3 Proof of the main result

for Laplacian operators

Xiang Gao

Abstract. In this paper, we consider the characterization of eigenfunc- tions for Laplacian operators on some Riemannian manifolds. Firstly we prove that for the space form (M_Kⁿ, gK) with the constant sectional curva- tureK, the first eigenvalue of Laplacian operatorλ1(M_Kⁿ) is greater than the limit of the first Dirichlet eigenvalue of Laplacian operatorλ^D₁ (BK(p, r)).

Based on this, we then present a characterization of the Ricci soliton being ann-dim space form by the eigenfunctions corresponding to the first eigen- value of Laplacian operator, which gives a generalization of an interesting result by Cheng in [3] from 2-dim to n-dim. Moreover, this result also gives a partly proof of a conjecture by Hamilton that a compact gradient shrinking Ricci soliton with positive curvature operator must be Einstein.

M.S.C. 2010: 58G25, 35P05.

Key words: Laplacian operator; Dirichlet eigenvalue; eigenfunction.

1 Introduction and main results

Suppose that (Mⁿ, g) is ann-dimC^∞ complete Riemannian manifold, and let ∆ de- note the Laplacian operator. If the manifold is compact, it is well known that the eigenvalue problem−∆ϕ=λϕhas discrete eigenvalues, and we list them as

0 =λ0(Mⁿ)< λ1(Mⁿ)≤λ2(Mⁿ)≤ · · ·.

We callλi(Mⁿ) theith eigenvalue and call a function satisfying ∆ϕ=−λiϕan ith eigenfunction.

Recall that the first eigenvalueλ1(Mⁿ) for the closed Riemannian manifold Mⁿ is defined as follows:

(1.1) λ1(Mⁿ) = inf

f∈Ω

R

Mⁿ|∇f|²dµ R

Mⁿf²dµ ,

Balkan Journal of Geometry and Its Applications, Vol.17, No.2, 2012, pp. 46-53.∗

°c Balkan Society of Geometers, Geometry Balkan Press 2012.

where Ω is the completing Hilbert space of Ω0=

½

f ∈C^∞(Mⁿ)

¯¯

¯¯ Z

Mⁿ

f dµ= 0

¾

under the norm

kfk²₁= Z

Mⁿ

f²dµ+ Z

Mⁿ

|∇f|²dµ.

For the first eigenvalue λ1(Mⁿ) for closed Riemannian manifolds, we have the following famous theorem For the first eigenvalue λ1(Mⁿ) for closed Riemannian manifolds, we have the following famous theorem in [4]:

Theorem 1.1 (Lichnerowicz-Obata). Let (Mⁿ, g)be a closed Riemannian mani- fold satisfyingRc≥(n−1)K >0. Then

λ1(Mⁿ)≥nK,

and the equality holds iff(Mⁿ, g)is isometric to the space form (M_Kⁿ, g_K)with con- stant sectional curvature K and the eigenfunction

f(x) =Acos³√ Kr´

+Bsin³√ Kr´

, wherer=d(p, x).

On the other hand, we denote the open geodesic ball with center pand radiusr byB(p, r), and letBK(p, r) denote the geodesic ball with radiusrin then-dim simply connected space form (M_Kⁿ, gK) with constant sectional curvatureK. Then the first Dirichlet eigenvalueλ^D₁ (B(p, r)) ofB(p, r) can be denoted as:

(1.2) λ^D₁ (B(p, r)) = inf

f∈H₀²(B(p,r))

R

B(p,r)|∇f|²dµ R

B(p,r)f²dµ ,

whereH₀²(B(p, r)) is the completing Hilbert space ofC₀^∞(B(p, r)) under the norm kfk²₁=

Z

B(p,r)

f²dµ+ Z

B(p,r)

|∇f|²dµ.

Forλ^D₁ (B(p, r)) we have the following famous theorem in [2]:

Theorem 1.2 (Cheng). Let (Mⁿ, g) be a complete Riemannian manifold satisfy- ingRc≥(n−1)K. Then

λ^D₁ (B(p, r))≤λ^D₁ (BK(p, r)) and the equality holds iffB(p, r)is isometric to BK(p, r).

Moreover by using Theorem 1.2 we have the following corollary:

Corollary 1.3 (Cheng). Let (Mⁿ, g) be a compact Riemannian manifold satisfy- ingRc≥0. Then

λ1(Mⁿ)≤λ^D₁ µ

B µ

p,dMⁿ

2

¶¶

≤ Cn

d²_Mn

, whereCn = 2n(n+ 4)anddMⁿ is the diameter ofMⁿ.

In this paper, by using Theorem 1.1 and 1.2 we firstly prove a useful result, which is one of the main results as follows:

Theorem 1.4.Let(M_Kⁿ, gK)be a space form with constant sectional curvature K, and λ^D₁ (M_Kⁿ) denote the first Dirichlet eigenvalue ofM_Kⁿ which is defined by

λ^D₁ (M_Kⁿ) = lim

r→√^π K

λ^D₁ (BK(p, r)),

whereλ^D₁ (BK(p, r))is the first Dirichlet eigenvalue of Laplacian operator ofBK(p, r), then

(1.3) λ^D₁ (M_Kⁿ) = lim

r→√^π K

λ^D₁ (B_K(p, r))≤λ₁(M_Kⁿ)≤λ^D₁ µ

B_K µ

p,dM_Kⁿ

2

¶¶

.

One of the basic problems in Riemannian geometry is to relate curvature and topology, in [1] B¨ohm and Wilking prove that n-dimensional closed Riemannian manifolds with 2-positive curvature operators are diffeomorphic to spherical space forms, i.e., they admit metrics with constant positive sectional curvature. Moreover it is a well-known theorem of Tachibana in [7] that any compact Einstein manifold with positive sectional curvature must be of constant curvature. Since Einstein manifolds are special Ricci solitons with constant potential functions, hence, inspired by his own work in [5] and [6], Hamilton made the following conjecture:

Conjecture 1.5 (Hamilton). A compact gradient shrinking Ricci soliton with pos- itive curvature operator must be Einstein.

Recall that the definition of the gradient Ricci soliton is as follows:

Definition 1.1. A complete Riemannian manifold (Mⁿ, g) is called a gradient Ricci soliton if there is a smooth functionf :Mⁿ→R, such that

Rc+∇∇f+ε 2g= 0,

where Rc is the Ricci curvature tensor andεis a real number. Moreover (i) Ifε <0, it is called a shrinking gradient Ricci soliton;

(ii) Ifε= 0, it is called a steady one;

(iii) Ifε >0, it is called an expanding one.

For the compact gradient Ricci soliton, Hamilton proved the following fact:

Theorem 1.6 (Hamilton). A compact gradient steady or expanding Ricci soliton must be Einstein.

On the other hand, for the special caseS², in [3] Cheng derived an interesting result by using the approach of tensor analysis and Gauss-Bonnet Theorem as follows:

Theorem 1.7 (Cheng). Suppose thatM²is homeomorphic toS²andϕ1,ϕ2andϕ3

are three first eigenfunctions such that their square sum is a constant. Then M² is actually isometric to a sphere with constant sectional curvature.

Thus for the manifolds withdim≥3, it is natural to ask the following question:

Problem 1.2. Is Theorem 1.7 also true whendim≥3 ?

Although the general answer is hard, recall that each manifold with dim = 2 is actually an Einstein manifold, which is a special Ricci soliton with the constant po- tential function. Based on this observation, we will give an affirmative answer for the special case of Ricci solitons. In fact, we present a characterization of a Ricci soliton being ann-dim space form by the first eigenfunctions of Laplacian operator, which also gives a partly proof of Conjecture 1.5 of Hamilton as follows:

Theorem 1.8. Let (Mⁿ, g) be a compact gradient Ricci soliton with positve Ricci curvature, and suppose that there exists a geodesic ballB(p, r)with center p and radius r such that the eigenfunctions{ϕi}^m_i=1corresponding to the first Dirichlet eigenvalue of Laplacian operatorλ^D₁ (B(p, r))satisfyP

i

ϕ²_i ≡C, where C is a nonzero constant. Let µ= inf{λ∈R|∇∇f ≤λg},

then

(i) (Mⁿ, g)is a shrinking Ricci soliton,

(ii) (Mⁿ, g) is locally isometric to the space formM_Kⁿ with constant sectional curva- tureK=−^µ+_n−1^ε2.

The paper is organized as follows: In section 2, we prove Theorem 1.4. In section 3, we present the proof of Theorem 1.8 by using the approach of tensor analysis and Theorem 1.1, 1.2 and 1.4.

2 Proof of Theorem 1.4

Proof of Theorem 1.4. Firstly let points p, q ∈ M_Kⁿ such that d(p, q) = dM_Kⁿ, then we consider the geodesic balls BK

³

p,^{dM n}₂^K´

and BK

³

q,^{dM n}₂^K´

in the n-dim simply connected space formM_Kⁿ. We denoteuandvas the first Dirichlet eigenfunctions of Laplacian operator corresponding toB_K³

p,^{dM n}₂^K´

andB_K³

q,^{dM n}₂^K´

, and define the following two functions:

e u(x) =





u(x), x∈BK

³

p,^{dM n}₂^K´ 0, x∈M_Kⁿ\BK

³ p,^{dM n}₂^K

´

and

e v(x) =





v(x), x∈BK

³ q,^{dM n}₂^K

´

0, x∈M_Kⁿ\BK

³

q,^{dM n}₂^K´ . Thus

(2.1) R

M_Kⁿ|∇eu|²dµ R

M_Kⁿue²dµ = R

BK

µ p,^dMn₂^K

¶|∇u|²dµ

R

BK

µ p,^dMn₂^K

¶u²dµ =λ^D₁ µ

BK

µ p,dM_Kⁿ

2

¶¶

and

(2.2) R

M_Kⁿ|∇ev|²dµ R

M_Kⁿve²dµ = R

BK

µ q,^dMn₂^K

¶|∇u|²dµ

R

BK

µ q,^dMn₂^K

¶u²dµ =λ^D₁ µ

BK

µ q,dM_Kⁿ

2

¶¶

.

Then we choose a constantCsuch that Z

M_Kⁿ

(eu+Cev)dµ= 0,

and by the definition of the first eigenvalue of Laplacian operatorλ1(M_Kⁿ) we have

λ1(M_Kⁿ)≤ R

M_Kⁿ|∇(eu+Cev)|²dµ R

M_Kⁿ(eu+Cev)²dµ . SinceM_Kⁿ is a space form with diameterdM_Kⁿ, we have

λ^D₁ µ

BK

µ p,dM_Kⁿ

2

¶¶

=λ^D₁ µ

BK

µ q,dM_Kⁿ

2

¶¶

and

V ol µ

BK

µ p,dM_Kⁿ

2

¶

∩BK

µ q,dM_Kⁿ

2

¶¶

= 0.

Thus

λ1(M_Kⁿ)≤ R

M_Kⁿ|∇(eu+Cev)|²dµ R

M_Kⁿ(eu+Cev)²dµ

= R

M_Kⁿ|∇eu|²dµ+C²R

M_Kⁿ|∇ev|²dµ R

M_Kⁿue²dµ+C²R

M_Kⁿev²dµ

=λ^D₁ µ

BK

µ p,dM_Kⁿ

2

¶¶

=λ^D₁ µ

BK

µ q,dM_Kⁿ

2

¶¶

, for the last two equalities we use (2.1) and (2.2).

For the other inequality, since the metric of space formM_Kⁿ has the form gM_Kⁿ =dr²+sK(r)g_Sⁿ⁻¹,

and by Theorem 1.1 we can choose the function ϕ(x) =Acos³√

Kr´

+Bsin³√ Kr´

wherer=d(p, x) such that

λ1(M_Kⁿ) = R

M_Kⁿ|∇ϕ|²dµ R

M_Kⁿϕ²dµ

= R^√π_K

0 |∇ϕ(r)|²sK(r)ⁿ⁻¹drR

Sⁿ⁻¹dµSⁿ⁻¹

R√^π

0 Kϕ(r)²s_K(r)ⁿ⁻¹drR

Sⁿ⁻¹dµ_Sn−1

= R^√π

0 K|∇ϕ(r)|²sK(r)ⁿ⁻¹dr R^√π

0 Kϕ(r)²sK(r)ⁿ⁻¹dr

= lim

s→^√π

K

R_s

0 |∇ϕ(r)|²sK(r)ⁿ⁻¹dr R_s

0ϕ(r)²sK(r)ⁿ⁻¹dr ,

where we useM_Kⁿ is a space form with diameterdM_Kⁿ. Then we define a functionϕs∈ H₀²(BK(p, s)) such thatϕs(x) =ϕ(x) for anyx∈BK(p, s). Then by the definition of the first Dirichlet eigenvalueλ^D₁ (BK(p, s)) it follows that

R_s

0 |∇ϕ(r)|²s_K(r)ⁿ⁻¹dr R_s

0ϕ(r)²sK(r)ⁿ⁻¹dr = R_s

0 |∇ϕ(r)|²s_K(r)ⁿ⁻¹drR

Sⁿ⁻¹dµ_Sn−1

R_s

0ϕ(r)²sK(r)ⁿ⁻¹drR

Sⁿ⁻¹dµ_Sⁿ⁻¹

= R

BK(p,s)|∇ϕs|²dµ R

BK(p,s)ϕ²_sdµ

≥λ^D₁ (BK(p, s)). Consequently

λ1(M_Kⁿ) = lim

s→√^π K

R_s

0 |∇ϕ(r)|²sK(r)ⁿ⁻¹dr R_s

0 ϕ(r)²sK(r)ⁿ⁻¹dr ≥ lim

s→√^π K

λ^D₁ (BK(p, s)) =λ^D₁ (M_Kⁿ).

¤

3 Proof of the main result

By using Theorem 1.1, 1.2 and 1.4, we now turn to prove our main result Theorem 1.8.

Proof of Theorem 1.8. (i) If (Mⁿ, g) is steady or expanding Ricci soliton, which sat- isfies

Rc+∇∇f+ε 2g= 0

such thatε≥0, then by using Theorem 1.6 we have (Mⁿ, g) is an Einstein manifold and the Ricci potential functionfis a constant. Thus

Rc=−ε 2g≤0,

which leads a contradiction to the positive Ricci curvature.

(ii) Note that the assumption of Theorem 1.8 says that







∆ϕi+λ^D₁ (B(p, r))ϕi= 0, i= 1,· · · , m P

i

ϕ²_i ≡C,

thus

0 = ∆ ÃX

i

ϕ²_i

!

=X

i

2|∇ϕi|²+2X

i

ϕi∆ϕi

=X

i

2|∇ϕi|²−2λ^D₁ (B(p, r))X

i

ϕ²_i,

which implies

(3.1) X

i

|∇ϕi|²=Cλ^D₁ (B(p, r)). Recall that the Bochner formula (see [4]) says that

∆|∇f|²= 2|∇∇f|²+ 2h∇f,∇∆fi+ 2Rc(∇f,∇f),

whereRcdenotes the Ricci curvature and∇f is the vector field. Taking Laplacian of both sides of (3.1) we have

0 =X

i

∆|∇ϕi|²= 2X

i

|∇∇ϕi|²+ 2X

i

h∇ϕi,∇∆ϕii+ 2X

i

Rc(∇ϕi,∇ϕi)

= 2X

i

|∇∇ϕi|²−2λ^D₁ (B(p, r))²X

i

ϕ²_i + 2X

i

Rc(∇ϕi,∇ϕi)

= 2X

i

| ∇∇ϕi|²−2λ^D₁ (B(p, r))²X

i

ϕ²_i −2X

i

∇∇f(∇ϕi,∇ϕi)−εX

i

| ∇ϕi|²

≥2X

i

| ∇∇ϕ_i|²−2λ^D₁ (B(p, r))²X

i

ϕ²_i −2µX

i

| ∇ϕ_i|²−εX

i

| ∇ϕ_i|²

≥ 2 n

X

i

|∆ϕi|²−2λ^D₁ (B(p, r))²X

i

ϕ²_i −2µX

i

| ∇ϕi|²−εX

i

| ∇ϕi|²

=−2 µ

1− 1 n

¶

λ^D₁ (B(p, r))²C−2³ µ+ε

2

´

λ^D₁ (B(p, r))C.

SinceC >0 we have

λ^D₁ (B(p, r))≥ − 1

¡1−_n¹¢³ µ+ε

2

´ .

For the space form with constant sectional curvatureK=−^µ+_n−1^ε2, we have Rc=−(µ+ε

2)g.

Then by using Theorem 1.1 and 1.4 we have

λ^D₁ (B_K(p, r))≤λ₁(M_Kⁿ) =− 1

¡1−¹_n¢³ µ+ε

2

´ ,

whereλ1(M_Kⁿ) is the first eigenvalue of Laplacian operator for the space formM_Kⁿ. This implies

(3.2) λ^D₁ (B(p, r))≥λ^D₁ (BK(p, r)). On the other hand, by the definition ofµwe have

Rc=−

³

∇∇f+ε 2g

´

≥ −

³ µ+ε

2

´

g= (n−1) µ

−µ+^ε₂ n−1

¶ , and by Theorem 1.2, it follows that

(3.3) λ^D₁ (B(p, r))≤λ^D₁ (BK(p, r)), whereK=−^µ+_n−1^ε2.

Then by (3.2) and (3.3) we derive that

(3.4) λ^D₁ (B(p, r)) =λ^D₁ (BK(p, r)),

thus by using the equality condition in Theorem 1.2, we complete the proof. ¤ Acknowledgment. I would especially like to thank the referee and editor for mean- ingful suggestions that led to improvements of the article.

References

[1] C. B¨ohm, B. Wilking, Manifolds with positive curvature operators are space forms. Ann. Math., 167, (2008), 1079-1097.

[2] S. Y. Cheng, Eigenvalue comparison theorems and its geometric application.

Math. Z., 143, (1975), 289-297.

[3] S. Y. Cheng,A characterization of the 2-sphere by eigenfunctions. Proc. Amer.

Math. Soc., 55, (1976). 379-381.

[4] B. Chow, P. Lu, L. Ni, Hamilton’s Ricci Flow. Graduate Studies in Mathemat- ics, 77, American Mathematical Society, Providence, RI, 2006.

[5] R.S. Hamilton, Four manifolds with positive curvature operator, J. Diff. Geom., 24, (1986), 153-179.

[6] R.S. Hamilton, The Harnack estimate for the Ricci flow, J. Diff. Geom., 37, (1993), 225-243.

[7] S. Tachibana.A theorem of Riemannian manifolds of positive curvature operator.

Proc. Japan Acad., 50, (1974), 301-302.

Author’s address:

Xiang Gao

School of Mathematical Sciences, Ocean University of China, Lane 238, Songling Road, Laoshan District, Qingdao City, Shandong Province, 266100, People’s Republic of China.

E-mail: [email protected]