Existence of finite global norm of potential vector field in a Ricci soliton
aa r X i v : . [ m a t h . G M ] F e b EXISTENCE OF FINITE GLOBAL NORM OF POTENTIAL VECTORFIELD IN A RICCI SOLITON
ABSOS ALI SHAIKH ∗ , CHANDAN KUMAR MONDAL AND PROSENJIT MANDAL Abstract.
In this article, we investigate global norm of potential vector field in Ricci soliton.In particular, we have deduced certain conditions so that the potential vector field has finiteglobal norm in expanding Ricci soliton. We have also showed that if the potential vector fieldhas finite global norm in complete non-compact Ricci soliton having finite volume, then thescalar curvature becomes constant. Introduction and preliminaries
A Riemannian manifold (
M, g ) is called the Ricci soliton [6] if its Ricci tensor
Ric satisfiesthe following equation:(1) 2
Ric + £ X g = 2 λg, where £ X g is the lie derivative of the metric tensor g with respect to the vector field X . Intensors of local coordinates system, (1) can also be expressed in the following form(2) 2 R ij + ∇ i X j + ∇ j X i = 2 λg ij , where R ij denotes the components of the Ricci tensor. The Ricci soliton is said to be expanding(resp., steady and shrinking) if λ < λ = 0 and λ > X is knownas potential vector field of the Ricci soliton and if X is the gradient of some smooth function f ∈ C ∞ ( M ), then f is called the potential function and in this case the Ricci soliton is calledthe gradient Ricci soliton. After normalization, gradient expanding Ricci soliton reduces to(3) R ij + ∇ i ∇ j f = − g ij . The study of curvature estimation in Ricci soliton is a well known research area in differentialgeometry. In 2009, Zhang [13] proved that the scalar curvature is non-negative in case of steady ∗ Corresponding author.
Mathematics Subject Classification: 53C20; 53C21; 53C44.Key words and phrases: Expanding Ricci soliton, scalar curvature, Global finite norm, Riemannian manifold. and shrinking gradient Ricci soliton and the scalar curvature is bounded below in case ofexpanding gradient Ricci soliton. Later, Chow et. al [5] improved this estimation for shrinkinggradient Ricci soliton. Petersen and Wylie [9] proved that a gradient expanding Ricci solitonwith constant scalar curvature R satisfies − n ≤ R ≤ R = − n , the metric becomesEinstein, and also Pigola-Rimoldi-Setti [10] and Zhang [14] independently showed that for sucha soliton the scalar curvature R ≥ − n . Again, Mondal and Shaikh [7] studied gradient Riccisoliton having non-negative Ricci curvature and realizing convex potential which fulfills finiteweighted Dirichlet condition and proved that in such a case scalar curvature vanishes. Recently,Chen [4] proved that if the Ricci curvature in a gradient expanding Ricci soliton satisfieslim l ( x ) →∞ l ( x ) | Ric | = 0 , where l ( x ) is the distance of x from a fixed point, then the scalar curvature is non-negative, andalso showed that if the Ricci curvature in a gradient expanding Ricci soliton is non-positive,then the scalar curvature is negative. In the present paper, we have deduced certain conditionsso that the potential vector field has finite global norm in expanding Ricci soliton. We havealso showed that if the potential vector field has finite global norm in complete non-compactRicci soliton having finite volume, then the scalar curvature becomes constant. Throughoutthis article, by the notation ( M, g ) or M we denote the Riemannian manifold M with theRiemannian metric g .For any k , 0 ≤ k ≤ n , let Λ k ( M ) be the vector space of all smooth k -forms in M . For any ω, η ∈ Λ k ( M ), the local inner product of ω and η is denoted by ( ω, η ) and is defined by, see [8,p. 149], ( ω, η ) = ω i , ··· ,i k η i , ··· ,i k , where ω = ω i , ··· ,i k dx i ∧ · · · ∧ dx i k , η = η i , ··· ,i k dx i ∧ · · · ∧ dx i k and η i , ··· ,i k = g i j · · · g i k j k η i , ··· ,i k .For a fixed k ≥
0, the Hodge star operator ∗ : Λ k ( M ) → Λ n − k ( M ) is defined by ∗ ω = sgn ( I, J ) ω i , ··· ,i k dx j ∧ · · · ∧ dx j n − k , for ω = ω i , ··· ,i k dx i ∧ · · · ∧ dx i k ∈ Λ k ( M ). Here j < · · · < j n − k is the rearrangement of thecomplements of i < · · · < i k in the set { , ..., n } in ascending order and sgn ( I, J ) is the sign of
XISTENCE OF FINITE GLOBAL NORM OF POTENTIAL VECTOR FIELD 3 the permutation i , ..., i k , j , ..., j n − k . For an oriented Riemannian manifold M , the global innerproduct in Λ k ( M ) is defined by h ω, η i = Z M ω ∧ ∗ η, for ω, η ∈ Λ k ( M ). We define the global norm of ω ∈ Λ k ( M ) by k ω k = h ω, ω i and remark that k ω k ≤ ∞ . There is a natural adjoint operator of the exterior derivative d : Λ k ( M ) → Λ k +1 ( M )called the co-differential operator δ : Λ k ( M ) → Λ k − ( M ) and is defined by δ = ( − k ∗ − d ∗ = ( − n ( k +1)+1 ∗ d ∗ , so that the following diagram commutesΛ k ( M ) ∗ −−−→ Λ n − k ( M ) y δ y d Λ k − ( M ) ( − k ∗ −−−−→ Λ n − k +1 ( M )For a 1-form ω ∈ Λ ( M ), we have( dω ) ij = ∇ i ω j − ∇ j ω i and( δω ) = −∇ i ω i , where ∇ i = g ij ∇ j . For a detailed discussion on Hodge operator and co-differential operatorsee [8]. By the notation Λ k ( M ), we denote the subspace of Λ k ( M ) containing all k -forms in M with compact support. Also the completion of Λ k ( M ) with respect to the global inner product h , i is denoted by L s ( M ).In this article, a vector field and its dual 1-form with respect to the Riemannian metric g will be denoted by the same letter, i.e., for a vector field X = X i ∂ i , the dual 1-form is X = X i dx i = g ij X j dx i . 2. Main results
Definition 2.1. [12] A vector field on a manofold M is said to have finite global norm if itsdual 1-form with respect to g belongs to L ( M ) ∩ Λ ( M ). A. A. SHAIKH, C. K. MONDAL AND P. MANDAL
Let o ∈ M be a fixed point and l ( x ) be the distance from o to x for each x ∈ M . Theopen ball with center o and radius r > B ( r ). Then there exists a Lipschitzcontinuous function ω r such that for some constant K >
0, (see [11]), | dω r | ≤ Kr almost everywhere on M ≤ ω r ( x ) ≤ ∀ x ∈ Mω r ( x ) = 1 ∀ x ∈ B ( r )supp ω r ⊂ B ( r )Then taking limit, we get lim r →∞ ω r = 1. Now we calculate the global norm of ω r dX and ω r δX when R ij ( ∇ i X j ) = − λR . Then we have g ( dX, dX ) = 12 { ( ∇ i X j − ∇ j X i )( ∇ i X j − ∇ j X i ) } = 12 { ∇ i X j )( ∇ i X j ) + 4 R ij ( ∇ i X j ) − λ ∇ i X i } = 4 g ( ∇ X, ∇ X ) + 2 R ij ( ∇ i X j ) − λ ( λn − R )= 4 g ( ∇ X, ∇ X ) − nλ , (4) g ( δX, δX ) = ( ∇ i X i )( ∇ j X j )= g ( λn − R, λn − R ) = | λn − R | . (5)The above two equations (4) and (5) together imply the following lemma: Lemma 2.1. If R ij ( ∇ i X j ) = − λR , then the potential vector field X of the Ricci soliton (2)satisfies the following: (6) k ω r dX k B (2 r ) = 4 k ω r ∇ X k B (2 r ) − n k ω r λ k B (2 r ) , (7) k ω r δX k B (2 r ) = k ω r ( λn − R ) k B (2 r ) . Combining lemma 2 and lemma 3 of [12], we have
XISTENCE OF FINITE GLOBAL NORM OF POTENTIAL VECTOR FIELD 5
Lemma 2.2. [12]
For any X ∈ Λ ( M ) , we have (8) 4 h ω r dω r ⊗ X, ∇ X i B (2 r ) + h ω r ∇ X, ω r X i B (2 r ) + 2 h ω r ∇ X, ω r ∇ X i B (2 r ) = 0 . h ω r ℜ X, ω r X i B (2 r ) = h ω r ∇ X, ω r X i B (2 r ) + h ω r dX, ω r dX i B (2 r ) +2 h ω r dX, dω r ∧ X i B (2 r ) + h ω r δX, ω r δX i B (2 r ) − h ω r δX, ∗ ( dω r ∧ ∗ X ) i B (2 r ) , (9) where ( ∇ X ) i = ∇ j ∇ j X i , ( ∇ X ) ij = ∇ i X j and ( ℜ X ) i = R ji X j is the Ricci transformation on Λ ( M ) . Lemma 2.3. [1]
For any X ∈ Λ k ( M ) , there exists a positive constant A independent of r suchthat k dω r ⊗ X k B (2 r ) ≤ Ar k X k B (2 r ) , k dω r ∧ X k B (2 r ) ≤ Ar k X k B (2 r ) and k dω r ∧ ∗ X k B (2 r ) ≤ Ar k X k B (2 r ) . From Lemma 2.1 and Lemma 2.3, we have the following: | h ω r dX, dω r ∧ X i B (2 r ) | ≤ k ω r dX k B (2 r ) k dω r ∧ X k B (2 r ) ≤ k ω r dX k B (2 r ) + 4 k dω r ∧ X k B (2 r ) ≤ k ω r ∇ X k B (2 r ) − n k ω r λ k B (2 r ) + 4 Ar k X k B (2 r ) , | h ω r δX, ∗ ( dω r ∧ ∗ X ) i B (2 r ) | ≤ k ω r δX k B (2 r ) k dω r ∧ ∗ X k B (2 r ) ≤ k ω r δX k B (2 r ) + 5 k dω r ∧ ∗ X k B (2 r ) ≤ k ω r ( λn − R ) k B (2 r ) + 5 Ar k X k B (2 r ) . (10) A. A. SHAIKH, C. K. MONDAL AND P. MANDAL
Thus using Lemma 2.2, we calculate for Ricci soliton (2): h ω r ℜ X, ω r X i B (2 r ) = − h ω r dω r ⊗ X, ∇ X i B (2 r ) − h ω r ∇ X, ω r ∇ X i B (2 r ) + h ω r dX, ω r dX i B (2 r ) + 2 h ω r dX, dω r ∧ X i B (2 r ) + h ω r δX, ω r δX i B (2 r ) − h ω r δX, ∗ ( dω r ∧ ∗ X ) i B (2 r ) ≥ − k ω r ∇ X k B (2 r ) − Ar k X k B (2 r ) − k ω r ∇ X k B (2 r ) + 4 k ω r ∇ X k B (2 r ) − n k ω r λ k B (2 r ) − k ω r ∇ X k B (2 r ) + n k ω r λ k B (2 r ) − Ar k X k B (2 r ) + k ω r ( λn − R ) k B (2 r ) − k ω r ( λn − R ) k B (2 r ) − Ar k X k B (2 r ) = 12 k ω r ∇ X k B (2 r ) − Ar k X k B (2 r ) − n k ω r λ k B (2 r ) + 45 k ω r ( λn − R ) k B (2 r ) . (11)If the Ricci curvature is non-positive then we can writelim sup r →∞ h ω r ℜ X, ω r X i B (2 r ) ≤ . Hence (11) reduces to the following inequality:5 k∇ X k − n k λ k + 8 k ( λn − R ) k ≤ , which yields 5 k∇ X k + Z M ( − nλ + 8 λ n − λnR + 8 R ) dv ≤ . Therefore, for n ≥
2, the above inequality gives(12) 5 k∇ X k + Z M ( − λnR + 8 R ) dv ≤ . Hence we get(13) 5 k∇ X k ≤ λn Z M R. Theorem 2.4.
Let ( M, g, X ) be an expanding Ricci soliton which is complete and non-compact.If the Ricci curvature is non-positive and the following conditions hold:(i) R ij ( ∇ i X j ) = R/ , and XISTENCE OF FINITE GLOBAL NORM OF POTENTIAL VECTOR FIELD 7 (ii) R M R ≥ − c for some positive constant c , then the potential vector field has finite globalnorm. Moreover, if the manifold has finite volume, then Z M R ≤ ncV ol ( M ) . Proof.
Since (
M, g, X ) is a complete non-compact expanding Ricci soliton, we can take λ = − .Now putting the value of λ in (13) and using the given conditions, we get our first result. Also,(12) implies that Z M R ≤ − n Z M R ≤ ncV ol ( M ) . (cid:3) Theorem 2.5.
Let ( M, g, X ) be a complete non-compact Ricci soliton with finite volume. If thepotential vector field X is of finite global norm, then the scalar curvature R must be constant.Proof. For any r >
0, we have1 r Z B (2 r ) | X | dV ≤ (cid:16) Z B (2 r ) h X, X i dV (cid:17) / (cid:16) Z B (2 r ) (cid:16) r (cid:17) dV (cid:17) / ≤ k X k B (2 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 (2 r ) | X | dV = 0 . Again we have (cid:12)(cid:12)(cid:12) Z B (2 r ) ω r divXdV (cid:12)(cid:12)(cid:12) ≤ Cr Z B (2 r ) | X | dV, for some constant C. In view of (1) the last relation yields Z M ( nλ − R ) dV = 0 , which gives R = nλ . This completes the proof. (cid:3) Acknowledgment
The third author gratefully acknowledges to the CSIR(File No.:09/025(0282)/2019-EMR-I),Govt. of India for financial assistance.
A. A. SHAIKH, C. K. MONDAL AND P. MANDAL
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