Critical metrics of the total scalar curvature functional on 4-manifolds
aa r X i v : . [ m a t h . DG ] M a y CRITICAL METRICS OF THE TOTAL SCALAR CURVATUREFUNCTIONAL ON 4-MANIFOLDS
A. BARROS, B. LEANDRO & E. RIBEIRO JR.A
BSTRACT . The purpose of this paper is to investigate the critical points of thetotal scalar curvature functional restricted to space of metrics with constant scalarcurvature of unitary volume, for simplicity CPE metrics. It was conjectured in1980’s that every CPE metric must be Einstein. We prove that a 4-dimensionalCPE metric with harmonic tensor W + must be isometric to a round sphere S .
1. I
NTRODUCTION
A fundamental problem in differential geometry is to find Riemannian metricson a given manifold that provides constant curvature. An useful tool in this direc-tion is to analyze the critical points of the total scalar curvature functional. Moreprecisely, let M n be a compact oriented smooth manifold and M the set of smoothRiemannian structures on M n of unitary volume. Given a metric g ∈ M we definethe total scalar curvature, or Einstein-Hilbert functional S : M → R by(1.1) S ( g ) = Z M R g dM g , where R g and dM g stand, respectively, for the scalar curvature of M n and the vol-ume form. Einstein and Hilbert have proved that the critical points of the functional S are Einstein, for more details see Theorem 4.21 in [4]. The Einstein-Hilbert func-tional restricted to a given conformal class is just the Yamabe functional, whosecritical points are constant scalar curvature metrics in that class.A result obtained by combining results due to Aubin [2], Schoen [17], Trudinger[19] and Yamabe [21] gives the existence of a constant scalar curvature metric inevery conformal class of Riemannian metrics on a compact manifold M n . There-fore, it is interesting to consider the set C = { g ∈ M ; R g is constant } . In [12], Koiso showed that, under generic condition, C is an infinite dimensionalmanifold (cf. Theorem 4.44 in [4] p. 127). Date : May 5, 2015.2000
Mathematics Subject Classification.
Primary 53C25, 53C20, 53C21; Secondary 53C65.
Key words and phrases.
Total scalar curvature functional, critical point equation, Einstein metric.A. Barros was partially supported by grant from CNPq/Brazil.B. Leandro was partially supported by grant from CNPq/Brazil Proc. 149896/2012-3.E. Ribeiro Jr acknowledges partial support by CNPq/Brazil and Funcap/Brazil.
It is well-known that the formal L -adjoint L ∗ g of the linearization L g of thescalar curvature operator at g is given by(1.2) L ∗ g ( f ) = − ( D g f ) g + Hess g f − f Ric g , where f is a smooth function on M n . Moreover, at any given metric g in the spaceof the metrics with constant scalar curvature the map L ∗ g defined on C ¥ to M is anover determined elliptic operator (cf. [1]). Formally the Euler-Lagrangian equationof Hilbert-Einstein action restricted to C may be written as the following criticalpoint equation(1.3) L ∗ g ( f ) = Ric − Rn g , where f is a smooth function on M n . From this, it is easy to check that (1.3) be-comes(1.4)
Ric − Rn g = Hess f − (cid:0) Ric − Rn − g (cid:1) f , where Ric , R and Hess stand, respectively, for the Ricci tensor, the scalar curvatureand the Hessian form on M n . Obviously Einstein metrics are recovered when f = . Moreover, the existence of a non constant solution is only known in the roundsphere for some height function. It has been conjectured that the critical points ofthe total scalar curvature functional S restricted to C are Einstein. For more details,we refer the reader to [4] p. 128. Definition 1.
A CPE metric is a 3-tuple ( M n , g , f ) , where ( M n , g ) is a compactoriented Riemannian manifold of dimension n ≥ with constant scalar curvaturewhile f is a smooth potential satisfying equation (1.4). Note that, computing the trace in (1.4), we obtain(1.5) D f + R ( n − ) f = . Whence f is an eigenfunction of the Laplacian and then R is positive.The conjecture proposed in [4] in the middle of 1980’s may be restated in termsof CPE (see e.g. [3], [5], [10], [11], [14] and [16]). More exactly, the quotedauthors proposed the following conjecture. Conjecture 1 (1980’s, [4]) . A CPE metric is always Einstein.
In the last years many authors have been tried to settle up this conjecture, butonly partial results were achieved. Among them, we detach the next ones: La-fontaine [13] in 1983 proved it under locally conformally flat assumption and
Ker L ∗ g ( f ) = Ker L ∗ g ( f ) , (see Corollary 1.3 in [6]); moreover, they also settled it for metricswith parallel Ricci tensor, see [7]. In 2000, Hwang [10] was able to obtain theconjecture provided f ≥ − RITICAL METRICS ON 4-MANIFOLDS 3 such that
Ker L ∗ g ( f ) = S and in 2010Chang, Hwang and Yun [5] showed that under these last conditions the conjectureis true. In 2014, Chang, Hwang and Yun [6] proved the conjecture under harmoniccurvature assumption.It is well-known that 4-dimensional compact Riemannian manifolds have spe-cial behavior. In large part, this is because the bundle of 2-forms on a 4-dimensionalcompact oriented Riemannian manifold can be invariantly decomposed as a directsum; many relevant facts may be found in [4] and [18]. For instance, on an orientedRiemannian manifold ( M , g ) , the Weyl curvature tensor W is an endomorphismof the bundle of 2-forms L = L + ⊕ L − such that W = W + ⊕ W − , where W ± : L ± −→ L ± are called of the self-dual and anti-self-dual parts of W . Half conformally flat metrics are also known as self-dual or anti-self-dual if W − or W + = , respectively. Barros and Ribeiro Jr [3] showed that Conjecture 1 isalso true for 4-dimensional half conformally flat manifolds. We highlight that CP endowed with Fubini-Study metric shows that the half-conformally flat conditionis weaker than locally conformally flat condition in dimension 4. While Qing andYuan [16] obtained a positive answer for Bach-flat manifolds in any dimension.Proceeding, viewing W + as a tensor of type ( , ) , we say that the tensor W + is harmonic if d W + = , where d is the formal divergence defined for any ( , ) -tensor T by d T ( X , X , X ) = trace g { ( Y , Z ) (cid:209) Y T ( Z , X , X , X ) } , where g is the metric of M . It is important to highlight that in dimension 4 wehave | d W | = | d W + | + | d W − | . Moreover, it should be emphasized that every 4-dimensional Einstein manifold hasharmonic tensor W + (cf. 16.65 in [4], see also Lemma 6.14 in [9]). We alsorecall that every oriented 4-dimensional manifolds with W + harmonic satisfies thefollowing relation(1.6) D | W + | = | (cid:209) W + | + R | W + | −
36 det W + . For details see, for instance, Proposition 16.73 in [4]. Therefore, it is natural to askwhich geometric implications has the assumption of the harmonicity of the tensor W + on a CPE metric.Inspired by the historical development on the study of CPE conjecture, in thispaper, we prove that, in dimension 4, the assumption that ( M , g ) is locally confor-mally flat, considered in [13] and [6], as well as half conformally flat consideredin [3], can be replaced by the weaker condition that M has harmonic tensor W + . More precisely, we have the following result.
Theorem 1.1.
Conjecture 1 is true for -dimensional manifolds with harmonictensor W + . A. BARROS, B. LEANDRO & E. RIBEIRO JR.
The main tools in the proof of the above result is to analyze the behavior ofthe level sets of the potential function f which defines a CPE metric combinedwith some pointwise arguments. Obviously if we change the condition d W + = d W − = M has harmonic tensor W + . This article is organized as follows. In Section 2, we review some classicaltensors which we shall use here. Moreover, we outline some useful informationsabout 4-dimensional manifolds. In Section 3, we prove the main result.2. B
ACKGROUND
Throughout this section we recall some basic tensors and informations that willbe useful in the proof of our main result. First of all, for operators S , T : H → H defined over an n -dimensional Hilbert space H the Hilbert-Schmidt inner productis defined according to(2.1) h S , T i = tr (cid:0) ST ⋆ (cid:1) , where tr and ⋆ denote, respectively, the trace and the adjoint operation. Moreover,if I denotes the identity operator on H the traceless operator of T is given by(2.2) ˚ T = T − trT n I . In particular the norm of ˚ T satisfies(2.3) | ˚ T | = | T | − ( trT ) n . Now, we recall that for a Riemannian manifold ( M n , g ) , n ≥ , the Weyl tensor W is defined by the following decomposition formula R i jkl = W i jkl + n − (cid:0) R ik g jl + R jl g ik − R il g jk − R jk g il (cid:1) − R ( n − )( n − ) (cid:0) g jl g ik − g il g jk (cid:1) , (2.4)where R i jkl stands for the Riemannian curvature operator. Moreover, the Cottontensor C is given according to(2.5) C i jk = (cid:209) i R jk − (cid:209) j R ik − ( n − ) (cid:0) (cid:209) i Rg jk − (cid:209) j Rg ik ) . These two tensors are related as follows(2.6) C i jk = − ( n − )( n − ) (cid:209) l W i jkl , provided n ≥ . Finally, the Schouten tensor A is given by(2.7) A i j = n − (cid:0) R i j − R ( n − ) g i j (cid:1) . For more details about these tensors we address to [4].
RITICAL METRICS ON 4-MANIFOLDS 5
In what follows M will denote an oriented 4-dimensional manifold and g is aRiemannian metric on M . As it was previously mentioned 4-manifolds are fairlyspecial. For instance, following the notations used in [9] (see also [20] p. 46),given any local orthogonal frame { e , e , e , e } on open set of M with dual basis { e , e , e , e } , there exists a unique bundle morphism ∗ called Hodge star (actingon bivectors), such that ∗ ( e ∧ e ) = e ∧ e . This implies that ∗ is an involution, i.e. ∗ = Id . In particular, this implies thatthe bundle of 2-forms on a 4-dimensional oriented Riemannian manifold can beinvariantly decomposed as a direct sum L = L + ⊕ L − . This allows us to concludethat the Weyl tensor W is an endomorphism of L = L + ⊕ L − such that(2.8) W = W + ⊕ W − . We recall that dim R ( L ) = dim R ( L ± ) = . Also, it is well-known that(2.9) L + = span n e ∧ e + e ∧ e √ , e ∧ e + e ∧ e √ , e ∧ e + e ∧ e √ o and(2.10) L − = span n e ∧ e − e ∧ e √ , e ∧ e − e ∧ e √ , e ∧ e − e ∧ e √ o . From this, the bundles L + and L − carry natural orientations such that the bases(2.9) and (2.10) are positive-oriented. Furthermore, if R denotes the curvature of M we get a matrix(2.11) R = W + + R Id ˚ Ric ˚ Ric ⋆ W − + R Id , where ˚ Ric : L − → L + stands for the Ricci traceless operator of M . Recalling that the Weyl tensor is trace-free on any pair of indices we have(2.12) W + pqr s = (cid:0) W pqr s + W pqr s (cid:1) , where ( r s ) , for instance, stands for the dual of ( r s ) , that is, ( r s r s ) = s ( ) forsome even permutation s in the set { , , , } (cf. Equation 6.17, p. 466 in [9]).In particular, we have W + = (cid:0) W + W (cid:1) . For more details we refer to [9] and [20].
A. BARROS, B. LEANDRO & E. RIBEIRO JR.
3. P
ROOF OF THE MAIN RESULT
In order to set the stage for the proof to follow let us recall an useful resultobtained in [3] for any dimension.
Lemma 1 ([3]) . Let (cid:0) M n , g , f ) be a CPE metric. Then: ( f + ) C i jk = W i jks (cid:209) s f − R ( n − ) (cid:0) (cid:209) j f g ik − (cid:209) i f g jk (cid:1) + ( n − )( n − ) (cid:0) R ik (cid:209) j f − R jk (cid:209) i f (cid:1) − ( n − ) (cid:0) R is (cid:209) s f g jk − R js (cid:209) s f g ik (cid:1) . Next, following the notation used in [3] we define the tensor T as follows T i jk = ( n − )( n − ) (cid:0) R ik (cid:209) j f − R jk (cid:209) i f (cid:1) − ( n − ) (cid:0) R is (cid:209) s f g jk − R js (cid:209) s f g ik (cid:1) − R ( n − ) (cid:0) (cid:209) j f g ik − (cid:209) i f g jk (cid:1) . (3.1)Taking into account this definition we deduce from Lemma 1 that(3.2) ( f + ) C i jk = W i jks (cid:209) s f + T i jk . Next, we need the following results by Hwang et al. [10, 11].
Proposition 1 ([10, 11]) . Let ( M n , g , f ) be a CPE metric with f non-constant.Then, the set { x ∈ M n : f ( x ) = − } has measure zero. We now denote by
Crit ( f ) the set { x ∈ M n ; (cid:209) f ( x ) = } . With this setting, wehave the following proposition.
Proposition 2 ([6]) . Let ( M n , g , f ) , be a CPE metric. Then Crit ( f ) has zero n-dimensional measure. Now we are ready to prove Theorem 1.1.3.1.
Proof of Theorem 1.1.
Proof.
To begin with, we shall compute the value of d W + on a 4-dimensional CPEmetric. In fact, since a CPE metric has constant scalar curvature, Equation (2.5)becomes C kl j = (cid:209) k R l j − (cid:209) l R k j . So, as an immediate consequence of (2.6) we have(3.3) 2 d W + jkl = (cid:0) (cid:209) k R jl − (cid:209) l R jk (cid:1) + (cid:0) (cid:209) k R jl − (cid:209) l R jk (cid:1) . From Lemma 1 and (3.1) we already know that(3.4) ( f + ) C i jk = W i jks (cid:209) s f + T i jk . This combined with (3.3) gives4 ( + f ) d W + jkl = ( + f ) (cid:0) (cid:209) k R jl − (cid:209) l R jk (cid:1) + ( + f ) (cid:0) (cid:209) k R jl − (cid:209) l R jk (cid:1) RITICAL METRICS ON 4-MANIFOLDS 7 = ( + f )( C kl j + C kl j )= h W kl js (cid:209) s f + W k l j s (cid:209) s f + T lk j + T l k j i (3.5)Now, using that d W + = − W kl js (cid:209) s f − W k l j s (cid:209) s f = T kl j + T kl j . (3.6)Hence we have0 = − (cid:16) W kl js (cid:209) s f + W k l j s (cid:209) s f (cid:17) (cid:209) j f = (cid:16) T kl j + T kl j (cid:17) (cid:209) j f , (3.7)so that (cid:16) T i jk + T i j k (cid:17) (cid:209) k f = . (3.8)On the other hand, Equation (3.1) allows us to deduce T i jk (cid:209) k f = ( R ik (cid:209) j f − R jk (cid:209) i f ) (cid:209) k f − ( R is (cid:209) s f g jk − R js (cid:209) s f g ik ) (cid:209) k f − R ( (cid:209) j f g ik − (cid:209) i f g jk ) (cid:209) k f = ( R ik (cid:209) j f − R jk (cid:209) i f ) (cid:209) k f . (3.9)Then, from (3.8) we arrive at ( R ik (cid:209) j f − R jk (cid:209) i f ) (cid:209) k f + ( R ik (cid:209) j f − R jk (cid:209) i f ) (cid:209) k f = . (3.10)In the sequel, we consider an orthonormal frame { e , e , e , e } diagonalizing Ric at a point q , such that (cid:209) f ( q ) = , with associated eigenvalues l k , ( k = , . . . , ) , respectively. It is important to highlight that the regular points of M , denoted by { p ∈ M n : (cid:209) f ( p ) = } , is dense in M . Otherwise, f must be constant in an openset of M ; in fact, from (1.5) this constant is zero, but from standard theory ofnodal sets f can not vanish in an open set, see for instance [8]. From now on, up toexplicit mention, we restrict our attention to regular points. So, a straightforwardcomputation using (3.10) gives the following useful system ( l − l ) (cid:209) f (cid:209) f + ( l − l ) (cid:209) f (cid:209) f = , ( l − l ) (cid:209) f (cid:209) f + ( l − l ) (cid:209) f (cid:209) f = , ( l − l ) (cid:209) f (cid:209) f + ( l − l ) (cid:209) f (cid:209) f = . (3.11)We now claim that (cid:209) f , whenever nonzero, is an eigenvector for Ric . In fact,taking into account that (cid:209) f ( p ) = ( (cid:209) j f ) = ≤ j ≤
4. If this occurs for exactly one of them, then (cid:209) f = ( (cid:209) j f ) e j for some j ,which gives that Ric ( (cid:209) f ) = l j (cid:209) f . On the other hand, if we have ( (cid:209) j f ) = (cid:209) f = (cid:209) f = (cid:209) f = (cid:209) f = . Then, from (3.11) we have l = l = l . In a such case wehave (cid:209) f = ( (cid:209) f ) e + ( (cid:209) f ) e . From this, we obtain
Ric ( (cid:209) f ) = Ric (( (cid:209) f ) e + ( (cid:209) f ) e ) = ( (cid:209) f ) Ric ( e ) + ( (cid:209) f ) Ric ( e )= ( (cid:209) f ) l e + ( (cid:209) f ) l e = l (cid:209) f . (3.12) A. BARROS, B. LEANDRO & E. RIBEIRO JR.
Next, the case ( (cid:209) j f ) = ( (cid:209) j f ) = j = , , , . In this case we use once more (3.11)to obtain ( l − l ) ( (cid:209) f (cid:209) f ) + ( l − l ) ( (cid:209) f (cid:209) f ) +( l − l ) ( (cid:209) f (cid:209) f ) + ( l − l ) ( (cid:209) f (cid:209) f ) +( l − l ) ( (cid:209) f (cid:209) f ) + ( l − l ) ( (cid:209) f (cid:209) f ) = . (3.13)Therefore, l = l = l = l . From here it follows that (cid:209) f is an eigenvector for Ric , which proves our claim.Proceeding, we study the level sets of the potential function f which defines aCPE metric. To that end we denote S c = { p ∈ M n : f ( p ) = c } . At regular pointsthe vector field e = (cid:209) f | (cid:209) f | is normal to S c and { e , e , e } is an orthonormal frameon S c . With this notation in mind, since
Ric ( (cid:209) f ) = l (cid:209) f and (cid:209) e a f = h (cid:209) f , e a i = a = { , , } , we immediately deduce from (3.1) that T abc = { a , b , c } = { , , } . Whence, in S c , (3.2) becomes(3.14) ( f + ) C abc = W abcs (cid:209) s f . We notice that, for an arbitrary Y ∈ T p S c , we have ˚ Ric ( (cid:209) f , Y ) = . So, we canuse the fundamental equation to infer (cid:209) e a | (cid:209) f | = ( f + ) ˚ Ric ( (cid:209) f , e a ) − R fn ( n − ) g ( (cid:209) f , e a ) = . This implies that | (cid:209) f | is constant in S c . On the other hand, it is well-known that the second fundamental form of S isgiven by(3.15) h ab = h (cid:209) e a e b , e i = − f ab | (cid:209) f | , where { a , b } = { , , } . Moreover, its mean curvature is H S = | (cid:209) f | (cid:0) f − D f (cid:1) . We then combine the fundamental equation with (3.15) to conclude that near thepoint p the second fundamental form is given by h ab = m g ab , where m is equal to the mean curvature. But, since | (cid:209) f | is constant in S c followsthat H is constant on S , so is m (see also [3]). Then, we may use Codazzi’s equationto infer R abc = (cid:209) S b h ac − (cid:209) S a h bc = p when { a , b , c } = { , , } . In particular, from (2.4) we infer W abc = ( f + ) C abc = . RITICAL METRICS ON 4-MANIFOLDS 9
We then use Proposition 1 to conclude that C abc = { a , b , c } = { , , } . Next, we already know from (3.3) that4 d W + jkl = C kl j + C ¯ k ¯ l j . (3.17)Therefore, since d W + = C abc = C ¯ a ¯ bc = . More precisely, we have0 = C ¯1¯2 c = C c , = C ¯1¯3 c = − C c and 0 = C ¯2¯3 c = C c . (3.18)Similarly, we can use (3.8) jointly with (3.2) to conclude that C i j = { i , j } = { , , , } . This allows us to conclude that C i jk = { i , j , k } = { , , , } . Whence, we have from (2.6) that d W = M . Since M has constant scalarcurvature, then M has harmonic curvature. Finally, it suffices to invoke Theorem1.2 in [6] to conclude that M is isometric to a round sphere S . In addition, wealso conclude that f is a height function on S . This finishes the proof of Theorem 1.1. (cid:3)
Acknowledgement.
The authors want to thank the referees for their careful read-ing and helpful suggestions. Moreover, the third author wish to express his grat-itude for the excellent support during his stay at Department of Mathematics -Lehigh University, where part of this paper was carried out. R EFERENCES [1] Anderson, M.: Scalar curvature, metric degenerations and the static vacuum Einstein equationson 3-manifolds. Geom. and Funct. Anal., 9 (1999), 855-967.[2] Aubin, T.: ´Equation diff´erentielles non-lin´eares et probl´eme de Yamabe concernant la courburescalaire. J. Math. Pures Appl. 55 (1976) 269-296.[3] Barros, A. and Ribeiro Jr., E.: Critical point equation on four-dimensional compact manifolds.Math. Nachr. 287, N. 14-15, (2014) 1618-1623.[4] Besse, A.: Einstein manifolds, Springer-Verlag, Berlin Heidelberg (1987).[5] Chang, J., Hwang, S. and Yun, G.: Rigidity of the critical point equation. Math. Nachr. 283(2010) 846-853.[6] Chang, J., Hwang, S. and Yun, G.: Total scalar curvature and harmonic curvature. TaiwaneseJ. Math. 18 (2014) 1439-1458.[7] Chang, J., Hwang, S. and Yun, G.: Critical point metrics of the total scalar curvature. Bull.Korean Math. Soc. 49 (2012) 655-667.[8] Cheng, S.: Eigenvalues and nodal sets. Comment. Math. Helv. 51 (1976) 43-55.[9] Dillen, F. and Verstralen, L.: Handbook of Differential Geometry. Elsevier Science B. V. vol. 1(2000).[10] Hwang, S.: Critical points of the total scalar curvature functional on the space of metrics ofconstant scalar curvature. Manuscripta Math. 103, (2000) 135 - 142 .[11] Hwang, S.: The critical point equation on a three-dimensional compact manifold. Proc. Amer.Math. Soc. 131 (2003) 3221-3230.[12] Koiso, N.: A decomposition of the space of Riemannian metrics on a manifolds. Osaka J. ofMath. 16, (1979) 423-429.[13] Lafontaine, J.: Sur la g´eom´etrie d’une g´en´eralisation de l’´equation diff´erentielle d’Obata. J.Math. Pures Appliqu´ees. 62 (1983) 63-72.[14] Leandro Neto, B.: A note on critical point metrics of the total scalar curvature functional.Journal of Math. Analysis and App. 424, (2015) 1544-1548.[15] Obata, M.: Certain conditions for a Riemannian manifold to be isometric with a sphere. J.Math. Soc. Japan. 14, (1962) 333-340. ∼ jeffv/courses/865 Fall 2011.pdf[21] Yamabe, H.: On deformation of Riemannian structures on compact manifolds. Osaka J. Math.12 (1960), 21-37.(A. Barros) U NIVERSIDADE F EDERAL DO C EAR ´ A - UFC, D EPARTAMENTO DE M ATEM ´ ATICA ,C AMPUS DO P ICI , A V . H UMBERTO M ONTE , B
LOCO
ORTALEZA / CE, B
RAZIL . E-mail address : [email protected] (B. Leandro) U NIVERSIDADE DE B RAS ´ ILIA - UNB, D
EPARTAMENTO DE M ATEM ´ ATICA , 70910-900, B
RAS ´ ILIA / DF, B
RAZIL . E-mail address : [email protected] (E. Ribeiro Jr) U
NIVERSIDADE F EDERAL DO C EAR ´ A - UFC, D EPARTAMENTO DE M ATEM ´ ATICA ,C AMPUS DO P ICI , A V . H UMBERTO M ONTE , B
LOCO
ORTALEZA / CE, B
RAZIL . E-mail address ::