m-quasi Einstein manifolds with convex potential
Absos Ali Shaikh, Prosenjit Mandal, Chandan Kumar Mondal, Akram Ali
aa r X i v : . [ m a t h . DG ] F e b m -QUASI EINSTEIN MANIFOLDS WITH CONVEX POTENTIAL ABSOS ALI SHAIKH ∗ , PROSENJIT MANDAL , CHANDAN KUMAR MONDAL † , AKRAM ALI Abstract.
The main objective of this paper is to investigate the m -quasi Einstein manifoldwhen the potential function becomes convex. In this article, it is proved that an m -quasiEinstein manifold satisfying some integral conditions with vanishing Ricci curvature along thedirection of potential vector field has constant scalar curvature and hence the manifold turnsout to be an Einstein manifold. It is also shown that in an m -quasi Einstein manifold thepotential function agrees with Hodge-de Rham potential up to a constant. Finally, it is provedthat if a complete non-compact and non-expanding m -quasi Einstein manifold has boundedscalar curvature and the potential vector field has global finite norm, then the scalar curvaturevanishes. Introduction and preliminaries
A Riemannian manifold (
M, g ) of dimension n ( >
2) is called an m -quasi Einstein manifold(see e.g. [3, 4, 17]), if its Ricci tensor Ric satisfies the following relation:(1)
Ric + 12 £ X g − m X ♭ ⊗ X ♭ = λg, where X is a smooth vector field on M , X ♭ is the dual 1-form of X with respect to themetric g and m, λ are scalar such that 0 < m ≤ ∞ . If we omit the lie derivative term in(1), then we get the notion of quasi Einstein manifold (see e.g. [8, 21, 20] and also referencestherein) which appeared in the literature while considering the investigation of exact solutionof Einstein field equation and also during the consideration of quasiumbilical hypersurfaces,studied by Cartan. In [9] it is shown that 3-dimensional Cartan hypersurfaces are quasi-Einstein manifolds. Throughout the paper we will consider the m -quasi Einstein manifolds([4]), which is a generalization of Ricci soliton in case of m -Bakry-Emery tensor. An m -quasiEinstein manifold is said to be expanding (resp., steady or shrinking), if λ < λ = 0 or Mathematics Subject Classification: 53C20; 53C21; 53C44.Key words and phrases: Quasi Einstein Manifold; scalar curvature; convex function; Einstein Manifold. λ > X is the gradient of a smooth function f , then (1) reduces to the following form:(2) Ric + ∇ f − m df ⊗ df = λg. The function f is called potential function of the m -quasi Einstein manifold. The tensor of theleft hand side of (2) is called the m -Bakry-Emery tensor. Further, if m = ∞ , then (2) reducesto the equation of gradient Ricci soliton (see e.g. [7, 11, 15, 16, 19]) with the potential function f . If m and λ are smooth functions on M , then (2) is called the generalized m -quasi Einsteinmanifold [5]. Moreover, taking the trace of (2), we deduce(3) ∆ f = λn − R + 1 m |∇ f | , where R denotes the scalar curvature of the manifold. Case et al. [4] proved that a compact m -quasi Einstein metric with constant scalar curvature is trivial. They also showed that all2-dimensional m -quasi Einstein metrics on compact manifolds are trivial. In 2012, Barros andRibeiro [3] proved that in a complete non-compact m -quasi Einstein manifold if λn ≥ R and |∇ f | ∈ L ( M ), then the manifold becomes Einstein. In 2015, Hu et al. [12] classified m -quasiEinstein manifolds with parallel Ricci tensor, and also in [13], they classified m -quasi Einsteinmanifolds with all eigenvalues of the Ricci tensor as constant. Kim and Shin [14] also classified3-dimensional m -quasi Einstein manifolds. Barros and Gomes [2] studied compact m -quasiEinstein manifolds and also showed that if such a manifold is Einstein, then its potential vectorfield vanishes. We note that the investigation of various structures of Riemannian manifoldsthrough the convexity revel several geometrical and topological properties of such manifolds.Hence by motivating the above studies as well as the study of [1], in the present paper, weprove the following: Theorem 1.1.
Suppose ( M, g ) is an m -quasi Einstein manifold which is complete and non-compact and endowed with a positive convex potential function f and Ric ≥ − ( n − K forsome positive constant K . If the Ricci curvature of the manifold vanishes along ∇ f and f satisfies (4) Z M − B ( p,r ) fd ( x, p ) < ∞ , -QUASI EINSTEIN MANIFOLDS WITH CONVEX POTENTIAL 3 where B ( p, r ) is an open ball with center p and radius r , and d ( x, p ) is the shortest distancebetween x and p in M , then the following holds:(i) The manifold is Einstein,(ii) The scalar curvature of the manifold is a non-positive constant,(iii) The Ricci curvature of the manifold is non-positive everywhere in M . Corollary 1.1.1.
Let ( M, g ) be a complete non-compact m -quasi Einstein manifold with posi-tive convex potential function f . If the manifold is radially Ricci flat and satisfies the condition(4), then the manifold is Einstein with non-positive Ricci curvature. Theorem 1.2. If ( M, g ) is a compact oriented m -quasi Einstein manifold such that f is apotential function on M , then f agrees, upto a constant, with the Hodge-de Rham potential. Theorem 1.3.
Let ( M, g ) be a complete non-compact and non-expanding m -quasi Einsteinmanifold with finite volume. If the potential vector field X is of finite global norm, then thereexists an open ball where the scalar curvature R ≥ λn . Corollary 1.3.1.
Let ( M, g ) be a complete non-compact and non-expanding m -quasi Einsteinmanifold with finite volume. If the potential vector field X is of finite global norm and the scalarcurvature is bounded by λn , then the scalar curvature vanishes in M . Proof of the results
To prove Theorem 1.1 we need the following result of Hu et al. [12], which we state at first.
Theorem 2.1. [12]
In an m -quasi Einstein manifold ( M, g ) with constant scalar curvature, λ ≤ . Proof of Theorem 1.1 . Since f ∈ C ∞ ( M ), ring of smooth functions on M , is a non-constantconvex function on M , it follows that [23], M is non-compact. Let us consider the cut-offfunction, studied in [6], ϕ r ∈ C ( B ( p, r )) for r >
0, where C ( B ( p, r )) is a class of sec-ond order continuously differentiable functions with compact support and C being a constant, A. A. SHAIKH, P. MANDAL, C. K. MONDAL, A. ALI ( B ( p, r )) ⊆ M , p ∈ M such that ϕ r = 1 in B ( p, r )0 ≤ ϕ r ≤ B ( p, r )∆ ϕ r ≤ Cr in B ( p, r ) . Then for r → ∞ , we have ∆ ϕ r → ϕ r ≤ Cr . The convexity of f implies that f is alsosubharmonic [10], i.e., ∆ f ≥
0. In view of integration by parts, we obtain(5) Z M ∆ f ϕ r = Z M f ∆ ϕ r . Since ϕ r ≡ B ( p, r ), using (5), we get Z B ( p,r ) ∆ f = 0 . Again, in view of integration by parts and also by our assumption, we obtain0 ≤ Z B ( p, r ) ϕ r ∆ f = Z B ( p, r ) − B ( p,r ) f ∆ ϕ r ≤ Z B ( p, r ) − B ( p,r ) f Cr . But the right hand side tends to zero as r → ∞ . Hence we get Z M ∆ f = 0 . Thus, the subharmonocity of f implies that ∆ f = 0 in M . Therefore, (3) entails that(6) R − λn = 1 m |∇ f | , which implies that R ≥ λn . Taking u = log f and then simplifying we obtain − ∆ u = |∇ u | = |∇ f | f . -QUASI EINSTEIN MANIFOLDS WITH CONVEX POTENTIAL 5 Let l ( x ) be the distance of x ∈ M from the fixed point p . For any r >
0, consider the function η : [0 , ∞ ) → [0 ,
1] satisfying the following properties: η ( t ) = 1 for t ≤ rη ( t ) = 0 for t ≥ rη ′ ≤ η ′ ) ≤ ( η ′ ) η ≤ Cr | η ′′ | ≤ Cr , for some constant C < ∞ . Now define the function η on M by η ( x ) = η ( l ( x )) for x ∈ M . Thenthe function u satisfies the following inequality: |∇∇ u | ≥ n (∆ u ) = 1 n |∇ u | . By virtue of above inequality and the Bochner formula, we obtain12 ∆( η |∇ u | ) = 12 ∆ η |∇ u | + ∇ η ∇|∇ u | + 12 η ∆ |∇ u | = 12 ∆ η |∇ u | + ∇ η ∇|∇ u | + η |∇∇ u | + η ∇ u ∇ (∆ u ) + ηRic ( ∇ u, ∇ u )= 12 ∆ η |∇ u | + η ∇ η ∇|∇ u | + η |∇∇ u | − η ∇ u ∇|∇ u | + ηRic ( ∇ u, ∇ u ) ≥ (cid:16) η ∆ η − η |∇ η | + 1 η ∇ η ∇ u (cid:17) η |∇ u | + (cid:16) η ∇ η − ∇ u (cid:17) ∇ ( η |∇ u | )+ 1 n η |∇ u | + ηRic ( ∇ u, ∇ u ) . Again, calculation shows that η satisfies the following inequality:(7) |∇ η ∇ u | ≤ η |∇ u | n + n η |∇ η | . Since |∇ u | is non-zero, there is a point at which η |∇ u | is maximum where η is smooth, andhence ∆( η |∇ u | ) ≤ ∇ ( η |∇ u | ) = 0 such that0 ≥
12 ∆ η − η |∇ η | + ∇ η ∇ u + 1 n η |∇ u | + η Ric ( ∇ u, ∇ u ) . A. A. SHAIKH, P. MANDAL, C. K. MONDAL, A. ALI
By using (7), we have − ∆ η + nη |∇ η | + 2 η |∇ η | = − ∆ η + nη |∇ η | + 2 η |∇ η | + 2 ∇ η ∇ u − ∇ η ∇ u (8) ≥ nη |∇ η | + 2 ∇ η ∇ u + 2 n η |∇ u | + 2 η Ric ( ∇ u, ∇ u )(9) ≥ n η |∇ u | + 2 η Ric ( ∇ u, ∇ u ) . (10)Therefore, we get − η ′ ∆ d − η ′′ + n + 2 η ( η ′ ) ≥ n η |∇ u | + 2 η Ric ( ∇ u, ∇ u ) . Since
Ric ≥ − ( n − K , the Laplace comparison theorem implies that∆ d ≤ n − d (1 + √ Kd ) ≤ n − r + ( n − √ K, where η ′ = 0. Hence, we obtain1 n η |∇ u | + 2 η Ric ( ∇ u, ∇ u ) ≤ C ′ ( n ) r + C ′′ ( n ) r √ K, where C ′ ( n ) and C ′′ ( n ) are positive constants depend only on n . Consequently, we get1 n η |∇ u | ≤ C ′ ( n ) r + C ′′ ( n ) r √ K − η Ric ( ∇ u, ∇ u ) . Taking limit as r → ∞ in both sides, yields for all x ∈ M , |∇ u | x ≤ − nRic x ( ∇ u, ∇ u )= − nf Ric x ( ∇ f, ∇ f ) . (11)But Ricci curvature vanishes along ∇ f , i.e., Ric ( ∇ f, ∇ f ) = 0, hence, we have |∇ u | = 0, whichshows that f is constant. Therefore, (2) implies that the manifold becomes Einstein and R is equal to λn which is a constant. Again, Theorem 2.1 implies that in an m -quasi Einsteinmanifold λ ≤
0. Hence, R = λn ≤ (cid:3) Proof of Theorem 1.2 . Let (
M, g ) is a Riemannian manifold which is compact and oriented.If X is a vector field on M , the by virtue of Hodge-de Rham decomposition theorem, (see e.g.[22]), X can be written as(12) X = W + ∇ ξ, -QUASI EINSTEIN MANIFOLDS WITH CONVEX POTENTIAL 7 where div W = 0 and ξ is a smooth function called the Hodge-de Rham potential. We consideran m -quasi Einstein manifold ( M, g ) such that equation (1) yields(13) R + divX − m | X | = λn. Hence (12) entails that divX = ∆ ξ and consequently (13) takes the form(14) R + ∆ ξ − m | X | = λn. Again from (2) we have R + ∆ f − m |∇ f | = λn. From (14), it follows that(15) ∆( f − ξ ) = 0 , which implies that f = ξ + C , for some constant C , this proves the result. (cid:3) Let (
M, g ) be an oriented Riemannian manifold and Λ k ( M ) be the set of all k -th differentialforms in M . Now for any integer k ≥
0, the global inner product in Λ k ( M ) is defined by h η, ω i = Z M η ∧ ∗ ω, for η, ω ∈ Λ k ( M ). Here ‘ ∗ ′ is the the Hodge star operator. Hence, we define the global norm of η ∈ Λ k ( M ) by k η k = h η, η i and remark that k η k ≤ ∞ . Proof of Theorem 1.3 . For any r > r Z B ( p, r ) | X | dV ≤ (cid:16) Z B ( p, r ) h X, X i dV (cid:17) / (cid:16) Z B ( p, r ) (cid:16) r (cid:17) dV (cid:17) / ≤ k X k B ( p, r ) r (cid:16) V ol ( M ) (cid:17) / , where V ol ( M ) denotes the volume of M . Thus we obtainlim inf r →∞ r Z B ( p, r ) | X | dV = 0 . A. A. SHAIKH, P. MANDAL, C. K. MONDAL, A. ALI
Again, there exists a Lipschitz continuous function ω r such that for some constant K >
0, (see[24]), | dω r | ≤ Kr almost everywhere on M ≤ ω r ( x ) ≤ ∀ x ∈ Mω r ( x ) = 1 ∀ x ∈ B ( p, r )supp ω r ⊂ B ( p, r ) . Then taking limit, we get lim r →∞ ω r = 1. Therefore, by using the function ω r , we have (cid:12)(cid:12)(cid:12) Z B ( p, r ) ω r divXdV (cid:12)(cid:12)(cid:12) ≤ Cr Z B ( p, r ) | X | dV. In view of the m -quasi Einstein manifold equation (1), we get Z M { λn − R + 1 m | X | } dV = 0 , which yields(16) Z M ( λn − R ) dV ≤ . Since R is continuous, there is an open ball where R ≥ λn . (cid:3) Proof of Corollary 1.3.1 . Our assumption and (16) together imply that R = λn in M .Again, using Theorem 2.1, we get λ ≤
0. But the manifold is non-expanding m -quasi Einstein.Therefore, R = λn = 0 in M . (cid:3) Acknowledgment
The authors extend their appreciation to the deanship of scientific research at King KhalidUniversity for funding this work through research groups program under grant number R.G.P.1/50/42.
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