Bach-flat noncompact steady quasi-Einstein manifolds
aa r X i v : . [ m a t h . DG ] D ec BACH-FLAT NONCOMPACT STEADYQUASI-EINSTEIN MANIFOLDS
M. RANIERI AND E. RIBEIRO JR
Abstract.
The goal of this article is to study the geometry of Bach-flat noncompactsteady quasi-Einstein manifolds. We show that a Bach-flat noncompact steady quasi-Einstein manifold ( M n , g ) with positive Ricci curvature such that its potential functionhas at least one critical point must be a warped product with Einstein fiber. In addition,the fiber has constant curvature if n = 4 . Introduction
In 1958, Ren´e Thom posed the following well-known question: “
Are there any best Rie-mannian structures on a smooth manifold? ”. The best Riemannian structures on a givenmanifold are those of constant curvature. In this spirit, a Riemannian manifold of dimen-sion greater than 2 with constant Ricci curvature is called
Einstein.
Hilbert and Einsteinproved that the critical metrics of the total scalar curvature functional restricted to the setof smooth Riemannian structures on a compact manifold of unitary volume are Einstein. Weremark that Einstein manifolds are not only fascinating in themselves but are also relatedto many important topics of Riemannian geometry. For a comprehensive reference on sucha subject, we refer the reader to [6].A classical problem in Riemannian geometry is to construct new explicit examples ofEinstein metrics. According to “Besse’s book” [6], a promising way for that purpose is thatof warped products. The m -Bakry-Emery Ricci tensor, which appeared previously in [2] and[20], is useful as an attempt to better understand Einstein warped product. More precisely,the m -Bakry-Emery Ricci tensor is given by(1.1) Ric mf = Ric + ∇ f − m df ⊗ df, where f is a smooth function on M n and ∇ f stands for the Hessian of f. We highlight thatit is also used to study the weighted measure dµ = e − f dx, where dx is the Riemann-Lebesguemeasure determined by the metric.A complete Riemannian manifold ( M n , g ) , n ≥ , will be called m - quasi-Einstein mani-fold , or simply quasi-Einstein manifold , if there exist a smooth potential function f on M n Date : September 21, 2016.2010
Mathematics Subject Classification.
Primary 53C25, 53C20, 53C21; Secondary 53C65.
Key words and phrases.
Einstein manifolds; warped product; Bach-flat metrics.E. Ribeiro was partially supported by CNPq/Brazil.M. Ranieri was partially supported by CAPES/Brazil. and a constant λ satisfying the following fundamental equation(1.2) Ric mf = Ric + ∇ f − m df ⊗ df = λg, where ∇ f stands for the Hessian of f. It is easy to see that a ∞ -quasi-Einstein manifold means a gradient Ricci soliton. Riccisolitons model the formation of singularities in the Ricci flow and correspond to self-similarsolutions, i.e., solutions which evolve along symmetries of the flow, see [10] and referencestherein for more details on this subject. On the other hand, when m is a positive integer itcorresponds to a warped product Einstein metric, see, for instance, [12, 17]. We also remarkthat 1-quasi-Einstein manifolds are more commonly called static metrics and such metricshave connections to scalar curvature, the positive mass theorem and general relativity. Recallthat a quasi-Einstein metric g on a manifold M n will be called expanding , steady or shrinking ,respectively, if λ < , λ = 0 or λ >
0. Moreover, a quasi-Einstein manifold will be called trivial if its potential function f is constant, otherwise it will be nontrivial . Notice that thetriviality implies that M n is an Einstein manifold.According to [5, 6, 12, 17] and [22] the remarkable motivation to study quasi-Einsteinmetrics on a Riemannian manifold is its direct relation with the existence of Einstein warpedproduct, which also have different properties compared with the gradient Ricci solitons. Inthis sense, it is important to recall that, on a quasi-Einstein manifold, there is an indispens-able constant µ such that(1.3) ∆ f − |∇ f | = mλ − mµe m f . See [5, 17, 22] and [24] for a comprehensive treatment of this feature.Qian [20] proved that shrinking quasi-Einstein manifolds must be compact. Moreover,from Kim-Kim [17] the converse statement remains true. Thereby, it is now well-knownthat a quasi-Einstein manifold is compact if and only if λ > . An example of nontrivialquasi-Einstein manifold with λ > , m > µ > µ as well as steady quasi-Einsteinmanifolds with µ > µ ≤ M n , g ) , n ≥ , which was introduced to study conformal relativity in [1], is definedin terms of the components of the Weyl tensor W ikjl as follows(1.4) B ij = 1 n − ∇ k ∇ l W ikjl + 1 n − R kl W i k j l , while for n = 3 it is given by(1.5) B ij = ∇ k C kij , where C ijk stands for the Cotton tensor. We say that ( M n , g ) is Bach-flat when B ij = 0 . Itis straightforward to check that locally conformally flat metrics as well as Einstein metricsare Bach-flat. In addition, for dimension n = 4 , it is well-known that half-conformallyflat or locally conformally to an Einstein manifold implies Bach-flat. However, Leistner INSTEIN WARPED PRODUCTS 3 and Nurowski [18] obtained a large class of Bach-flat examples which are not conformallyEinstein; for more details we address to [6].Recently, Cao and Chen [8] have studied Bach-flat gradient Ricci solitons. They obtaineda stronger classification for shrinking gradient Ricci solitons under the Bach-flat assumption.Afterward, Cao, Catino, Chen, Mantegazza and Mazzieri [7] were able to show that any n -dimensional ( n ≥
4) complete Bach-flat gradient steady Ricci soliton with positive Riccicurvature such that the scalar curvature R attains its maximum at some interior point mustbe isometric to the Bryant soliton. For more details, we refer the reader [7, 8] and [9]. TheBach-flat assumption was also studied in another special metrics, see, for instance [4, 11, 15]and [21].In light of the previous results, it is natural to ask what occurs on quasi-Einstein man-ifolds. As it was previously mentioned a quasi-Einstein manifold is compact if and onlyif λ > . In that case, Chen and He [14] proved that a Bach-flat shrinking quasi-Einsteinmanifold is either Einstein or a finite quotient of a warped product with ( n − u = e − fm (cf. Lemma 3).After these preliminary remarks we may announce our first result as follows. Theorem 1.
Let ( M n , g, f, m > , n ≥ , be a Bach-flat noncompact steady quasi-Einstein manifold with positive Ricci curvature such that f has at least one critical point.Then M n has harmonic Weyl tensor and W ijkl ∇ l f = 0 . At the same time, it is worth to point out that 4-dimensional manifolds have specialbehavior; see [6] for more information about this specific dimension. In such a dimensionwe have established the following result.
Theorem 2.
Let ( M , g, f, m > be a -dimensional Bach-flat noncompact steady quasi-Einstein manifold with positive Ricci curvature such that f has at least one critical point.Then M is locally conformally flat. Next, as an application of Theorems 1 and 2, jointly with Theorem 1.2 in [16], we havethe following classification result.
Corollary 1.
Let ( M n , g, f, m > , n ≥ , be a Bach-flat noncompact steady quasi-Einstein manifold with positive Ricci curvature such that f has at least one critical point.Then ( M n , g ) is a warped product with g = dt + ψ ( t ) g L and f = f ( t ) , where g L is Einstein of non-negative Ricci curvature. In addition, the fiber has constantcurvature if n = 4 . M. RANIERI AND E. RIBEIRO JR Background and Key Lemmas
In this section we shall present a couple of lemmas that will be useful in the proof of ourmain results. We begin recalling that the Weyl curvature W ijkl is defined by the followingdecomposition formula R ijkl = W ijkl + 1 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) , (2.1)where R ijkl stands for the Riemann curvature tensor. Moreover, the Cotton tensor C ijk isgiven by(2.2) C ijk = ∇ i R jk − ∇ j R ik − n − (cid:0) ∇ i Rg jk − ∇ j Rg ik ) . It is easy to check that C ijk is skew-symmetric in the first two indices and trace-free in anytwo indices. We also remember that W ijkl and C ijk are related as follows(2.3) − ( n − n − C ijk = ∇ l W ijkl . Moreover, taking into account (2.3) we may extend the definition of the Bach tensor for n ≥ B ij = 1 n − (cid:0) ∇ k C kij + R kl W ikjl (cid:1) . Since W ≡ n = 3 we have B ij = ∇ k C kij . Following the notation employed in [14], in the spirit of [8], we recall that the covariant3-tensor D is given by D ijk = 1 n − R jk ∇ i f − R ik ∇ j f ) + 1( n − n −
2) ( R il ∇ l f g jk − R jl ∇ l f g ik ) − R ( n − n −
2) ( g jk ∇ i f − g ik ∇ j f ) . (2.5)It is not difficult to check that the tensor D ijk is skew-symmetric in their first two indicesand trace-free in any two indices:(2.6) D ijk = − D jik and g ij D ijk = g ik D ijk = 0 . In order to set the stage for the proof to follow let us recall some useful results obtainedin [14]. Indeed, taking into account (2.5) we shall show a relation between the Cotton tensorand the Weyl tensor on a quasi-Einstein manifold.
Lemma 1 (Chen-He [14]) . Let ( M n , g, f ) be a quasi-Einstein manifold. Then we have: (2.7) C ijk = m + n − m D ijk − W ijkl ∇ l f. We also need of the following results by Chen-He [14].
INSTEIN WARPED PRODUCTS 5
Lemma 2 (Chen-He [14]) . Let ( M n , g, f ) be a quasi-Einstein manifold. Assume that Σ isa level set of f with ∇ f ( p ) = 0 . Then we have: (2.8) | D | = 2 |∇ f | ( n − n X a,b =2 | h ab − Hn − g ab | + m n − n − m − |∇ Σ R | , where h ab stands for the second fundamental form of Σ and H is its mean curvature. The next result shows that the vanishing of the tensor D ijk implies interesting rigidityproperties about the geometry of the level surfaces of the potential function. Proposition 1 (Chen-He [14]) . Let ( M n , g, f, m > be a quasi-Einstein manifold with D ijk = 0 . Let c be a regular value of f and Σ = { p ∈ M | f ( p ) = c } be a level hypersurfaceof f. We also consider e = ∇ f |∇ f | and choose an orthonormal frame { e , ..., e n } tangent to Σ . Then: (1) the scalar curvature R and |∇ f | of ( M n , g, f ) are constant on Σ;(2) R a = 0 for ≥ and e is an eigenvector of Ric ;(3) on Σ , the Ricci tensor either has a unique eigenvalue or, two distinct eigenvalueswith multiplicity and n − , moreover the eigenvalue with multiplicity is in thedirection of ∇ f ;(4) the second form fundamental h ab of Σ is h ab = Hn − g ab ; (5) the mean curvature H is constant on Σ;(6) R abc = 0 , for a, b, c ∈ { , ..., n } . Before preceeding, it is important to remember some classical equations concerning quasi-Einstein manifolds. First of all, considering the function u = e − fm on M n , we immediatelyget ∇ u = − um ∇ f as well as(2.9) Hessf − m df ⊗ df = − mu Hessu. Taking into account (1.2) and (1.3), it is easy to obtain(2.10) u m ( R − λn ) + ( m − |∇ u | = − λu + µ We also remember that, by Wang [24], if λ ≤
0, then R ≥ λn. Now we turn our attentionfor steady case. Whence, it follows from (2.10) that(2.11) |∇ u | ≤ µm − u R ≤ mµ. In the sequel we investigate the asymptotic behavior of the function u = e − fm . Moreprecisely, we prove a pinching estimate which plays a central role in this work.
M. RANIERI AND E. RIBEIRO JR
Lemma 3.
Let ( M n , g, f, m > be a complete noncompact steady quasi-Einstein manifoldwith positive Ricci curvature such that f has at least one critical point. Then, there existpositive constants c and c such that the function u = e − fm satisfies the following estimates (2.13) c r ( x ) − c ≤ u ( x ) ≤ r µm − r ( x ) + | u ( p ) | , where p is a critical point of f and r ( x ) is the distance function from p. Proof.
The proof will follow [9] (cf. Proposition 2.3 in [9]). Firstly, notice that by (2.11)the upper bound in (2.13) in fact occurs for noncompact steady quasi-Einstein manifolds ingeneral.Now, we deal of the lower bound. To do so, we first notice that (1.2) and (2.10) yields(2.14)
Ric = mu Hessu. We now assume that p is a critical point of f. So, taking into account that M n has positiveRicci curvature and u > , we immediately deduce from (2.14) that u is a strictly convexfunction. We then consider any minimizing normal geodesic γ ( s ) , ≤ s ≤ s , for sufficientlylarge s > , starting from the point p = γ (0) . Further, denote by X ( s ) = ˙ γ ( s ) the unittangent vector along γ and Dudt = ˙ u = ∇ X u ( γ ( s )) . With these notations in mind, we mayuse (2.14) to achieve(2.15) ∇ X ˙ u = ∇ X ∇ X u = um Ric ( X, X ) . Remembering that ∇ u = − um ∇ f, it follows that a critical point of f is also critical point of u. Therefore, upon integrating (2.15) along γ, for s ≥ , we arrive at(2.16) ˙ u ( γ ( s )) = Z s um Ric ( X, X ) ds ≥ Z um Ric ( X, X ) ds ≥ c , where c = cm min B p (1) u ( x )and c > B p (1) . Proceeding, on integrating (2.16) from 1 to s we get u ( γ ( s )) = Z s ˙ u ( s ) ds + u ( γ (1)) ≥ c s − c + u ( γ (1)) ≥ c s − c . This is what we wanted to prove. (cid:3)
As an immediate application of Lemma 3 we have the following result.
Corollary 2.
Let ( M n , g, f, m > be a complete noncompact steady quasi-Einstein man-ifold with positive Ricci curvature such that f has at least one critical point. Then M n isdiffeomorphic to R n . INSTEIN WARPED PRODUCTS 7
Proof.
From Eq. (2.13) we immediately have that u is a proper function. In particular, Eq.(2.14) implies that u is strictly convex and then it is well-known that M n is diffeomorphicto R n . (cid:3) Now, we use Lemma 3 to prove the main result of this section. It plays a crucial role inthe proof of Theorem 1.
Lemma 4.
Let ( M n , g, f, m > , n ≥ , be a Bach-flat noncompact steady quasi-Einsteinmanifold with positive Ricci curvature such that f has at least one critical point. Then thetensor D vanishes identically.Proof. To start with, we combine (2.4) and (2.7) to obtain( n − B ij = ∇ k C kij + W ikjl R kl = ∇ k (cid:18) m + n − m D kij − W kijl ∇ l f (cid:19) + W ikjl R kl = m + n − m ∇ k D kij − ( ∇ k W kijl ) ∇ l f − W kijl ∇ k ∇ l f + W ikjl R kl . Then, using (1.2) and (2.3), we arrive at( n − B ij = m + n − m ∇ k D kij + n − n − C lji ∇ l f − m W kijl ∇ k f ∇ l f. (2.17)Recall that ∇ u = − m u ∇ f and this substituted into (2.17) yields(2.18) ( n − B ij ∇ i u ∇ j ue − u u = m + n − m ( ∇ k D kij ) ∇ i u ∇ j ue − u u . On the other hand, a straightforward computation gives ∇ k ( D kij ∇ i u ∇ j ue − u u ) = ( ∇ k D kij ) ∇ i u ∇ j ue − u u + D kij ( ∇ k ∇ i u ) ∇ j ue − u u + D kij ∇ i u ( ∇ k ∇ j u ) e − u u = ( ∇ k D kij ) ∇ i u ∇ j ue − u u + D kij (cid:16) um R ki (cid:17) ∇ j ue − u u + D kij (cid:16) um R kj (cid:17) ∇ i ue − u u , where we have used Eq. (2.14) in the last step. Therefore, returning to Eq. (2.18) weimmediately achieve( n − B ij ∇ i u ∇ j ue − u u = m + n − m ∇ k ( D kij ∇ i u ∇ j ue − u u ) − m + n − m D kij e − u u ( R ki ∇ j u + R kj ∇ i u ) . (2.19)Notice that ∇ u = − um ∇ f substituted into (2.5) provides M. RANIERI AND E. RIBEIRO JR − um D ijk = 1 n − R jk ∇ i u − R ik ∇ j u )+ 1( n − n − h R il ∇ l ug jk − R jl ∇ l ug ik − R ( g jk ∇ i u − g ik ∇ j u ) i . (2.20)Moreover, since the tensor D is skew-symmetric in the two first indices, it is not difficult tosee that D kij R ki ∇ j u = 0 and then comparing with (2.20) we infer D kij ( R ki ∇ j u + R kj ∇ i u ) = 12 D kij R kj ∇ i u − D ikj R kj ∇ i u = − D kij ( R ij ∇ k u − R kj ∇ i u )= n − m u | D | . (2.21)Next, upon integrating (2.19) over the ball B p ( s ) , we use (2.21) together with the diver-gence theorem to deduce Z B p ( s ) B ( ∇ u, ∇ u ) e − u u dV g = m + n − m ( n − h Z ∂B p ( s ) D kij ∇ i u ∇ j ue − u u ν k dσ − n − m Z B p ( s ) u | D | e − u dV g i , (2.22)where ν denotes the outward unit normal to ∂B p ( s ) . Moreover, since g has positive Riccicurvature, then | R ij | ≤ R. This jointly with (2.20) yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∂B p ( s ) uD kij ∇ i u ∇ j ue − u u ν k dσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z ∂B p ( s ) |∇ u | ( | R ij | + R ) u e − u dσ ≤ C (cid:18)r µm − (cid:19) Z ∂B p ( s ) u Re − u dσ ≤ C (cid:18)r µm − (cid:19) mµ Z ∂B p ( s ) e − u dσ, (2.23)where we also have used (2.11) and (2.12). Moreover, we already know from (2.13) that − u ( x ) ≤ − c r ( x ) + c , where c and c are positive constants and r is the distance function. Thus, by (2.23) onehas(2.24) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∂B p ( s ) uD kij ∇ i u ∇ j ue − u u ν k dσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C e − s Area ( ∂B p ( s )) . The assumption of positive Ricci curvature allows to use the Bishop-Gromov theorem toinfer
Area ( ∂B p ( s )) ≤ C s n − . INSTEIN WARPED PRODUCTS 9
Hence, it follows from (2.24) that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∂B p ( s ) uD kij ∇ i u ∇ j ue − u u ν k dσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C e − s s n − . Therefore, by letting s → + ∞ in Eq. (2.22) we achieve Z M B ( ∇ u, ∇ u ) e − u u dV g = − m + n − m Z M u | D | e − u dV g . Finally, since M n is Bach-flat and u > D ijk = 0 . This finishes the proof ofthe lemma. (cid:3) Proof of the Main Results
Proof of Theorem 1.
Proof.
We follow the trend of Chen and He [14] (see also Cao and Chen [8]). To begin with,since M n is Bach-flat it follows from Lemma 4 that D ijk = 0 . Therefore, we may use (2.7)to get(3.1) C ijk = − W ijkl ∇ l f. At the same time, we already know that such a metric is real analytic (cf. Proposition2.4 in [16]). Therefore, taking into account (3.1) as well as (2.3), it suffices to show thatthe Cotton tensor C ijk vanishes at points p ∈ M n such that ∇ f ( p ) = 0 . So, we consider aregular point p ∈ M n , with associated level set Σ . Moreover, choose any local coordinates( θ , . . . , θ n ) on Σ and split the metric in the local coordinates ( f, θ , . . . , θ n ) as follows g = 1 |∇ f | df + g ab ( f, θ ) dθ a dθ b . Letting ∂ f = ∂ = ∇ f |∇ f | we immediately get ∇ f = 1 and ∇ a f = 0 , for a ≥ . From (3.1)and the symmetries of the Weyl tensor we have C ij = 0 . Next, by Proposition 1, we have R a = 0 and R abc = 0 for any integers 2 ≤ a, b, c ≤ n. Whence, it is easy to check that W abc = R abc = 0and use once more (3.1) to deduce C abc = − W abc |∇ f | = 0 . We now claim that C ab = 0 for all a, b ≥ . To prove our claim we apply the samearguments used in [14] (p. 324). Indeed, notice that C ab = 1 |∇ f | W ( ∇ f, ∂ a , ∇ f, ∂ b ) . On the other hand, from (2.1) we infer1 |∇ f | W ( ∇ f, ∂ a , ∇ f, ∂ b ) = 1 |∇ f | R ( ∇ f, ∂ a , ∇ f, ∂ b ) + R ( n − n − g ab − n − (cid:18) |∇ f | Ric ( ∇ f, ∇ f ) g ab + R ab (cid:19) . (3.2) Easily one verifies that h ab = Γ ab |∇ f | . Moreover, we also have Γ ab = − ∇ f ( g ab ) . Hence, itfollows that(3.3) h ab = − ∇ f |∇ f | ( g ab ) . Proceeding, we invoke Proposition 1 to deduce that |∇ f | is constant on Σ , which immediatelygives [ ∂ a , ∇ f ] = 0 , and then (cid:10) ∇ f |∇ f | , ∂ a (cid:11) = 0 , which implies ∇ ∇ f |∇ f | ∇ f |∇ f | = 0 . By these settingswe get(3.4) 1 |∇ f | R ( ∇ f, ∂ a , ∇ f, ∂ b ) = ∇ f ( n − |∇ f | Hg ab − H ( n − g ab . In particular, by tracing (3.4) with respect to a and b we obtain1 |∇ f | Ric ( ∇ f, ∇ f ) = ∇ f |∇ f | H − H ( n − . This substituted into (3.4) yields(3.5) R ( ∇ f, ∂ a , ∇ f, ∂ b ) = Ric ( ∇ f, ∇ f )( n − g ab . By using again Proposition 1 (3) we may consider |∇ f | Ric ( ∇ f, ∇ f ) = η and Ric ( ∂ a , ∂ b ) = κg ab , for a, b ≥ , where η and κ are the eigenvalues of the Ricci curvature. Therefore,substituting (3.5) into (3.2) we achieve C ab = 0 , which settles our claim.Finally, it is not difficult to see that C ijk = 0 whenever ∇ f ( p ) = 0 . Besides, we alreadyknow that g is analytic, which allows us to conclude that C ijk = 0 on M n . So, the proof iscompleted. (cid:3)
Proof of Theorem 2.
Proof.
First of all, we invoke Theorem 1 to conclude that C ≡ W ijkl ∇ l f = 0 . Moreover, we consider a point p ∈ M such that ∇ f ( p ) = 0 . Choosing an orthonormalframe { e , e , e , e } with e = ∇ f |∇ f | at the point p, we have W ijk = 0 for all 1 ≤ i, j, k ≤ W ijkl = 0 whenever ∇ f ( p ) = 0 . Then, since g is analytic, M is locally conformally flat. This is what we wantedto prove. (cid:3) Acknowledgement.
The authors would like to thank A. Barros, G. Catino and R. Batistafor fruitful conversations about this subject. Moreover, the authors want to thank the refereefor his careful reading and valuable suggestions.
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Universidade Federal de Alagoas - UFAL, Instituto de Matem´atica, CEP 57072-970-Macei´o / AL, Brazil
E-mail address : [email protected] (E. Ribeiro) Universidade Federal do Cear´a - UFC, Departamento de Matem´atica, Campus doPici, Av. Humberto Monte, Bloco 914, CEP 60455-760-Fortaleza / CE, Brazil
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