DSpace at VNU: Eigenfunctions of the weighted Laplacian and a vanishing theorem on gradient steady Ricci soliton

DSpace at VNU: Eigenfunctions of the weighted Laplacian and a vanishing theorem on gradient steady Ricci soliton tài liệ...

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www.elsevier.com/locate/jmaa

Eigenfunctions of the weighted Laplacian and a vanishing

theorem on gradient steady Ricci soliton

Nguyen Thac Dunga, ∗, Nguyen Thi Le Haib, Nguyen Thi Thanhc

a

Department of Mathematics, Mechanics and Informatics (MIM), Hanoi University of Sciences

(HUS-VNU), No 334, Nguyen Trai Road, Thanh Xuan, Hanoi, Viet Nam

bDepartment of Informational Technology, Hanoi University of Civil Engineering, No 55, Giai Phong

Road, Hai Ba Trung District, Hanoi, Viet Nam

cTran 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 a b s t r a c t

Article history:

Received 23 September 2013

Available online 3 March 2014

Submitted by H.R Parks

Keywords:

Bakry–Émery curvature

Eigenvalues

Eigenfunctions

Gradient steady Ricci soliton

Smooth metric measure spaces

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 L pon

such a soliton for any 2 < p < c(n).

© 2014 Elsevier Inc All rights reserved.

1 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 Δ f is given by

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 Δ f u = 0 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

* 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.

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  0 Denote the

k

n

f :=

M φ2

k e −f = 1.

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 < 0 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  0 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  0 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 L p -norm on such a space for any p > 0 if Ric f  0 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

Theorem 1.2 Let (M, g) be a gradient steady Ricci soliton of dimension n Suppose that the soliton has at

Vol

 Cr n Then

:=



ω: ω is a harmonic 1-form,



M



= 0

for any 2 < p < 2n 4n −1

This note is organized as follows In Section2, we prove the gradient estimate of the higher eigenfunctions

of the f -Laplacian on smooth metric measure spaces Then, in Section3, we prove the vanishing type theorem

on gradient steady Ricci solitons

2 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  0 and there exists a constant a > 0 such that |∇f|  a Let d be the diameter of M and V f 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

satisfying

Δf φ i=−λ i φ i ,



M

φ i φ j e −f = δ ij

For a given constant c, consider the function

where φ =k

i=1 b i φ i with b i ∈ R andk

i=1 b2

i = 1 Let

Assume that

(b1, ,b k)∈R k

b2 +···+b2

k=1

ψ(b1, , b k ).

Lemma 2.1 Let u =k

i=1 a i φ i then

|∇u|2+ Au2 A max

M u2,

4λ k +a2

.

Proof We follow the arguments in[7] Define

k

i=1

b2i − 1

.

Then, subject to the constraink

i=1 b2

i = 1, F achieves its maximum value at some point (a1, , a k , x0, α).

We now show

|∇u|2(x0) + cu2(x0) c max

M u2,

for c > 2λ k +a2+a √

4λ k +a2

By Wang and Zhou’s arguments used to prove Lemma 2.1 in[7], we have

and

k

j=1





= αa i

Suppose now that

|∇u|2(x0) + cu2(x0) > c max

M u2.

Then

and one can choose an orthonormal frame{e1, , e n } at x0 so that

∇u(x0) = u1(x0)e1.

We know that (see[7])

On the other hand, at the maximum point (a1, , a k , x0, α),

ΔF (a1, , a k , x0, α) 0

or equivalently,

Note that Δf(·) = Δ − ∇f, · By the Bochner formula, we have

From(2.2)and(2.3), we obtain

2

2

By Schwarz’s inequality,

∇ f, u2

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

Using the elementary inequality xy x2

4ε + εy2 for any ε > 0, the above inequality implies

4γ |∇u|2 0, for any β, γ > 0 Since Δ f u = −i=1 λ i a i φ i, we can compute

k

i,j=1

λ i a i a j ∇φ i , ∇φ j  − c

k

i,j=1

λ i a i a j φ i φ j

=−

k

i=1

λ i a i

k

j=1



=−α

k

i=1

λ i a2i

Hence, in the view of the inequality(2.1), if β is small, we have

k

i=1



4β − ca

4γ



This implies that

0c2− ac2



4β − ca

4γ





+



4β − ca

4γ



=





4β − ca

4γ − λ k



Now, we choose γ > 0 such that c2− ac2β − acγ − cλ k = 0, namely

a , 0 < β <

then the above inequality becomes

0



4

 1

ca



It is trivial to show that

0<β< c −λk ca

 1

ca



=c − λ k

2ac

and the minimal value is obtained at β = c−λ k

2ac Thus, we choose β = c−λ k

2ac then the inequality(2.5)reduces to

0





This is impossible if c > 2λ k +a2+a √

4λ k +a2

The proof is complete by letting c approach 2λ k +a2+a √

4λ k +a2

Theorem 2.2 If a > 0 then there exists a constant c(a, d, V f , n) such that

k , φ2 cλ n2

k

In particular,

k

2 For all k  1,

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  0 and |∇f|  a Let

J f (x, r2, ξ)

J f (x, r1, ξ)  e 2ad



r2

r1

n−1

V f (B x (r2))



r2

r1

n

.

Proof Let y ∈ B p (R) Let γ be the minimizing geodesic from x to y such that γ(0) = x and γ(r) = y Let

J f (x, r2, ξ)

J f (x, r1, ξ) 



r2

r1

n−1

exp

2

r1

r1



0

r2

r2



0

f (t) dt

.

Now, since |∇f| is bounded, we have

J f (x, r2, ξ)

J f (x, r1, ξ) 



r2

r1

n−1

exp

2

r1

r1

 

r2

r2

 



r2

r1

n−1

exp

2

r1

r1



0

r2

r2



0

at dt





r2

r1

n −1

exp(2ad).

The proof is complete 2

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 inLemma 2.3 2

3 Gradient steady Ricci solitons

A gradient steady Ricci soliton is a special smooth metric measure space (M n , g, e −f) satisfying

R ij + f ij = 0, where R ij 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 > 0

Δf + R = 0

where R is the scalar curvature of M After scaling we can assume a = 1 We write the potential function f

in polar coordinate

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

max

θ ∈S n −1

r



0



θ ∈S n −1

r



0



Vol

 Cr n

Note thatTheorem 3.1can 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

= 0

2n −1 .

Proof Our argument is close to the argument in [5] Let ω = n

i=1 a i dx i be any harmonic 1-form with



n

i=1

Let φ be a cut-off function on M such that φ = 1 on B p (r) (a geodesic ball centered at some fixed point p

of radius r), φ = 0 outside B p (2r) and |∇φ|  C

r Let q ∈ R such that

1

1

then

4n 2n + 1 < q < 2.

Using the integration by parts, we obtain



M

M

f ij a i a j φ2

=



M

(a i)j f i a j φ2+



M

f i a i a j

φ2

On the other hand, integrating by parts again, it follows that

−



M

(a i)j f i a j φ2= 1

2



M

(Δf ) |ω|2φ2+1

2



M

|ω|2

Combining (3.6)and(3.7), we have



M



M

2



M

R |ω|2φ2−12



M

|ω|2

Now, the Bochner formula implies

Δ

|ω|2

= 2Ric(ω, ω) + 2 |∇ω|2

 2Ric(ω, ω) + 2∇| ω |2

where we used first Kato inequality in the last inequality Multiplying this by φ2, then using (3.8) and integration by parts, we infer

2



M

∇| ω |2



M



M



M

|ω|2



M





M

∇| ω |2



M



M

|ω|2



M

This implies that

M

∇| ω |2



M



M



M

|ω|2



M

 Ca

M

for some constant C > 0, where in the last inequality, we have used that |∇f|  a.

From(3.9)and Hölder inequality, we obtain

 

M

∇| ω |q

2 +

 

M

R q |ω| q φ q

2

 Cr n(2−1) 

M

∇| ω |2



M



 Car n(2−1)

M

 Car n(2−1) r n(1 −2

p)

 

M

2|ω| p

2

p

 Ca r 2n(

2−1)

r

 

M

2

p

where the constant C > 0 might be different from line to line Let r → ∞ and using that ω ∈ L p (M ), we

conclude that

∇| ω |q

= R q |ω| q = 0.

This infers that either ω = 0 or; |ω| = C and R = 0 If ω = 0 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| = 0 The proof is

complete 2

Corollary 3.3 Let (M n , g) be a gradient Ricci soliton whose satisfying the Euclidean volume growth condition

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] H.D Cao, D.T Zhou, On complete gradient shrinking Ricci solitons, J Differential Geom 85 (2010) 175–186.

[2] N.T Dung, C.J Sung, Weighted f -harmonic 1-form on smooth metric measure spaces, preprint.

[3] J Jost, Riemannian Geometry and Geometric Analysis, fifth ed., Springer, 2008.

[4]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.

[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.

[9] N Yang, A note on nonnegative Bakry–Émery Ricci curvature, Arch Math 93... 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. .. comparison inLemma 2.3

3 Gradient steady Ricci solitons

A gradient steady Ricci soliton is a special smooth metric measure space (M n , g, e −f) satisfying