DSpace at VNU: Eigenfunctions of the weighted Laplacian and a vanishing theorem on gradient steady Ricci soliton tài liệ...
J Math Anal Appl 416 (2014) 553–562 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Eigenfunctions of the weighted Laplacian and a vanishing theorem on gradient steady Ricci soliton Nguyen Thac Dung a,∗ , Nguyen Thi Le Hai b , Nguyen Thi Thanh c a Department of Mathematics, Mechanics and Informatics (MIM), Hanoi University of Sciences (HUS-VNU), No 334, Nguyen Trai Road, Thanh Xuan, Hanoi, Viet Nam b Department of Informational Technology, Hanoi University of Civil Engineering, No 55, Giai Phong Road, Hai Ba Trung District, Hanoi, Viet Nam c Tran Phu High School for the Gifted, No 12, Tran Phu Street, Ngo Quyen District, Hai Phong City, Viet Nam a r t i c l e i n f o Article history: Received 23 September 2013 Available online March 2014 Submitted by H.R Parks Keywords: Bakry–Émery curvature Eigenvalues Eigenfunctions Gradient steady Ricci soliton Smooth metric measure spaces a b s t r a c t The aim of this note has two folds First, we show a gradient estimate of the higher eigenfunctions of the weighted Laplacian on smooth metric measure spaces In the second part, we consider a gradient steady Ricci soliton and prove that there exists a positive constant c(n) depending only on the dimension n of the soliton such that there is no nontrivial harmonic 1-form (hence harmonic function) which is in Lp on such a soliton for any < p < c(n) © 2014 Elsevier Inc All rights reserved Introduction A smooth metric measure space (M, g, e−f dv) is a Riemannian manifold (M, g) together with a weighted volume form e−f dv, where f is a smooth function on M and dv is the volume element induced by the Riemannian metric g The associated weighted Laplacian Δf is given by Δf u = Δu − ∇f, ∇u It is easy to see that Δf is a self-adjoint operator on the space L2 (M, e−f dv) of square integrable functions on M with respect to the measure e−f dv A function u is said to be f -harmonic if Δf u = Moreover, let ω be a 1-form on M then ω is said to be a f -harmonic 1-form if ω is a closed form and δf ω := δω + ω, df = * Corresponding author E-mail addresses: dungmath@yahoo.co.uk (N.T Dung), lehaidhxd@yahoo.com.vn (N.T Le Hai), thanhchuyentp@gmail.com (N.T Thanh) http://dx.doi.org/10.1016/j.jmaa.2014.02.054 0022-247X/© 2014 Elsevier Inc All rights reserved 554 N.T Dung et al / J Math Anal Appl 416 (2014) 553–562 where δ is the operator which is adjoint to d (see [3] or [2]) It is easy to see that if u is a f -harmonic function on M then du is a f -harmonic 1-form On a smooth metric measure space, we can introduce the Bakry–Émery curvature Ric f associated as follows Ric f = Ric + Hess(f ), where Ric denotes the Ricci curvature and Hess(f ) denotes the Hessian of f In the first part of this note, we obtain the following theorem Theorem 1.1 Let (M n , g, e−f ) be a compact smooth metric measure space with Ric f Denote the diameter, the weighted volume of M by d, Vf , respectively Assume that |∇f | a for some constant a > 0, then there is a constant c(a, d, Vf , n) such that for all k 1, |∇φk | n+2 cλk , λk |φk | n cλk4 c−1 k n Here φk is an eigenfunction of the f -Laplacian with respect to the eigenvalue λk and ||φk ||2f := M φ2k e−f = This result is very much motivated by and follows the ideas in the paper of Wang and Zhou [7] where they showed the lower bound for the higher eigenvalues of the Hodge Laplacian on a Riemannian manifold with Ricci curvature bounded from below Our argument is close to the argument used in [7] which was earlier developed by Li in his beautiful paper [4] In the second part of this note, we will study gradient steady Ricci solitons that are special cases of smooth metric measure spaces Recall that (M, g) is called a gradient Ricci soliton if there is a smooth function f : M → R and a constant λ ∈ R so that Ric + Hess f = λg The function f is called a potential function for g The soliton is referred as shrinking, steady, expanding if λ > 0, λ = 0, λ < respectively Ricci solitons are self-similar solutions of the Ricci flow, and play an important role in the study of singularity formation They are also a generalization of Einstein manifolds In this part we want to prove a vanishing theorem on such a soliton It is well-known that vanishing type theorems are important results in geometric analysis Recently, there are several interesting vanishing type theorems on smooth metric measure spaces or gradient Ricci solitons For example, in [6], Munteanu and Wang considered a smooth metric measure space with Ric f If the potential function f is of sublinear growth then any positive f -harmonic function on M must be constant They also shown that there does not exist a nontrivial f -harmonic function on M provided Ric f and the boundedness of f Later, in [2], the first author and Sung gave a vanishing type theorem on a complete noncompact smooth metric measure space with the same assumption In detail, we pointed out that there is no nontrivial f -harmonic function with finite Lp -norm on such a space for any p > if Ric f and f is bounded In [5], Munteanu and Sesum proved that if (M, g) is a gradient shrinking Kähler–Ricci soliton, and u is a harmonic function with finite energy on M then u has to be a constant function It turns out that there is only at most one nonparabolic end on a gradient shrinking Kähler–Ricci soliton Moreover, they also proved an other vanishing theorem on a gradient steady Ricci soliton According to their result, there is no nontrivial harmonic function with finite energy on such a soliton Motivated by the above results, in this paper, we will prove the following theorem N.T Dung et al / J Math Anal Appl 416 (2014) 553–562 555 Theorem 1.2 Let (M, g) be a gradient steady Ricci soliton of dimension n Suppose that the soliton has at most Euclidean volume growth, namely, for any x ∈ M , there exists r0 > such that for r r0 , Crn Vol B(x, r) Then H Lp (M ) := |ω|p < ∞ ω: ω is a harmonic 1-form, =0 M for any < p < 4n 2n−1 This note is organized as follows In Section 2, we prove the gradient estimate of the higher eigenfunctions of the f -Laplacian on smooth metric measure spaces Then, in Section 3, we prove the vanishing type theorem on gradient steady Ricci solitons Eigenfunctions of f -Laplacian Let (M n , g, e−f dv) be a compact oriented smooth metric measure space without boundary Suppose that Ric f and there exists a constant a > such that |∇f | a Let d be the diameter of M and Vf be the weighted volume with respect to weighted measure e−f dv First, we consider the eigenfunctions of the weighted Laplacian Let us denote the eigenvalues of the f -Laplacian by = λ0 < λ1 λ2 · · · λk · · · with the corresponding eigenfunctions φi , i = 0, 1, 2, , satisfying φi φj e−f = δij Δf φi = −λi φi , M For a given constant c, consider the function Q(x) = |∇φ|2 + cφ2 k i=1 bi φi where φ = with bi ∈ R and k i=1 bi = Let ψ(b1 , , bk ) := max Q(x) x∈M Assume that ψ(a1 , , ak ) = Lemma 2.1 Let u = k i=1 max (b1 , ,bk )∈Rk b21 +···+b2k =1 φi then |∇u|2 + Au2 where A = ψ(b1 , , bk ) √ 2λk +a2 +a 4λk +a2 A max u2 , M N.T Dung et al / J Math Anal Appl 416 (2014) 553–562 556 Proof We follow the arguments in [7] Define k F (b1 , , bk , x, λ) = Q(x) − λ b2i − i=1 Then, subject to the constrain We now show k i=1 bi = 1, F achieves its maximum value at some point (a1 , , ak , x0 , α) |∇u|2 (x0 ) + cu2 (x0 ) √ c max u2 , M for c > 2λk +a +a2 4λk +a By Wang and Zhou’s arguments used to prove Lemma 2.1 in [7], we have α = Q(u, x0 ) = |∇u|2 (x0 ) + cu2 (x0 ) and k aj ∇φi , ∇φj + caj φi φj = αai j=1 Suppose now that |∇u|2 (x0 ) + cu2 (x0 ) > c max u2 M Then ∇u(x0 ) = and one can choose an orthonormal frame {e1 , , en } at x0 so that ∇u(x0 ) = u1 (x0 )e1 We know that (see [7]) |∇∇u|2 u211 = c2 u2 (2.1) On the other hand, at the maximum point (a1 , , ak , x0 , α), ΔF (a1 , , ak , x0 , α) or equivalently, Δ|∇u|2 + cΔu2 (2.2) Note that Δf (·) = Δ − ∇f, · By the Bochner formula, we have Δf |∇u|2 = 2|∇∇u|2 + ∇Δf u, ∇u + 2Ric f (∇u, ∇u) (2.3) N.T Dung et al / J Math Anal Appl 416 (2014) 553–562 557 From (2.2) and (2.3), we obtain |∇∇u|2 + ∇Δf u, ∇u + Ric f (∇u, ∇u) + c ∇f, ∇|∇u|2 + cuΔf u + c|∇u|2 + ∇f, ∇u2 2 (2.4) By Schwarz’s inequality, 2|∇u||∇f | ∇|∇u| ∇f, ∇|∇u|2 ∇f, u 2|u||∇u||∇f | 2a|∇u||∇∇u| 2a|u||∇u|, where we used the Kato inequality (|∇|∇u|| |∇∇u|) and the boundedness of |∇f | By the lower bound of Ric f , (2.1) and (2.4), we conclude that |∇∇u|2 + ∇Δf u, ∇u + cuΔf u + c|∇u|2 − a|∇u||∇∇u| − ca|u||∇u| x2 4ε Using the elementary inequality xy + εy for any ε > 0, the above inequality implies (1 − aβ)|∇∇u|2 + ∇Δf u, ∇u + cuΔf u + c|∇u|2 − for any β, γ > Since Δf u = − i=1 a ca |∇u|2 − caγu2 − |∇u|2 4β 4γ 0, λi φi , we can compute k k ∇Δf u, ∇u + cuΔf u = − λi aj ∇φi , ∇φj − c i,j=1 k =− k k aj ∇φi , ∇φj + caj φi φj = −α λi i=1 λi aj φi φj i,j=1 j=1 λi a2i i=1 Hence, in the view of the inequality (2.1), if β is small, we have k (1 − aβ)c2 u2 − α λi a2i − caγu2 + c − i=1 ca a − |∇u|2 4β 4γ This implies that c2 − ac2 β − acγ u2 − λk α + c − ca a − |∇u|2 4β 4γ c2 − ac2 β − acγ u2 − λk |∇u|2 + cu2 + c − = c2 − ac2 β − acγ − cλk u2 + c − ca a − |∇u|2 4β 4γ ca a − − λk |∇u|2 4β 4γ Now, we choose γ > such that c2 − ac2 β − acγ − cλk = 0, namely γ= c − λk − acβ , a 0 then there exists a constant c(a, d, Vf , n) such that |∇φ|2 n+2 n φ2 cλk2 |φk | cλk4 cλk , In particular, |∇φk | For all k n+2 cλk , n 1, λk c−1 k n To prove this theorem, first we show a volume comparison theorem Lemma 2.3 Let (M, g, e−f dv) be a compact smooth metric measure space with Ric f and |∇f | d be the diameter of M Then along any minimizing geodesic starting from x ∈ Bp (R) we have Jf (x, r2 , ξ) Jf (x, r1 , ξ) e2ad r2 r1 a Let n−1 for any < r1 < r2 < R In particular, for any < r1 < r2 Vf (Bx (r2 )) Vf (Bx (r1 )) e2ad r2 r1 n Here Jf = e−f J(x, r, ξ) is the f -volume in geodesic polar coordinates Proof Let y ∈ Bp (R) Let γ be the minimizing geodesic from x to y such that γ(0) = x and γ(r) = y Let f (t) := f (γ(t)), by the formula (2.5) in the proof of Lemma 2.1 in [6] (also see [9]), we have Jf (x, r2 , ξ) Jf (x, r1 , ξ) r2 r1 n−1 exp r1 r1 r2 f (t) dt − r2 f (t) dt Now, since |∇f | is bounded, we have Jf (x, r2 , ξ) Jf (x, r1 , ξ) r2 r1 n−1 exp r1 r1 f (t) − f (0) dt − r2 r2 f (t) − f (0) dt N.T Dung et al / J Math Anal Appl 416 (2014) 553–562 r2 r1 n−1 r2 r1 n−1 exp r1 r1 r2 at dt + r2 at dt The proof is complete 559 exp(2ad) ✷ Proof of Theorem 2.2 The proof is similar to the proof of Theorem 2.2 in [7] with note that the Bishop volume comparison theorem in [7] is now replaced by the volume comparison in Lemma 2.3 ✷ Gradient steady Ricci solitons A gradient steady Ricci soliton is a special smooth metric measure space (M n , g, e−f ) satisfying Rij + fij = 0, where Rij are Ricci curvatures and f is a smooth function A gradient steady Ricci soliton satisfies the following |∇f |2 + R = a2 , for some constant a > Δf + R = R where R is the scalar curvature of M After scaling we can assume a = We write the potential function f in polar coordinate f (r, θ) = −r + ϕ(r, θ) where r(·) = d(x, ·) for some x ∈ M , θ ∈ S n−1 In [8], Wei and Wu gave an estimation of the Euclidean volume of the Ricci soliton Theorem 3.1 (See [8].) Let (M n , g, f ) be a complete gradient steady Ricci soliton satisfying the normalized condition a = Assume that there exist constants C1 , C2 such that r r ϕ(r, θ) − ϕ(t, θ) dt max θ∈S n−1 ϕ(r, θ) − ϕ(t, θ) dt + C1 r C1 θ∈S n−1 0 for sufficiently large r Then the soliton has at most Euclidean volume growth, namely, for any x ∈ M , there exists r0 > such that for r r0 , Vol B(x, r) Crn Note that Theorem 3.1 can be considered as an analogue of volume growth theorem for gradient shrinking Ricci soliton of [1] In this section, we only investigate the gradient steady Ricci soliton with at most Euclidean volume growth Our theorem is stated as follows Theorem 3.2 Let (M, g) be a gradient steady Ricci soliton of dimension n Suppose that the soliton has at most Euclidean volume growth Then H Lp (M ) = for any < p < 4n 2n−1 N.T Dung et al / J Math Anal Appl 416 (2014) 553–562 560 Proof Our argument is close to the argument in [5] Let ω = |ω|p < ∞ Since ω is harmonic, we have that M ∂aj ∂ai = , ∂xj ∂xi n i=1 n ∀i, j = 1, 2, , n, and i=1 dxi be any harmonic 1-form with ∂ai = ∂xi Let φ be a cut-off function on M such that φ = on Bp (r) (a geodesic ball centered at some fixed point p of radius r), φ = outside Bp (2r) and |∇φ| Cr Let q ∈ R such that 1 + =1 p q then 4n < q < 2n + Using the integration by parts, we obtain Ric(ω, ω)φ2 = − M fij aj φ2 M (ai )j fi aj φ2 + = M fi aj φ2 j (3.6) |ω|2 ∇f, ∇φ2 (3.7) M On the other hand, integrating by parts again, it follows that − (ai )j fi aj φ2 = M (Δf )|ω|2 φ2 + M M Combining (3.6) and (3.7), we have df, ω · ω, dφ2 + Ric(ω, ω)φ2 = M M R|ω|2 φ2 − M |ω|2 ∇f, ∇φ2 (3.8) M Now, the Bochner formula implies Δ |ω|2 = 2Ric(ω, ω) + 2|∇ω|2 2Ric(ω, ω) + ∇|ω| where we used first Kato inequality in the last inequality Multiplying this by φ2 , then using (3.8) and integration by parts, we infer ∇|ω| φ2 + M R|ω|2 φ2 − M ∇|ω|2 , ∇φ2 + M M ∇|ω| φ2 + M This implies that |ω|2 ∇f, ∇φ2 − M |ω|2 |∇φ|2 + M df, ω · ω, ∇φ2 |ω|2 ∇f, ∇φ2 − M df, ω · ω, ∇φ2 M N.T Dung et al / J Math Anal Appl 416 (2014) 553–562 ∇|ω| φ2 + M R|ω|2 φ2 M |ω|2 |∇φ|2 + M |ω|2 ∇f, ∇φ2 − M 561 df, ω · ω, ∇φ2 M |ω|2 |∇φ| Ca (3.9) M for some constant C > 0, where in the last inequality, we have used that |∇f | From (3.9) and Hölder inequality, we obtain q ∇|ω| φq M q q R |ω|q φq + q Crn( q −1) 2 M ∇|ω| φ2 + M Car a n( q2 −1) R|ω|2 φ2 M |ω|2 |∇φ| M Carn( q −1) rn(1− p ) p |∇φ| |ω|p p M Ca r 2n( q2 −1) |ω| p r p M where the constant C > might be different from line to line Let r → ∞ and using that ω ∈ Lp (M ), we conclude that ∇|ω| q q = R |ω|q = This infers that either ω = or; |ω| = C and R = If ω = we are done Assume that R = 0, we can use Theorem 1.11 in [5] about the infinite volume for steady solitons to conclude that |ω| = The proof is complete ✷ Corollary 3.3 Let (M n , g) be a gradient Ricci soliton whose satisfying the Euclidean volume growth condition 4n then there is no nontrivial harmonic function which is in Lp (M ) for any < p < 2n−1 Remark 3.4 When p = 2, there is a similar result proved by Munteanu and Sesum in [5] without the volume growth condition Acknowledgment A part of this paper was done during a visit of the first author to Vietnam Institute for Advanced Study in Mathematics (VIASM) He would like to express his deep thanks to staffs there for the excellent working conditions, and support References [1] [2] [3] [4] H.D Cao, D.T Zhou, On complete gradient shrinking Ricci solitons, J Differential Geom 85 (2010) 175–186 N.T Dung, C.J Sung, Weighted f -harmonic 1-form on smooth metric measure spaces, preprint J Jost, Riemannian Geometry and Geometric Analysis, fifth ed., Springer, 2008 P Li, On the Sobolev constant and the p-spectrum of a compact Riemannian manifold, Ann Sci Éc Norm Super 13 (1980) 451–468 [5] O Munteanu, N Sesum, On gradient Ricci solitons, J Geom Anal 23 (2013) 539–561, arXiv:0910.1105v1 [6] O Munteanu, J Wang, Smooth metric measure spaces with non-negative curvature, Comm Anal Geom 19 (3) (2011) 451–486 562 N.T Dung et al / J Math Anal Appl 416 (2014) 553–562 [7] J Wang, L Zhou, Gradient estimate for eigenforms of Hodge Laplacian, Math Res Lett 19 (2012) 575–588, arXiv: 1109.4968v3 [math.DG] [8] Q.F Wei, Peng Wu, On volume growth of gradient steady Ricci solitons, Pacific J Math 265 (2013) 233–241, arXiv: 1208.2040 [9] N Yang, A note on nonnegative Bakry–Émery Ricci curvature, Arch Math 93 (2009) 491–496 ... formation They are also a generalization of Einstein manifolds In this part we want to prove a vanishing theorem on such a soliton It is well-known that vanishing type theorems are important... estimation of the Euclidean volume of the Ricci soliton Theorem 3.1 (See [8].) Let (M n , g, f ) be a complete gradient steady Ricci soliton satisfying the normalized condition a = Assume that there... nonparabolic end on a gradient shrinking Kähler Ricci soliton Moreover, they also proved an other vanishing theorem on a gradient steady Ricci soliton According to their result, there is no nontrivial