is the tensorial product. As an application of the Weitzenb¨ ock type formula obtained in Theorem 3 we have the following corollary. Corollary 1. Let (M 4 , g, f ) be a four-dimensional simply connected, compact Miao-Tam critical metric with nonnegative scalar curvature and boundary isometric to a standard sphere S3 . Suppose that (1.7)

˚ ⊙ df ⊗ df, W idMg ≥ 2 hRic 3 M

Z

Z

M

√ ˚ 2 + 4|W |2 f 2 |W |dMg , 6|Ric|

˚ stands for the traceless Ricci tensor. Then M 4 is isometric to a geodesic where Ric ball in a simply connected space form R4 or S4 . 2. Background and Key Lemmas In this section we shall establish the standard notation and terminology that we follow throughout this paper. Moreover, we shall present some key lemmas which will be useful for the establishment of the desired results. We start remembering that the fundamental equation of a Miao-Tam critical metric is given by (2.1)

− (∆f )g + Hessf − f Ric = g.

We also mention that a Riemannian manifold (M n , g) for which there exists a nontrivial function f satisfying (2.1) must have constant scalar curvature R (cf. Proposition 2.1 in [14] and Theorem 7 in [20]). Taking the trace of (2.1) we arrive at (2.2)

∆f +

fR + n = 0. n−1


6

Putting these facts together, we get Rf + 1 g. n−1 Furthermore, it is not difficult to verify the following identities ∇2 f − f Ric = −

(2.3)

1 R n ∆f 2 + f2 + f = |∇f |2 2 n−1 n−1

(2.4) and

R n 1 ∆f 3 + f3 + f 2 = 2f |∇f |2 . 3 n−1 n−1 Also, it easy to check from (2.2) that

(2.5)

˚ , ˚ = Hessf f Ric

(2.6)

˚ stands for the traceless of T. where T For what follows, it is important to remember that the Weyl tensor W is defined by the following decomposition formula 1 Rijkl = Wijkl + Rik gjl + Rjl gik − Ril gjk − Rjk gil n−2 R − (2.7) gjl gik − gil gjk , (n − 1)(n − 2) where Rijkl stands for the Riemann curvature tensor, whereas the Cotton tensor C is given by 1 ∇i Rgjk − ∇j Rgik ). (2.8) Cijk = ∇i Rjk − ∇j Rik − 2(n − 1)

Notice that Cijk is skew-symmetric in the first two indices and trace-free in any two indices. We also remember that W ≡ 0 in dimension three. Following the notation employed in [5], we recall that the covariant 3-tensor Tijk is given by

n−1 1 (Rik ∇j f − Rjk ∇i f ) + (gik Rjs ∇s f − gjk Ris ∇s f ) n−2 n−2 R (2.9) − (gik ∇j f − gjk ∇i f ). n−2 This tensor is closely tied to the Cotton tensor, and it played a fundamental role in the previous work [5] on classifying Bach-flat critical metrics of the volume functional. The tensor Tijk is also skew-symmetric in their first two indices and trace-free in any two indices. In order to set the stage for the proof to follow let us recall an useful result obtained previously in [[5], Lemma 1]. Tijk

=

Lemma 1 ([5]). Let (M n , g) be a connected, smooth Riemannian manifold and f is a smooth function on M n satisfying Eq. (1.1). Then we have: f (∇i Rjk − ∇j Rik ) = Rijkl ∇l f +

R (∇i f gjk − ∇j f gik ) − (∇i f Rjk − ∇j f Rik ). n−1


7

At the same time, we remember a result obtained by Hamilton [[17], Lemma 7.2] which plays a crucial role in this article. Lemma 2 ([17]). For any metric gij the curvature tensor Rijkl satisfies the identity ∆Rijkl (2.10)

=

∇i (∇k Rjl − ∇l Rjk ) − ∇j (∇k Ril − ∇l Rik ) −2Q(R)ijkl + (Rpjkl Rpi − Rpikl Rpj ),

where Q(R)ijkl = Bijkl − Bijlk − Biljk + Bikjl and Bijkl = Ripjq Rkplq . After these preliminary remarks we may state our first key lemma as follows. Lemma 3. Let (M n , g, f ) be a connected, smooth Riemannian manifold and f is a smooth function on M n satisfying (1.1). Then we have: div(f ∇Rijkl ) = (2.11)

R 2Rf + 2 2 Rijkl − ∇ f ⊙ Ric − g n−1 n−1 ijkl

+Cjil ∇k f + Cijk ∇l f + Clkj ∇i f + Ckli ∇j f − 2f Q(R)ijkl ,

where Cijk is the Cotton tensor. Proof. First of all, we need to express f ∆Rijkl in terms of the Cotton tensor. To that end, we first use Lemma 2 to infer f ∆Rijkl

= f ∇i (∇k Rjl − ∇l Rjk ) − f ∇j (∇k Ril − ∇l Rik ) +f (Rpjkl Rpi − Rpikl Rpj ) − 2f Q(R)ijkl = ∇i [f (∇k Rjl − ∇l Rjk )] − (∇k Rjl − ∇l Rjk )∇i f −∇j [f (∇k Ril − ∇l Rik )] + (∇k Ril − ∇l Rik )∇j f +f (Rpjkl Rpi − Rpikl Rpj ) − 2f Q(R)ijkl ,

and therefore by Lemma 1 we obtain

f ∆Rijkl

R (∇k f gjl − ∇l f gjk ) − (∇k f Rjl − ∇l f Rjk )] n−1 R −∇j [Rklip ∇p f + (∇k f gil − ∇l f gik ) − (∇k f Ril − ∇l f Rik )] n−1 −(∇k Rjl − ∇l Rjk )∇i f + (∇k Ril − ∇l Rik )∇j f

= ∇i [Rkljp ∇p f +

+f (Rpjkl Rpi − Rpikl Rpj ) − 2f Q(R)ijkl . Rearranging the terms we get


8

f ∆Rijkl

R (∇i ∇k f gjl − ∇i ∇l f gjk ) n−1 −(∇i ∇k f Rjl + ∇k f ∇i Rjl − ∇i ∇l f Rjk − ∇l f ∇i Rjk ) R (∇j ∇k f gil − ∇j ∇l f gik ) −∇j Rklip ∇p f − Rklip ∇j ∇p f − n−1 +(∇j ∇k f Ril + ∇k f ∇j Ril − ∇j ∇l f Rik − ∇l f ∇j Rik )

=

∇i Rkljp ∇p f + Rkljp ∇i ∇p f +

−(∇k Rjl − ∇l Rjk )∇i f + (∇k Ril − ∇l Rik )∇j f +f (Rpjkl Rpi − Rpikl Rpj ) − 2f Q(R)ijkl =

∇i Rjpkl ∇p f + ∇j Rpikl ∇p f + Rjpkl ∇i ∇p f − Ripkl ∇j ∇p f R + (∇i ∇k f gjl − ∇i ∇l f gjk − ∇j ∇k f gil + ∇j ∇l f gik ) n−1 −(∇i ∇k f Rjl − ∇i ∇l f Rjk − ∇j ∇k f Ril + ∇j ∇l f Rik ) +(∇j Ril − ∇i Rjl )∇k f + (∇i Rjk − ∇j Rik )∇l f −(∇k Rjl − ∇l Rjk )∇i f + (∇k Ril − ∇l Rik )∇j f +f (Rpjkl Rpi − Rpikl Rpj ) − 2f Q(R)ijkl .

Thus, it follows from (2.8) that

f ∆Rijkl

(2.12)

=

∇i Rjpkl ∇p f + ∇j Rpikl ∇p f + Rjpkl ∇i ∇p f − Ripkl ∇j ∇p f R (∇2 f ⊙ g)ijkl − (∇2 f ⊙ Ric)ijkl + n−1 +Cjil ∇k f + Cijk ∇l f + Clkj ∇i f + Ckli ∇j f +f (Rpjkl Rpi − Rpikl Rpj ) − 2f Q(R)ijkl ,

where ⊙ is the Kulkarni-Nomizu product. In particular, (2.12) can be rewritten as

f ∆Rijkl

(2.13)

= ∇i Rjpkl ∇p f + ∇j Rpikl ∇p f + Rjpkl ∇i ∇p f − Ripkl ∇j ∇p f R − ∇2 f ⊙ Ric − g + Cjil ∇k f + Cijk ∇l f + Clkj ∇i f n−1 ijkl +Ckli ∇j f + f (Rpjkl Rpi − Rpikl Rpj ) − 2f Q(R)ijkl .

Proceeding, we use Bianchi identity and Eq. (2.3) to arrive at


f ∆Rijkl

=

=

9

−∇p Rijkl ∇p f + Rjpkl ∇i ∇p f − Ripkl ∇j ∇p f R 2 g + Cjil ∇k f + Cijk ∇l f + Clkj ∇i f − ∇ f ⊙ Ric − n−1 ijkl

+Ckli ∇j f + f (Rpjkl Rpi − Rpikl Rpj ) − 2f Q(R)ijkl Rf + 1 Rf + 1 gip − Ripkl f Rjp − gjp −∇p Rijkl ∇p f + Rjpkl f Rip − n−1 n−1 R − ∇2 f ⊙ Ric − g + Cjil ∇k f + Cijk ∇l f + Clkj ∇i f n−1 ijkl +Ckli ∇j f + f (Rpjkl Rpi − Rpikl Rpj ) − 2f Q(R)ijkl .

Of which we deduce

f ∆Rijkl

(Rf + 1) Rijkl = −∇p Rijkl ∇p f − f (Rpjkl Rpi − Rpikl Rpj ) + 2 n−1 R g + Cjil ∇k f + Cijk ∇l f + Clkj ∇i f − ∇2 f ⊙ Ric − n−1 ijkl +Ckli ∇j f + f (Rpjkl Rpi − Rpikl Rpj ) − 2f Q(R)ijkl ,

which can rewritten succinctly as

f ∆Rijkl (2.14)

=

2Rf + 2 R 2 Rijkl − ∇ f ⊙ Ric − g − ∇p Rijkl ∇p f n−1 n−1 ijkl

+Cjil ∇k f + Cijk ∇l f + Clkj ∇i f + Ckli ∇j f − 2f Q(R)ijkl .

Finally, we are ready to compute div(f ∇Rijkl ). To do so, it suffices to observe that div(f ∇Rijkl ) = f ∆Rijkl + ∇p Rijkl ∇p f, and this combined with (2.14) yields div(f ∇Rijkl ) =

R 2Rf + 2 Rijkl − ∇2 f ⊙ Ric − g n−1 n−1 ijkl

+Cjil ∇k f + Cijk ∇l f + Clkj ∇i f + Ckli ∇j f − 2f Q(R)ijkl . So, the proof of the lemma is finished.

To conclude this section we provide an useful expression for div(f 2 ∇|W |2 ), which is a key ingredient in the proof of Theorem 3. Before to do that, we mention that in the remainder of this section, we will always consider hS, T i = Sijkl T ijkl , for any (0, 4)-tensors S and T. Our convention differs from [25] by 14 .


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Lemma 4. Let (M n , g, f ) be a connected, smooth Riemannian manifold and f is a smooth function on M n satisfying (1.1). Then we have: div(f 2 ∇|W |2 ) = (2.15)

4f 2nf 2 4Rf 2 |W |2 + |W |2 − hRic ⊙ Ric, W i n−1 n−1 n−2 4(n − 1) +f h∇|W |2 , ∇f i + hRic ⊙ df ⊗ df, W i n−2 +8|ι∇f W |2 − 4f 2 hQ(W ), W i + 2f 2 |∇W |2 ,

where ι is the interior multiplication and ⊗ stands for the tensorial product. Proof. Since W is traceless, it is not difficult to see that hA ⊙ g, W i = 0 for any (0, 2)-tensor A. This data jointly with (2.14) gives f 2 h∆Rm, W i = =

f 2 ∆Rijkl Wijkl R 2Rf 2 + 2f 2 Wijkl Rijkl Wijkl − f ∇ f ⊙ Ric − g n−1 n−1 ijkl

−f ∇p Rijkl ∇p f Wijkl + f Cijk ∇l f Wijkl + f Cjil ∇k f Wijkl +f Ckli ∇j f Wijkl + f Clkj ∇i f Wijkl − 2f 2 Q(R)ijkl Wijkl .

Consequently, it follows from (2.7) that f 2 h∆Rm, W i = f 2 ∆Rijkl Wijkl =

2f 2Rf 2 |W |2 + |W |2 − f (∇2 f ⊙ Ric)ijkl Wijkl n−1 n−1 −f ∇p Rijkl ∇p f Wijkl + 4f Cijk ∇l f Wijkl − 2f 2 Q(R)ijkl Wijkl .

and hence, using Eq. (1.3) of [12] and once more (2.7), we obtain f 2 h∆Rm, W i

=

2Rf 2 2f |W |2 + |W |2 − f (∇2 f ⊙ Ric)ijkl Wijkl n−1 n−1 −f ∇p Wijkl ∇p f Wijkl + 4f Cijk ∇l f Wijkl − 2f 2 Q(W )ijkl Wijkl −

4f 2 (Rik Rjl − Ril Rjk )Wijkl . n−2

Thereby, from Lemma 2 of [5] and (2.1) we achieve

f 2 h∆Rm, W i =

(2.16)

2Rf 2 2f |W |2 + |W |2 − f (∇2 f ⊙ Ric)ijkl Wijkl n−1 n−1 −f ∇p Wijkl ∇p f Wijkl + 4Tijk ∇l f Wijkl + 4|ι∇f W |2 −2f 2 hQ(W ), W i −

2f 2 (Ric ⊙ Ric)ijkl Wijkl , n−2

where ι is the interior multiplication and Tijk was defined in (2.9). In particular, the expression of (2.9) substituted into (2.16) immediately arrives


f 2 h∆Rm, W i =

11

2f 2Rf 2 |W |2 + |W |2 − f 2 (Ric ⊙ Ric)ijkl Wijkl n−1 n−1 4(n − 1) −f ∇p Wijkl ∇p f Wijkl + (Rik ∇j f − Rjk ∇i f )∇l f Wijkl n−2 2f 2 (Ric ⊙ Ric)ijkl Wijkl . +4|ι∇f W |2 − 2f 2 hQ(W ), W i − n−2

Summing up, we obtain f 2 h∆Rm, W i

(2.17)

=

2f nf 2 2Rf 2 |W |2 + |W |2 − (Ric ⊙ Ric)ijkl Wijkl n−1 n−1 n−2 4(n − 1) (Rik ∇j f ∇l f − Rjk ∇i f ∇l f )Wijkl −f ∇p Wijkl ∇p f Wijkl + n−2 +4|ι∇f W |2 − 2f 2 hQ(W ), W i.

Easily one verifies that (Ric ⊙ df ⊗ df )ijkl Wijkl = 2(Rik ∇j f ∇l f − Rjk ∇i f ∇l f )Wijkl , and plugging this in (2.17) we see that f 2 h∆Rm, W i =

= (2.18)

2Rf 2 2f nf 2 |W |2 + |W |2 − hRic ⊙ Ric, W i n−1 n−1 n−2 2(n − 1) 1 − f ∇p |W |2 ∇p f + (Ric ⊙ df ⊗ df )ijkl Wijkl 2 n−2 +4|ι∇f W |2 − 2f 2 hQ(W ), W i 2Rf 2 2f nf 2 |W |2 + |W |2 − hRic ⊙ Ric, W i n−1 n−1 n−2 1 2(n − 1) − f h∇|W |2 , ∇f i + hRic ⊙ df ⊗ df, W i 2 n−2 +4|ι∇f W |2 − 2f 2 hQ(W ), W i.

On the other hand, it is straightforward to verify that div(f 2 ∇|W |2 ) = = (2.19)

=

f 2 ∆|W |2 + h∇|W |2 , ∇f 2 i

2f 2 h∆W, W i + 2f 2 |∇W |2 + 2f h∇|W |2 , ∇f i

2f 2 h∆Rm, W i + 2f 2 |∇W |2 + 2f h∇|W |2 , ∇f i

Together, (2.18) and (2.19) yields

div(f 2 ∇|W |2 ) =

4Rf 2 4f 2nf 2 |W |2 + |W |2 − hRic ⊙ Ric, W i n−1 n−1 n−2 4(n − 1) hRic ⊙ df ⊗ df, W i +f h∇|W |2 , ∇f i + n−2 +8|ι∇f W |2 − 4f 2 hQ(W ), W i + 2f 2 |∇W |2 ,


12

which gives the desired result.

As already mentioned, 4-manifolds display peculiar features. For instance, the bundle of 2-forms on a four-dimensional oriented Riemannian manifold can be invariantly decomposed as a direct sum Λ2 = Λ+ ⊕ Λ− . This decomposition is conformally invariant. Moreover, it allows us to conclude that the Weyl tensor W is an endomorphism of Λ2 = Λ+ ⊕Λ− such that W = W + ⊕W − , where W + and W − stand for the self-dual and anti-self-dual parts of the Weyl tensor of M 4 , respectively. In particular, we also mention that a four-dimensional connected, smooth Riemannian manifold (M 4 , g) and a smooth function f on M 4 satisfying (1.1) must to fulfill

div(f 2 ∇|W ± |2 ) = (2.20)

4f 4Rf 2 |W ± |2 + |W ± |2 − 4f 2 h(Ric ⊙ Ric)± , W ± i 3 3 +f h∇|W ± |2 , ∇f i + 6h(Ric ⊙ df ⊗ df )± , W ± i

+8|ι∇f W ± |2 − 4f 2 hQ(W )± , W ± i + 2f 2 |∇W ± |2 .

We want to point out that the proof of (2.20) for the anti-self-dual and self-dual part of the Weyl tensor can be implemented in quite the same way of the proof of Lemma 4, so it is omitted. 3. Proof of the Main Results 3.1. Proof of Theorem 1. Proof. To begin with, we use (2.2) to arrive at div(f ∇|∇f |2 − |∇f |2 ∇f ) = =

f ∆|∇f |2 − |∇f |2 ∆f R n f ∆|∇f |2 + f |∇f |2 + |∇f |2 n−1 n−1

Upon integrating this expression over M n we use Stokes’s formula to obtain Z Z R f |∇f |2 dMg f ∆|∇f |2 dMg = |∇f |3 |∂M | − n−1 M M Z n (3.1) |∇f |2 dMg . − n−1 M

In the last expression, we have used that f vanishes on ∂M and |∇f | is constant on ∂M (cf. [7], p. 1538). We now recall the classical Böchner formula (cf. [13], p. 83): 1 ∆|∇f |2 = Ric(∇f, ∇f ) + h∇∆f, ∇f i + |Hess f |2 . 2


13

This jointly with (2.2) yields Z

M

f ∆|∇f |2 dMg

=

2

Z

f Ric(∇f, ∇f )dMg + 2

M

Z

M

f h∇∆f, ∇f idMg

Z +2 f |Hess f |2dMg M Z Z 2R f |∇f |2 dMg f Ric(∇f, ∇f )dMg − 2 n−1 M M Z f |Hess f |2dMg . +2

=

M

With aid of (2.6) we can rewritten this above expression as Z

M

f ∆|∇f |2 dMg

Z 2R f Ric(∇f, ∇f )dMg − f |∇f |2 dMg n−1 M M Z Z 2 3 ˚ 2 f (∆f )2 dMg . +2 f |Ric| dMg + n M M

=

2

(3.2)

Z

On the other hand, it is not hard to check that

2

Z

f Ric(∇f, ∇f )dMg

M

Z

=

M

div(f 2 Ric(∇f ))dMg −

Z

M

f 2 Rij ∇i ∇j f dMg ,

where we have used the twice contracted second Bianchi identity (2divRic = ∇R = ˚ = Ric − R g to get 0). Next, we use (2.1) and that Ric n 2

Z

M

f Ric(∇f, ∇f )dMg

=

= (3.3)

Z ˚ 2 dMg − 1 f 3 |Ric| R2 f 3 dMg n M M Z − Rf 2 (∆f + 1)dMg M Z Z 1 3 ˚ 2 R2 f 3 dMg − f |Ric| dMg + n(n − 1) M M Z 1 + Rf 2 dMg . n−1 M

−

Z

Substituting (3.3) into (3.2) and comparing with (3.1) we obtain |∇f |3 |∂M | =

Z ˚ 2 dMg − 1 Rf |∇f |2 dMg f 3 |Ric| n−1 M M Z Z 1 1 2 3 + R f dMg + Rf 2 dMg n(n − 1) M n−1 M Z Z n 2 2 f (∆f ) dMg + |∇f |2 dMg . + n M n−1 M

Z

Next, use (2.2) and (2.5) to arrive at


14

|∇f |3 |∂M | =

=

Z n−2 R2 f 3 dMg 2n(n − 1)2 M M 2 Z Z n−2 n 2 Rf 2 + dMg + f Rf dMg + 2(n − 1)2 M n M n−1 n−1 Z n + |∇f |2 dMg n−1 M Z Z ˚ 2 dMg + n + 2 f 3 |Ric| R2 f 3 dMg 2 2n(n − 1) M M Z Z 2n n+6 2 Rf dM + f dMg + g 2(n − 1)2 M (n − 1)2 M Z n |∇f |2 dMg . + n−1 M Z

˚ 2 dMg + f 3 |Ric|

1 Proceeding, since the mean curvature of the boundary is H = |∇f | (cf. Theorem 7 in [20], see also Eq. (3.3) of [7]), we may invoke (2.4) to obtain Z Z ˚ 2 dMg + n + 2 H 3 |∂M | = H 3 f 3 |Ric| R2 f 3 dMg 2n(n − 1)2 M M Z Z n(n + 2) 3 3(n + 2) 3 2 H H Rf dM + f dMg + g 2(n − 1)2 (n − 1)2 M M Z ˚ 2 dMg = H3 f 3 |Ric| M Z n+2 3 + H (3.4) [R2 f 3 + 3nRf 2 + 2n2 f ]dMg , 2n(n − 1)2 M

or equivalently (3.5)

|∂M | ≥

n+2 H 3 CR , 2n(n − 1)2

where CR is a constant given by Z R2 f 3 + 3nRf 2 + 2n2 f dMg . CR = M

Moreover, to see that CR is a positive constant it suffices to combine (2.2) and (2.4) in order to rewrite the last expression as Z Z 2 f ∆f dMg (Rf + n) f dMg − n(n − 1) CR = ZM ZM |∇f |2 dMg , (Rf + n)2 f dMg + n(n − 1) = M

M

which is clearly positive. Finally, from (3.4) we deduce that the equality holds in (3.5) if and only if M n is Einstein. Hence, we may apply Theorem 1.1 of [21] to conclude that M n is isometric to a geodesic ball in a simply connected space form Rn , Hn or Sn . The proof is completed.


15

3.2. Proof of Theorem 2. Proof. Firstly, it is easy to verify from (2.2) that Z n R V ol(M ). f dMg + |∇f ||∂M | = n−1 M n−1 Since M n has nonnegative scalar curvature, we have (3.6)

|∂M | ≥

nH V ol(M ), n−1

where H is the mean curvature of ∂M with respect to the outward unit normal. We then combine Theorem 1 with (3.6) to infer |∂M |n

n+2 H 3 CR |∂M |n−1 2n(n − 1)2 n−1 nH n+2 3 H CR V ol(M ) 2n(n − 1)2 n−1

≥ ≥

(3.7)

=

CR,H V ol(M )n−1 ,

where CR,H is a constant given by CR,H =

(n + 2)nn−2 n+2 H CR , 2(n − 1)n+1

and CR is defined in Theorem 1. In particular, the equality holds in (3.7) if and only if the equality holds in (3.6) as well as in Theorem 1. From this it follows that the scalar curvature must be zero and (M n , g) is isometric to a geodesic ball in Rn . This is what we wanted to prove. 3.3. Proof of Theorem 3. Proof. We start choosing n = 4 in Lemma 4 with respect to the anti-self-dual and self-dual part of the Weyl tensor (see also Eq. (2.20)) to obtain div(f 2 ∇|W ± |2 ) = (3.8)

4f 4Rf 2 |W ± |2 + |W ± |2 − 4f 2 h(Ric ⊙ Ric)± , W ± i 3 3 +f h∇|W ± |2 , ∇f i + 6h(Ric ⊙ df ⊗ df )± , W ± i

+8|ι∇f W ± |2 − 4f 2 hQ(W )± , W ± i + 2f 2 |∇W ± |2 .

On the other hand, it is known from the Berger curvature decomposition (cf. [25], p. 1090) that (3.9)

± Q(W )± ijkl Wijkl = 36 det W .

Moreover, from Lema 3.2 in [25] we have (3.10)

|ι∇f W ± |2 = |W ± |2 |∇f |2 ,


16

which substituted jointly with (3.9) into (3.8) gives 4Rf 2 4f ± |W ± |2 + |W ± |2 − 4f 2 h(Ric ⊙ Ric) , W ± i 3 3 ± +f h∇|W ± |2 , ∇f i + 6h(Ric ⊙ df ⊗ df ) , W ± i

div(f 2 ∇|W ± |2 ) =

+8|∇f |2 |W ± |2 − 144f 2 det W ± + 2f 2 |∇W ± |2 , or equivalently, div(f 2 ∇|W ± |2 ) = 2f 2 |∇W ± |2 +

4Rf 2 4f + + 8|∇f |2 |W ± |2 3 3

+f h∇|W ± |2 , ∇f i − 144f 2 det W ± ±

±

−4f 2 h(Ric ⊙ Ric) , W ± i + 6h(Ric ⊙ df ⊗ df ) , W ± i. So, the proof is finished.

3.4. Proof of Corollary 1. Proof. Upon integrating the expression obtained in Lemma 4 over M 4 , we use the Stokes’s formula and that the Weyl tensor W is trace-free to obtain 4

Z 4 ˚ ⊙ Ric, ˚ W idMg f 2 hRic Rf 2 + f |W |2 dMg − 4 3 3 M M Z Z Z ˚ ⊙ df ⊗ df, W idMg + 8 |ι∇f W |2 dMg hRic f h∇|W |2 , ∇f idMg + 6 + M M M Z Z (3.11) −4 f 2 |∇W |2 dMg . f 2 hQ(W ), W idMg + 2

0 =

Z

M

M

On the other hand, we may use (2.2) to deduce div f |W |2 ∇f =

f |W |2 ∆f + |W |2 |∇f |2 + f h∇|W |2 , ∇f i

4 R 2 f |W |2 − f |W |2 + |W |2 |∇f |2 + f h∇|W |2 , ∇f i. 3 3 Now, on integrating this expression over M n we get =

(3.12)

Z

−

2

M

f h∇|W | , ∇f idMg =

Z

M

4 f 2 + f − |∇f |2 |W |2 dMg . 3 3

R

Besides, it is not difficult to verify that ˚ ⊙ Ric| ˚ 2 = 8|Ric| ˚ 4 − 8|Ric ˚ 2 |2 |Ric

˚ 2 = |Ric| ˚ 2 , we immediately (cf. [11]). In particular, taking into account that trRic 2 2 ˚ 4 | Ric| ˚ | ≥ have |Ric , which implies 4

(3.13)

˚ ⊙ Ric| ˚ 2 ≤ 6|Ric| ˚ 4. |Ric

Returning to Eq. (3.11), we may use (3.12) and (3.13) to deduce


0 ≥

(3.14)

17

√ Z 8 ˚ 2 |W |dMg f 2 |Ric| Rf 2 + f − |∇f |2 |W |2 dMg − 4 6 3 M M 3 Z Z ˚ |ι∇f W |2 dMg hRic ⊙ df ⊗ df, W idMg + 8 +6 M M Z Z 2 f 2 |∇W |2 dMg . f hQ(W ), W idMg + 2 −4 Z

5

M

M

Easily one verifies that |hQ(W ), W i| ≤ 4|W |3 and |ι∇f W |2 = |W |2 |∇f |2 ; for more details see, for instance, Lemma 3.2 in [25]. These informations substituted into (3.14) achieves 0 ≥

(3.15)

√ Z 8 ˚ 2 |W |dMg f 2 |Ric| Rf 2 + f + 7|∇f |2 |W |2 dMg − 4 6 3 M 3 M Z Z ˚ f 2 |∇W |2 dMg hRic ⊙ df ⊗ df, W idMg + 2 +6 M M Z −16 f 2 |W |3 dMg . Z

5

M

In order to conclude it therefore suffices to use our assumption to infer (3.16)

Z

M

5

8 Rf 2 + f + 7|∇f |2 |W |2 dMg = 0. 3 3

Hence, since f and g are analytic (cf. Proposition 2.1 in [14]), Eq. (3.16) forces M 4 to be locally conformally flat, and we are in position to use Theorem 1.2 of [21] to conclude that M 4 is isometric to a geodesic ball in a simply connected space form R4 or S4 . This completes the proof of Corollary 1. Acknowledgement. The authors are grateful to R. Batista and P. Wu for fruitful conversations on the subject of this paper. References [1] Ambrozio, L.: On static three-manifolds with positive scalar curvature. J. Diff. Geom. 7 (2017) 1-45. [2] Baltazar, H. and Ribeiro Jr., E.: Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary. To appear in Pacific J. Math. (2018), arXiv:1703.01819 [math.DG]. [3] Baltazar, H. and Ribeiro Jr., E.: Critical metrics of the volume functional on manifolds with boundary. Proc. Amer. Math. Soc. 145 (2017) 3513-3523. [4] Barbosa, E., Lima, L. and Freitas, A.: The generalized Pohozaev-Schoen identity and some geometric applications. arXiv:1607.03073v1 [math.DG] (2016). [5] Barros, A., Di´ ogenes, R. and Ribeiro Jr., E.: Bach-Flat critical metrics of the volume functional on 4-dimensional manifolds with boundary. J. Geom. Anal. 25 (2015) 2698-2715. [6] Barros, A. and Da Silva, A.:

Rigidity for critical metrics of the volume functional.

arXiv:1706.07367 [math.DG] (2017).


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[7] Batista, R., Di´ ogentes, R., Ranieri, M. and Ribeiro Jr., E.: Critical metrics of the volume functional on compact three-manifolds with smooth boundary. J. Geom. Anal. 27 (2017) 1530-1547. [8] Berger, M.: Sur quelques vari´ et´ es dEinstein compactes. Ann. Mat. Pura Appl. 53 (1961) 89-95. [9] Besse, A.: Einstein manifolds. Springer-Verlag, Berlin Heidelberg (1987). [10] Boucher, W., Gibbons, G. and Horowitz, G.: Uniqueness theorem for anti-de Sitter spacetime. Phys. Rev. D (3) 30 (1984) 2447-2451. [11] Catino, G.: Integral pinched shrinking Ricci solitons. Advances in Math. 303 (2016) 279-294. [12] Catino, G. and Mantegazza, C.: The evolution of the Weyl tensor under the Ricci flow. Ann. Inst. Fourier. 61 (2012) 1407-1435. [13] Chavel, I.: Eigenvalues in Riemannian geometry. Pure appl. math., vol 115, Academic Press, New York, 1984. [14] Corvino, J., Eichmair, M. and Miao, P.: Deformation of scalar curvature and volume. Math. Annalen. 357 (2013) 551-584. [15] Derdzinski, A.: Self-dual K¨ ahler manifolds and Einstein manifolds of dimension four. Compositio Math. 49 (1983) 405-433. [16] Gompf, R. and Stipsicz, A.: 4-manifolds and Kirby calculus. Graduate Studies in Mathematics, vol. 20, American Mathematical Society (1999). [17] Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Diff. Geom. 17 (1982) 255306. [18] Kim, J. and Shin, J.: Four dimensional static and related critical spaces with harmonic curvature. To appear in Pacific J. Math. arXiv: 1604.03241v1 [math.DG] (2016). [19] Mazya. M.: Lectures on Isoperimetric and Isocapacitary Inequalities in the Theory of Sobolev Spaces. Contemp. Math. 338 (2003) 307-340. [20] Miao, P. and Tam, L.-F.: On the volume functional of compact manifolds with boundary with constant scalar curvature. Calc. Var. PDE. 36 (2009) 141-171. [21] Miao, P. and Tam, L.-F.: Einstein and conformally flat critical metrics of the volume functional. Trans. Amer. Math. Soc. 363 (2011) 2907-2937. [22] Ros, A.: The isoperimetric problem. Global theory of minimal surfaces. 2 (2001) 175-209. [23] Schoen, R. and Yau S.-T.: Lectures on Differential Geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, MA, 1994. [24] Shen, Y.: A note on Fischer-Marsdens conjecture. Proc. Amer. Math. Soc. 125 (1997) 901905. [25] Wu, P.: A Weitzenbock formula for canonical metrics on four-manifolds. Trans. Amer. Math. Soc. 369 (2017) 1079-1096. [26] Yuan, W.: Volume comparison with respect to scalar curvature. arXiv:1609.08849v1 [math.DG] (2016).

´ tica, (H. Baltazar) Universidade Federal do Piau´ı - UFPI, Departamento de Matema ´ 64049-550, Teresina, Piauı, Brazil. E-mail address: [email protected]

(R. Di´ ogenes) UNILAB, Instituto de Ciˆ encias Exatas e da Natureza, 62785-000, Acarape, ´ , Brazil. Ceara E-mail address: [email protected]


19

´ - UFC, Departamento de Matema ´ tica, (E. Ribeiro Jr) Universidade Federal do Ceara ´ , Brazil. Campus do Pici, Av. Humberto Monte, Bloco 914, 60455-760, Fortaleza, Ceara E-mail address: [email protected]