AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 237418 10.1155/2013/237418 237418 Research Article A Global Curvature Pinching Result of the First Eigenvalue of the Laplacian on Riemannian Manifolds Wang Peihe 1 Li Ying 2 Zou Wenming 1 School of Mathematical Sciences Qufu Normal University Shandong, Qufu 273165 China qfnu.edu.cn 2 College of Science University of Shanghai for Science and Technology Shanghai 200093 China usst.edu.cn 2013 28 3 2013 2013 08 12 2012 19 02 2013 09 03 2013 2013 Copyright © 2013 Peihe Wang and Ying Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The paper starts with a discussion involving the Sobolev constant on geodesic balls and then follows with a derivation of a lower bound for the first eigenvalue of the Laplacian on manifolds with small negative curvature. The derivation involves Moser iteration.

1. Introduction

The Laplacian is one of the most important operator on Riemannian manifolds, and the study of its first eigenvalue is also an interesting subject in the field of geometric analysis. In general, people would like to estimate the first eigenvalue of Laplacian in terms of geometric quantities of the manifolds such as curvature, volume, diameter, and injectivity radius. In this sense, the first interesting result is that of Lichnerowicz and Obata, which proved the following result in : let Mn be an n-dimensional compact Riemannian manifold without boundary with Ric(M)(n-1), then the first eigenvalue of Laplacian on Mn will satisfy that λ1(M)n, and the inequality becomes equality if and only if MnSn.

The above result implies that the first eigenvalue of the Laplacian will have a lower bound less than n if the Ricci curvature of manifolds involved has a lower bound n-1 except on a small part where the Ricci curvature satisfied that Ric(M)0. Now a natural question arises: what is the lower bound of the first eigenvalue of Laplacian on such a manifold? In , Petersen and Sprouse gave a lower bound under the assumption that the bad part of the manifolds is small in the sense of Lp-norm, where p is a constant larger than half of the dimension of the manifold. In this paper, we are interested in the lower bound of the first eigenvalue under the global pinching of the Ricci curvature and we obtain a universal estimate of this lower bound on a certain class of manifolds.

2. A Sobolev Constant on the Geodesic Ball

The Sobolev inequality is one of the most important tools in geometric analysis, and the Sobolev constant plays an important part in the study of this field. In this section, we will obtain a general Sobolev constant only depending on the dimension of the manifold on the geodesic ball with small radius.

Definition 1.

Let Bp(R)M be a geodesic ball with radius R; we define the Sobolev constant Cs(R) on it to be the infimum among all the constant C such that the inequality ||f||2n/(n-2)2C||f||22 holds for all fW01.2(Bp(R)).

Definition 2.

Let Bp(R)M be a geodesic ball with radius R; we define the isoperimetric constant C0(R) on it to be the supremum among all the constant α such that the inequality Area(Ω)αVol(Ω)1-(1/n) holds for all ΩBp(R) with smooth boundary.

For any fixed point p and radius R, Croke proves that the equality Cs(R)=4((n-1)/(n-2))2C0(R)-2 holds , but one expects the constant Cs(R) to be independent on the location of the point p, under some assumptions. In what follows, we will give an upper bound to Cs(R) independent of the point p.

Let M be an n-dimensional Riemannian manifold, SM is the unit tangent bundle of M, and π:SMM is the canonical projective map. ΩM,ξSΩ,γξ(t) is the normalized geodesic from π(ξ) with the initial velocity ξ. We define some notations as follows: (1)τ(ξ)=sup{τ>0γξ(t)Ω,t(0,τ)}.

C ( ξ ) is the arc length from π(ξ) to the cut locus point along γξ(t). Consider(2)UΩ={ξSΩC(ξ)τ(ξ)},Ux=(πUΩ)-1(x);ωx=μx(Ux)cn-1;ω=infxΩωx, where μx is the standard surface measure of the unit sphere,cn-1 is denoted to be the area of the unit sphere Sn-1.

Definition 3.

Using the Notation above, ω=infxΩωx is called the visibility angle of Ω.

If the manifold has Inj(M)i which ensures that any minimal geodesic starting from any point in Bp(i/2) will reach the boundary Bp(i/2) before it reaches its cut locus, then the visibility angle of Bp(i/2) for any point p which we denote by ω(i/2) satisfies ω(i/2)=1.

Lemma 4.

Let Mn be a closed Riemannian manifold with Inj (M)i, then for any pM, the following Sobolev inequality holds on Bp(i/2): ||f||2n/(n-2)2C||f||22, where fW01.2(Bp(i/2)) and C=C(n).

Proof.

Croke proved the following inequality : (3)Area(Ω)Vol(Ω)1-(1/n)cn-1(cn/2)1-(1/n)ω1+(1/n), where ΩBp(i/2),ΩC, and ω is just the visibility angle of the domain Ω.

As discussed above, we will have ω(Ω)=1 if ΩBp(i/2); then according to Croke’s inequality, we obtain C0(i/2)cn-1/(cn/2)1-(1/n)  . The relation between C0(i/2) and Cs(i/2) tells us that Cs(i/2)C(n), where C(n) is a constant only depending on the dimension n.

Proposition 5.

Let Mn be a closed n-dimensional Riemannian manifold with Inj (M)i, then for all pM, Vol (Bp(i/2))Cnin, where Cn is a constant only depending on the dimension n.

Proof.

Also take the inequality of Croke (4)Area(Ω)Vol(Ω)1-(1/n)cn-1(cn/2)1-(1/n)ω1+(1/n), then the result can easily be derived from the fact that ω(i/2)=1 and Area(Bp(r))=dVol(Bp(r))/dr after we integrate both sides of the inequality.

3. The First Eigenfunction and Eigenvalue

Let Mn be a closed n-dimensional Riemannian manifold; suppose that λ1(M) is the first eigenvalue of the Laplacian and u is the first eigenfunction. In other words, they will satisfy that Δu+λ1(M)u=0. By linearity, we can assume that -1u1 and infxMu=-1 for the linearity. For the convenience, we call it the normalized eigenfunction. Next we will study some properties of the normalized eigenfunction and the eigenvalue.

Lemma 6.

Let Mn be a closed n-dimensional Riemannian manifold with Ric(M)0 and Inj (M)i. Then, a constant C1(n,i)>0 can be found such that λ1(M)C1(n,i).

Proof.

One of the theorems of Yau and Schoen  shows that λ1(M)En/d2En/i2C1(n,i) if Ric(M)0, where d is the diameter of the manifold and En is a constant depending only on n.

We will now introduce some notation. Let Ric-(x) denote the lowest eigenvalue of the Ricci curvature tensor at x. For a function f(x) on Mn, we denote f+(x)=max{f(x),0}. Notice that a Riemannian manifold satisfies Ricn-1 if and only if ((n-1)-Ric-)+0.

The well-known Myers theorem shows that a closed manifold with Ricn-1 would have a bounded diameter dπ. In other words, one can deduce that dπ if one has (1/Vol(M))M((n-1)-Ric-)+dvol=0. We will show next a result analogous to the one in  which we will use in our estimation of the eigenvalue. The proof follows identically; so it will be omitted (the reader can refer to the aforementioned article).

Lemma 7.

Let Mn be a closed n-dimensional Riemannian manifold with Ric (M)0, then for any δ>0, there exists ϵ0=ϵ0(n,δ)>0 such that if (5)1 Vol (M)M((n-1)- Ric -)+d vol ϵ0(n,δ), then the diameter will satisfy d<π+δ. In particular, there exists ϵ1=ϵ1(n) such that if (6)1 Vol (M)M((n-1)- Ric -)+d vol ϵ1(n), then the diameter will satisfy d<2π. This fact, together with the volume comparison theorem, implies that Vol (M)C2(n), where C2(n) is also a constant only dependent of n.

Now, we can get a rough lower bound for the first eigenvalue.

Lemma 8.

For n, let ϵ1=ϵ1(n)>0 as above and suppose that Mn is a closed manifold with (7) Ric (M)0,1 Vol (M)M((n-1)- Ric -)+d vol ϵ1(n); then there exists a constant C3(n)>0 such that λ1(M)C3(n).

Proof.

The proof mainly belongs to Li and Yau . Let u be the normalized eigenfunction of M, set v=log(a+u) where a>1. Then, we can easily get that (8)Δv=-λ1(M)ua+u-|v|2.

Denote that Q(x)=|v|2(x), and we then have by the Ricci identity on manifolds with Ric(M)0: (9)ΔQ=2vij2+2vjvjii2vij2+2v,Δv. For the term vij2, we have (10)i,jvij2(Δv)2n1n(Q2+2λ1(M)ua+u), and for the term v,Δv, we have (11)v,Δv=-aλ1(M)a+uQ-v,Q. Therefore, assume x0M to be the maximum of Q; then at x0, we have (12)02nQ(x0)+(4λ1(M)n-2(n+2)aλ1(M)n(a-1)). Therefore, (13)Q(x)Q(x0)(n+2)aλ1(M)a-1.

Denote γ to be the minimizing unit speed geodesic joining the maximum and minimum points of u; then integrating Q1/2 along γ, one will get:(14)log(aa-1)log(a+maxua-1)d((n+2)aλ1(M)a-1)1/2.

Let t=(a-1)/a; then for any t(0    1), we have (n+2)λ1(M)t(d-2(log(1/t))2).

Considering the maximum of the right hand and the upper bound of the diameter derived in Lemma 7, we can deduce that a positive constant C3(n) can be found such that (15)λ1(M)4e-2(n+2)d2C3(n), where d is the diameter of the manifold.

Corollary 9.

If the manifold one discussed satisfies all the conditions in Lemma 8 and its injectivity radius satisfies Inj (M)i and if one let u to be the normalized eigenfunction, then there exists a constant C4(n,i)>0 such that |u|2C4(n,i).

Proof.

Set a=2 in the (13) from above. Then applying Lemma 6, one obtains (16)19|u|2|v|2(x)2C1(n,i)(n+2);

therefore, (17)|u|2C4(n,i).

Proposition 10.

Let Mn be a closed n-dimensional Riemannian manifold, u the first eigenfunction of the Laplacian, and λ1(M) the corresponding eigenvalue, then Δ|u|+λ1(M)|u|0 holds in the sense of distribution. Moreover, if Mn is compact with boundary, then the same conclusion holds for its Neumann boundary value problem.

Proof.

From the definition, we know that Δu+λ1(M)u=0 holds on M. Denote (18)M+={xMu(x)>0},M-={xMu(x)<0},M0={xMu(x)=0}. According to the maximum principle of elliptic equation and the discussion about nodal set and nodal regions in , we can conclude that M+=M-=M0 is a smooth manifold with dimension n-1.

For all ϕC(M),ϕ0, integrating by parts we then have (19)M|u|Δϕ+λ1(M)ϕ|u|=M+|u|Δϕ+λ1(M)ϕ|u|+M-|u|Δϕ+λ1(M)ϕ|u|=M+(uϕn+-ϕun+)+M+ϕ(Δu+λ1(M)u)=.-M-(uϕn--ϕun-)-M-ϕ(Δu+λ1(M)u)=M-ϕun--M+ϕun+0, where n+ and n- denote the outward normal direction with respect to the boundaries of M+ and M-, respectively. Note that u/n-0 on M- and u/n+0 on M+ for the definition of M- and M+. This completes the proof.

When M has boundary, we can apply the same reasoning, except that the test function will require ϕC0(M). This gives the proof.

As long as the given manifold is compact, one knows that the first normalized eigenfunction is then determined. This indicates that the first normalized eigenfunction of the Laplacian has a close relation with the geometry of the manifold. In particular, one would hope to bound the L2-norm of first normalized eigenvalue of Laplacian from below by the geometric quantities. In this sense, we have the following result.

Theorem 11.

Let Mn be a closed n-dimensional Riemannian manifold with Ric (M)0 and Inj (M)i. If u is the normalized eigenfunction of the Laplacian, then there exists a constant C5(n,i)>0 such that Mu2C5(n,i).

Proof.

We use Moser iteration to get the result. From Proposition 10, we know that Δ|u|+λ1(M)|u|0 holds on M in the sense of distribution. Set v=|u| and take the point pM such that u(p)=-1.

For a1, denote R=i/2;   ϕ is a cut-off function on Bp(R), then we have by integrating by parts: (20)λ1(M)Bp(R)ϕ2v2a=-Bp(R)ϕ2v2a-1Δv=2Bp(R)ϕv2a-1+(2a-1)Bp(R)ϕ2v2a-2|v|22Bp(R)ϕv2a-1+aBp(R)ϕ2v2a-2|v|2. However, using the identity (21)Bp(R)|(ϕva)|2=Bp(R)|ϕ|2v2a+2aBp(R)ϕv2a-1ϕ,v+a2Bp(R)ϕ2v2a-2|v|2, we have (22)λ1(M)aBp(R)v2aϕ2Bp(R)|(ϕva)|2-Bp(R)v2a|ϕ|2; therefore, using the Sobolev inequality in Lemma 4, (23)λ1(M)aBp(R)v2aϕ2+Bp(R)v2a|ϕ|2Bp(R)|(ϕva)|21C(Bp(R)(ϕva)2n/(n-2))(n-2)/n=1Cϕva2n/(n-2)  2.

Let (24)ϕ={1,in  Bp(ρ),ρ+σ-rσ,in  Bp(ρ+σ)Bp(ρ),0,in  Bp(R)Bp(ρ+σ)  .

Putting ϕ into the inequality above and we then have by splitting the integral into three parts and using the values of ϕ on each of them: (25)v2an/(n-2);Bp(ρ)[C(λ1(M)a+1σ2)]1/2av2a;Bp(ρ+σ), where we denote ||f||p;Ω=(Ωfp)1/p only for emphasizing the integral domain.

Set (26){aj}:a0=1,a1=nn-2,,aj=(nn-2)j,,{σj}:σ0=R4,σ1=R8,,σj=R2-(2+j),,{ρj}:ρ-1=R,,ρj=R-s=0jσl,.

And putting {aj},{σj},{ρj} into (25), we can derive after iteration that (27)v2aj+1;  Bp(ρj)[C(λ1(M)aj+1σj2]1/2ajv2aj;Bp(ρj-1)s=0j[C(λ1(M)as+1σs2)]1/2asv2;Bp(R).

Let j+, then (28)v;Bp(R/2)j=0[C(λ1(M)aj+1σj2]1/2ajv2;Bp(R).

The product can be estimated as follows: (29)j=0[C(λ1(M)aj+1σj2)]1/2ajj=0[C(λ1(M)+16R2)]1/2aj4j/2aj.

The right hand will converge to a fixed number by using the fact that j=0Bμj=Bμ/(μ-1) and the fact j=0jμ-j is finite for some BR,μ>1. From λ1(M)C1(n,i), we can find a positive constant C5(n,i)>0 such that (30)v;Bp(R/2)C5-1/2(n,i)v2;Bp(R)C5-1/2(n,i)v2;M.

Therefore, (31)Mu2C5(n,i).

4. The Lower Bound of the First Eigenvalue

Using the same notation as above, we can state the following result.

Theorem 12.

For n, i,δ+, there is an ϵ=ϵ(n,i,δ)>0 such that any closed manifold Mn with Ric (M)0, Inj (M)i and (32)1 Vol (M)M((n-1)- Ric -)+d vol ϵ will satisfy that λ1(M)n-δ.

Proof.

Assume that u is the normalized eigenfunction of Laplacian on Mn, let Q(x)=|u|2+(λ1(M)/n)u2, direct computation shows that (33)12ΔQ=uij2+ujuiij+Rijuiuj+λ1(M)n|u|2+λ1(M)nuΔu1-nnλ1(M)|u|2+Rijuiuj.

Integrating both sides on Mn, we have (34)01-nnλ1(M)M|u|2dvol+MRijuiujdvol1-nnλ1(M)M|u|2dvol+MRic-|u|2dvol=1-nnλ1(M)M|u|2dvol+(n-1)M|u|2dvol-M[(n-1)-Ric-]|u|2dvol; therefore, (35)λ1(M)n-Vol(M)λ1(M)M|u|2dvol1Vol(M)×M((n-1)-Ric-)+|u|2dvol

if we suppose that (36)1Vol(M)M((n-1)-Ric-)+dvolϵ1(n). If ϵ1 is the one obtained in Lemma 7, then one has: (37)λ1(M)n-C2C4C3C51Vol(M)M((n-1)-Ric-)+dvol.

Finally, if one chooses (38)0<ϵ=ϵ(n,i,δ)min{ϵ1,δC3C5C2C4}, then λ1(M)n-δ as long as (1/Vol(M))M((n-1)-Ric-)+dvolϵ, and this proves the theorem.

Acknowledgments

The authors owe great thanks to the referees for their careful efforts to make the paper clearer. Research of the first author was supported by STPF of University (no. J11LA05), NSFC (no. ZR2012AM010), the Postdoctoral Fund (no. 201203030) of Shandong Province, and Postdoctoral Fund (no. 2012M521302) of China. Part of this work was done while the first author was staying at his postdoctoral mobile research station of QFNU.

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