# A Global Curvature Pinching Result of the First Eigenvalue of the Laplacian on Riemannian Manifolds

Authors: Peihe Wang; Ying Li
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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.


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## Body

## 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 [1]: 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 Mn≅Sn.The above result implies that the first eigenvalue of the Laplacian will have a lower bound less thann 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 [2], 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.
LetBp(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)2≤C||∇f||22 holds for all f∈W01.2(Bp(R)).Definition 2.
LetBp(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 Ω⊆Bp(R) with smooth boundary.
For any fixed pointp and radius R, Croke proves that the equality Cs(R)=4((n-1)/(n-2))2C0(R)-2 holds [3], 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.
LetM be an n-dimensional Riemannian manifold, SM is the unit tangent bundle of M, and π:SM→M is the canonical projective map. Ω⊆M,ξ∈SΩ,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>0∣γξ(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={ξ∈SΩ∣C(ξ)≥τ(ξ)},Ux=(π∣UΩ)-1(x);ωx=μx(Ux)cn-1;ω=infx∈Ωωx,
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=infx∈Ωωx is called the visibility angle of Ω.
If the manifold hasInj(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.
LetMn be a closed Riemannian manifold with Inj(M)≥i, then for any p∈M, the following Sobolev inequality holds on Bp(i/2): ||f||2n/(n-2)2≤C||∇f||22, where f∈W01.2(Bp(i/2)) and C=C(n).Proof.
Croke proved the following inequality [4]:
(3)Area(∂Ω)Vol(Ω)1-(1/n)≥cn-1(cn/2)1-(1/n)ω1+(1/n),
where Ω⊆Bp(i/2),∂Ω∈C∞, and ω is just the visibility angle of the domain Ω.
As discussed above, we will haveω(Ω)=1 if Ω⊆Bp(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.
LetMn be a closed n-dimensional Riemannian manifold with Inj(M)≥i, then for all p∈M,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

LetMn 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 -1≤u≤1 and infx∈Mu=-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.
LetMn 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 [1] shows that λ1(M)≤En/d2≤En/i2≜C1(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. LetRic-(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 Ric≥n-1 if and only if ((n-1)-Ric-)+≡0.
The well-known Myers theorem shows that a closed manifold withRic≥n-1 would have a bounded diameter d≤π. In other words, one can deduce that d≤π if one has (1/Vol(M))∫M‍((n-1)-Ric-)+dvol=0. We will show next a result analogous to the one in [5] 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.
LetMn 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)1Vol(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)1Vol(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.
Forn∈ℕ, let ϵ1=ϵ1(n)>0 as above and suppose that Mn is a closed manifold with
(7)Ric(M)≥0,1Vol(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 [6]. 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 thatQ(x)=|∇v|2(x), and we then have by the Ricci identity on manifolds with Ric(M)≥0:
(9)ΔQ=2vij2+2vjvjii≥2vij2+2〈∇v,∇Δv〉.
For the term vij2, we have
(10)∑i,jvij2≥(Δv)2n≥1n(Q2+2λ1(M)ua+u),
and for the term 〈∇v,∇Δv〉, we have
(11)〈∇v,∇Δv〉=-aλ1(M)a+uQ-〈∇v,∇Q〉.
Therefore, assume x0∈M to be the maximum of Q; then at x0, we have
(12)0≥2nQ(x0)+(4λ1(M)n-2(n+2)aλ1(M)n(a-1)).
Therefore,
(13)Q(x)≤Q(x0)≤(n+2)aλ1(M)a-1.
Denoteγ 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.
Lett=(a-1)/a; then for any t∈(01), 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 Lemma7, we can deduce that a positive constant C3(n) can be found such that
(15)λ1(M)≥4e-2(n+2)d2≥C3(n),
where d is the diameter of the manifold.Corollary 9.
If the manifold one discussed satisfies all the conditions in Lemma8 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|2≤C4(n,i).Proof.
Seta=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|2≤C4(n,i).Proposition 10.
LetMn 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 thatΔu+λ1(M)u=0 holds on M. Denote
(18)M+={x∈M∣u(x)>0},M-={x∈M∣u(x)<0},M0={x∈M∣u(x)=0}.
According to the maximum principle of elliptic equation and the discussion about nodal set and nodal regions in [1], we can conclude that ∂M+=∂M-=M0 is a smooth manifold with dimension n-1.
For allϕ∈C∞(M),ϕ≥0, 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+(u∂ϕ∂n+-ϕ∂u∂n+)+∫M+ϕ(Δu+λ1(M)u)=.-∫∂M-(u∂ϕ∂n--ϕ∂u∂n-)-∫M-ϕ(Δu+λ1(M)u)=∫∂M-ϕ∂u∂n--∫∂M+ϕ∂u∂n+≥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.WhenM has boundary, we can apply the same reasoning, except that the test function will require ϕ∈C0∞(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 theL2-norm of first normalized eigenvalue of Laplacian from below by the geometric quantities. In this sense, we have the following result.Theorem 11.
LetMn 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 ∫Mu2≥C5(n,i).Proof.
We use Moser iteration to get the result. From Proposition10, we know that Δ|u|+λ1(M)|u|≥0 holds on M in the sense of distribution. Set v=|u| and take the point p∈M such that u(p)=-1.
Fora≥1, 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=2∫Bp(R)ϕv2a-1+(2a-1)∫Bp(R)ϕ2v2a-2|∇v|2≥2∫Bp(R)ϕv2a-1+a∫Bp(R)ϕ2v2a-2|∇v|2.
However, using the identity
(21)∫Bp(R)|∇(ϕva)|2=∫Bp(R)|∇ϕ|2v2a+2a∫Bp(R)ϕv2a-1〈∇ϕ,∇v〉+a2∫Bp(R)ϕ2v2a-2|∇v|2,
we have
(22)λ1(M)a∫Bp(R)v2aϕ2≥∫Bp(R)|∇(ϕva)|2-∫Bp(R)v2a|∇ϕ|2;
therefore, using the Sobolev inequality in Lemma 4,
(23)λ1(M)a∫Bp(R)v2aϕ2+∫Bp(R)v2a|∇ϕ|2≥∫Bp(R)|∇(ϕva)|2≥1C(∫Bp(R)(ϕva)2n/(n-2))(n-2)/n=1C∥ϕva∥2n/(n-2)2.
Let(24)ϕ={1,inBp(ρ),ρ+σ-rσ,inBp(ρ+σ)Bp(ρ),0,inBp(R)Bp(ρ+σ).
Puttingϕ 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)∥v∥2an/(n-2);Bp(ρ)≤[C(λ1(M)a+1σ2)]1/2a∥v∥2a;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)∥v∥2aj+1;Bp(ρj)≤[C(λ1(M)aj+1σj2]1/2aj∥v∥2aj;Bp(ρj-1)≤∏s=0j[C(λ1(M)as+1σs2)]1/2as∥v∥2;Bp(R).
Letj→+∞, then
(28)∥v∥∞;Bp(R/2)≤∏j=0∞[C(λ1(M)aj+1σj2]1/2aj∥v∥2;Bp(R).
The product can be estimated as follows:(29)∏j=0∞[C(λ1(M)aj+1σj2)]1/2aj≤∏j=0∞[C(λ1(M)+16R2)]1/2aj4j/2aj.
The right hand will converge to a fixed number by using the fact that∏j=0∞Bμj=Bμ/(μ-1) and the fact ∑j=0∞jμ-j is finite for some B∈R,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)∥v∥2;Bp(R)≤C5-1/2(n,i)∥v∥2;M.
Therefore,(31)∫Mu2≥C5(n,i).


## 4. The Lower Bound of the First Eigenvalue

Using the same notation as above, we can state the following result.Theorem 12.
Forn∈ℕ, i,δ∈ℝ+, there is an ϵ=ϵ(n,i,δ)>0 such that any closed manifold Mn with Ric(M)≥0,Inj(M)≥i and
(32)1Vol(M)∫M((n-1)-Ric-)+dvol≤ϵ
will satisfy that λ1(M)≥n-δ.Proof.
Assume thatu 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)nuΔu≥1-nnλ1(M)|∇u|2+Rijuiuj.
Integrating both sides onMn, we have
(34)0≥1-nnλ1(M)∫M|∇u|2dvol+∫MRijuiujdvol≥1-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-)+dvol≤ϵ1(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-)+dvol≤ϵ, and this proves the theorem.



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