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

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.

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 𝑀^𝑛 be an 𝑛-dimensional compact Riemannian manifold without boundary with Ric(𝑀)≥(𝑛−1), then the first eigenvalue of Laplacian on 𝑀^𝑛 will satisfy that 𝜆₁(𝑀)≥𝑛, and the inequality becomes equality if and only if 𝑀^𝑛≅𝑆^𝑛.

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 [2], Petersen and Sprouse gave a lower bound under the assumption that the bad part of the manifolds is small in the sense of 𝐿^𝑝-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 𝐵_𝑝(𝑅)⊆𝑀 be a geodesic ball with radius 𝑅; we define the Sobolev constant 𝐶_𝑠(𝑅) on it to be the infimum among all the constant 𝐶 such that the inequality ||𝑓||²_{2𝑛/(𝑛−2)}≤𝐶||∇𝑓||²₂ holds for all 𝑓∈𝑊₀^1.2(𝐵_𝑝(𝑅)).

Definition 2. Let 𝐵_𝑝(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⊆𝐵_𝑝(𝑅) with smooth boundary.
For any fixed point 𝑝 and radius 𝑅, Croke proves that the equality 𝐶_𝑠(𝑅)=4((𝑛−1)/(𝑛−2))²𝐶₀(𝑅)⁻² holds [3], but one expects the constant 𝐶_𝑠(𝑅) to be independent on the location of the point 𝑝, under some assumptions. In what follows, we will give an upper bound to 𝐶_𝑠(𝑅) independent of the point 𝑝.
Let 𝑀 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,𝛾_𝜉(𝑡) is the normalized geodesic from 𝜋(𝜉) with the initial velocity 𝜉. We define some notations as follows: 𝜏(𝜉)=sup𝜏>0∣𝛾_𝜉(𝑡)∈Ω,∀𝑡∈(0,𝜏). (1)
𝐶(𝜉) is the arc length from 𝜋(𝜉) to the cut locus point along 𝛾_𝜉(𝑡). Consider{(𝜉)(𝜉,𝑈𝑈Ω=𝜉∈𝑆Ω∣𝐶≥𝜏)}_𝑥=(𝜋∣𝑈Ω)⁻¹(𝑥);𝜔_𝑥=𝜇_𝑥𝑈_𝑥𝑐_𝑛−1;𝜔=inf_𝑥∈Ω𝜔_𝑥, (2) where 𝜇_𝑥 is the standard surface measure of the unit sphere,𝑐_𝑛−1 is denoted to be the area of the unit sphere 𝑆^𝑛−1.

Definition 3. Using the Notation above, 𝜔=inf_𝑥∈Ω𝜔_𝑥 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 𝐵_𝑝(𝑖/2) will reach the boundary 𝜕𝐵_𝑝(𝑖/2) before it reaches its cut locus, then the visibility angle of 𝐵_𝑝(𝑖/2) for any point 𝑝 which we denote by 𝜔(𝑖/2) satisfies 𝜔(𝑖/2)=1.

Lemma 4. Let 𝑀^𝑛 be a closed Riemannian manifold with Inj(𝑀)≥𝑖, then for any 𝑝∈𝑀, the following Sobolev inequality holds on 𝐵_𝑝(𝑖/2): ||𝑓||²_{2𝑛/(𝑛−2)}≤𝐶||∇𝑓||²₂, where 𝑓∈𝑊₀^1.2(𝐵_𝑝(𝑖/2)) and 𝐶=𝐶(𝑛).

Proof. Croke proved the following inequality [4]: Area(𝜕Ω)Vol(Ω)^1−(1/𝑛)≥𝑐_𝑛−1𝑐_𝑛/2^1−(1/𝑛)𝜔^1+(1/𝑛), (3) where Ω⊆𝐵_𝑝(𝑖/2),𝜕Ω∈𝐶^∞, and 𝜔 is just the visibility angle of the domain Ω.
As discussed above, we will have 𝜔(Ω)=1 if Ω⊆𝐵_𝑝(𝑖/2); then according to Croke’s inequality, we obtain 𝐶₀(𝑖/2)≥𝑐_𝑛−1/(𝑐_𝑛/2)^1−(1/𝑛). The relation between 𝐶₀(𝑖/2) and 𝐶_𝑠(𝑖/2) tells us that 𝐶_𝑠(𝑖/2)≤𝐶(𝑛), where 𝐶(𝑛) is a constant only depending on the dimension 𝑛.

Proposition 5. Let 𝑀^𝑛 be a closed 𝑛-dimensional Riemannian manifold with Inj(𝑀)≥𝑖, then for all 𝑝∈𝑀,Vol(𝐵_𝑝(𝑖/2))≥𝐶_𝑛𝑖^𝑛, where 𝐶_𝑛 is a constant only depending on the dimension 𝑛.

Proof. Also take the inequality of Croke Area(𝜕Ω)Vol(Ω)^1−(1/𝑛)≥𝑐_𝑛−1𝑐_𝑛/2^1−(1/𝑛)𝜔^1+(1/𝑛), (4) then the result can easily be derived from the fact that 𝜔(𝑖/2)=1 and Area(𝜕𝐵_𝑝(𝑟))=𝑑Vol(𝐵_𝑝(𝑟))/𝑑𝑟 after we integrate both sides of the inequality.

3. The First Eigenfunction and Eigenvalue

Let 𝑀^𝑛 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 inf_𝑥∈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. Let 𝑀^𝑛 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)≤𝐸_𝑛/𝑑²≤𝐸_𝑛/𝑖²≜𝐶₁(𝑛,𝑖) if Ric(𝑀)≥0, where 𝑑 is the diameter of the manifold and 𝐸_𝑛 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 𝑀^𝑛, 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 with Ric≥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. Let 𝑀^𝑛 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 1Vol(𝑀)_𝑀(𝑛−1)−Ric₋₊𝑑vol≤𝜖₀(𝑛,𝛿), (5) then the diameter will satisfy 𝑑<𝜋+𝛿. In particular, there exists 𝜖₁=𝜖₁(𝑛) such that if 1Vol(𝑀)_𝑀(𝑛−1)−Ric₋₊𝑑vol≤𝜖₁(𝑛), (6) 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∈ℕ, let 𝜖₁=𝜖₁(𝑛)>0 as above and suppose that 𝑀^𝑛 is a closed manifold with (𝑀)1Ric≥0,Vol(𝑀)_𝑀(𝑛−1)−Ric₋₊𝑑vol≤𝜖₁(𝑛); (7) 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 Δ𝑣=−𝜆₁(𝑀)𝑢−||𝑎+𝑢∇𝑣². (8)
Denote that 𝑄(𝑥)=|∇𝑣|²(𝑥), and we then have by the Ricci identity on manifolds with Ric(𝑀)≥0: Δ𝑄=2𝑣²_𝑖𝑗+2𝑣_𝑗𝑣_𝑗𝑖𝑖≥2𝑣²_𝑖𝑗+2⟨∇𝑣,∇Δ𝑣⟩. (9) For the term 𝑣²_𝑖𝑗, we have _𝑖,𝑗𝑣²_𝑖𝑗≥(Δ𝑣)²𝑛≥1𝑛𝑄²+2𝜆₁(𝑀)𝑢,𝑎+𝑢 (10) and for the term ⟨∇𝑣,∇Δ𝑣⟩, we have ⟨∇𝑣,∇Δ𝑣⟩=−𝑎𝜆₁(𝑀)⟨⟩.𝑎+𝑢𝑄−∇𝑣,∇𝑄 (11) Therefore, assume 𝑥₀∈𝑀 to be the maximum of 𝑄; then at 𝑥₀, we have 20≥𝑛𝑄𝑥₀+4𝜆₁(𝑀)𝑛−2(𝑛+2)𝑎𝜆₁(𝑀)𝑛().𝑎−1 (12) Therefore, 𝑄(𝑥)≤𝑄𝑥₀≤(𝑛+2)𝑎𝜆₁(𝑀).𝑎−1 (13)
Denote 𝛾 to be the minimizing unit speed geodesic joining the maximum and minimum points of 𝑢; then integrating 𝑄^1/2 along 𝛾, one will get:𝑎log𝑎−1≤log𝑎+max𝑢()𝑎−1≤𝑑𝑛+2𝑎𝜆₁(𝑀)𝑎−1^1/2. (14)
Let 𝑡=(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 Lemma 7, we can deduce that a positive constant 𝐶₃(𝑛) can be found such that 𝜆₁(𝑀)≥4𝑒⁻²(𝑛+2)𝑑²≥𝐶₃(𝑛), (15) 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|²≤𝐶₄(𝑛,𝑖).

Proof. Set 𝑎=2 in the (13) from above. Then applying Lemma 6, one obtains 19|∇𝑢|²≤|∇𝑣|²(𝑥)≤2𝐶₁(𝑛,𝑖)(𝑛+2); (16)
therefore, |∇𝑢|²≤𝐶₄(𝑛,𝑖). (17)

Proposition 10. Let 𝑀^𝑛 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 𝑀^𝑛 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 𝑀⁺={(𝑥)},𝑀𝑥∈𝑀∣𝑢>0₋={(𝑥)},𝑀𝑥∈𝑀∣𝑢<0₀={𝑥∈𝑀∣𝑢(𝑥)=0}. (18) 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 all 𝜙∈𝐶^∞(𝑀),𝜙≥0, integrating by parts we then have _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+𝜙Δ𝑢+𝜆₁(𝑀)𝑢−_𝜕𝑀−𝑢𝜕𝜙𝜕𝑛⁻−𝜙𝜕𝑢𝜕𝑛⁻−_𝑀−𝜙Δ𝑢+𝜆₁(𝑀)𝑢=_𝜕𝑀−𝜙𝜕𝑢𝜕𝑛⁻−_𝜕𝑀+𝜙𝜕𝑢𝜕𝑛⁺≥0, (19) 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 𝜙∈𝐶^∞₀(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 𝑀^𝑛 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 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 𝑝∈𝑀 such that 𝑢(𝑝)=−1.
For 𝑎≥1, denote 𝑅=𝑖/2; 𝜙 is a cut-off function on 𝐵_𝑝(𝑅), then we have by integrating by parts: 𝜆₁(𝑀)_{𝐵𝑝(𝑅)}𝜙²𝑣^2𝑎=−_{𝐵𝑝(𝑅)}𝜙²𝑣^2𝑎−1Δ𝑣=2_{𝐵𝑝(𝑅)}𝜙𝑣^2𝑎−1+(2𝑎−1)_{𝐵𝑝(𝑅)}𝜙²𝑣^2𝑎−2|∇𝑣|²≥2_{𝐵𝑝(𝑅)}𝜙𝑣^2𝑎−1+𝑎_{𝐵𝑝(𝑅)}𝜙²𝑣^2𝑎−2|∇𝑣|². (20) However, using the identity _{𝐵𝑝(𝑅)}∇𝜙𝑣^𝑎2=_{𝐵𝑝(𝑅)}∇𝜙²𝑣^2𝑎+2𝑎_{𝐵𝑝(𝑅)}𝜙𝑣^2𝑎−1∇𝜙,∇𝑣+𝑎²_{𝐵𝑝(𝑅)}𝜙²𝑣^2𝑎−2|∇𝑣|², (21) we have 𝜆₁(𝑀)𝑎_{𝐵𝑝(𝑅)}𝑣^2𝑎𝜙²≥_{𝐵𝑝(𝑅)}∇𝜙𝑣^𝑎2−_{𝐵𝑝(𝑅)}𝑣^2𝑎∇𝜙²; (22) therefore, using the Sobolev inequality in Lemma 4, 𝜆₁(𝑀)𝑎_{𝐵𝑝(𝑅)}𝑣^2𝑎𝜙²+_{𝐵𝑝(𝑅)}𝑣^2𝑎∇𝜙²≥_{𝐵𝑝(𝑅)}∇𝜙𝑣^𝑎2≥1𝐶_{𝐵𝑝(𝑅)}𝜙𝑣^{𝑎2𝑛/(𝑛−2)(𝑛−2)/𝑛}=1𝐶𝜙𝑣^𝑎2_{()2𝑛/𝑛−2}. (23)
Let 𝜙=1,in𝐵_𝑝𝜌,𝜌+𝜎−𝑟𝜎𝐵,in^𝑝𝜌+𝜎𝐵_𝑝𝜌,𝐵0,in^𝑝(𝑅)𝐵_𝑝.𝜌+𝜎 (24)
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: ‖𝑣‖_{2𝑎𝑛/(𝑛−2);𝐵𝑝(𝜌)}≤𝐶𝜆₁(𝑀)1𝑎+𝜎^21/2𝑎‖𝑣‖_{2𝑎;𝐵𝑝(𝜌+𝜎)}, (25) where we denote ||𝑓||_𝑝;Ω=(∫_Ω𝑓^𝑝)^1/𝑝 only for emphasizing the integral domain.
Set 𝑎_𝑗∶𝑎₀=1,𝑎₁=𝑛𝑛−2,…,𝑎_𝑗=𝑛𝑛−2^𝑗𝜎,…,_𝑗∶𝜎₀=𝑅4,𝜎₁=𝑅8,…,𝜎_𝑗=𝑅2^−(2+𝑗)𝜌,…,_𝑗∶𝜌₋₁=𝑅,…,𝜌_𝑗=𝑅−^𝑗_𝑠=0𝜎_𝑙,…. (26)
And putting {𝑎_𝑗},{𝜎_𝑗},{𝜌_𝑗} into (25), we can derive after iteration that ‖𝑣‖_{2𝑎𝑗+1;𝐵𝑝(𝜌𝑗)}≤𝐶(𝜆₁(𝑀)𝑎_𝑗+1𝜎²_𝑗^1/2𝑎𝑗‖𝑣‖_{2𝑎𝑗;𝐵𝑝(𝜌𝑗−1)}≤^𝑗_𝑠=0𝐶𝜆₁(𝑀)𝑎_𝑠+1𝜎²_𝑠^1/2𝑎𝑠‖𝑣‖_{2;𝐵𝑝(𝑅)}. (27)
Let 𝑗→+∞, then ‖𝑣‖_{∞;𝐵𝑝(𝑅/2)}≤^∞_𝑗=0𝐶(𝜆₁(𝑀)𝑎_𝑗+1𝜎²_𝑗^1/2𝑎𝑗‖𝑣‖_{2;𝐵𝑝(𝑅)}. (28)
The product can be estimated as follows: ^∞_𝑗=0𝐶𝜆₁(𝑀)𝑎_𝑗+1𝜎²_𝑗^1/2𝑎𝑗≤^∞_𝑗=0𝐶𝜆₁(𝑀)+16𝑅^21/2𝑎𝑗4^{𝑗/2𝑎𝑗}. (29)
The right hand will converge to a fixed number by using the fact that ∏^∞_𝑗=0𝐵^𝜇𝑗=𝐵^{𝜇/(𝜇−1)} and the fact ∑^∞_𝑗=0𝑗𝜇^−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 ‖𝑣‖_{∞;𝐵𝑝(𝑅/2)}≤𝐶₅^−1/2(𝑛,𝑖)‖𝑣‖_{2;𝐵𝑝(𝑅)}≤𝐶₅^−1/2(𝑛,𝑖)‖𝑣‖_2;𝑀. (30)
Therefore, _𝑀𝑢²≥𝐶₅(𝑛,𝑖). (31)

4. The Lower Bound of the First Eigenvalue

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

Theorem 12. For 𝑛∈ℕ, 𝑖,𝛿∈ℝ⁺, there is an 𝜖=𝜖(𝑛,𝑖,𝛿)>0 such that any closed manifold 𝑀^𝑛 with Ric(𝑀)≥0,Inj(𝑀)≥𝑖 and 1Vol(𝑀)_𝑀(𝑛−1)−Ric₋₊𝑑vol≤𝜖 (32) will satisfy that 𝜆₁(𝑀)≥𝑛−𝛿.

Proof. Assume that 𝑢 is the normalized eigenfunction of Laplacian on 𝑀^𝑛, let 𝑄(𝑥)=|∇𝑢|²+(𝜆₁(𝑀)/𝑛)𝑢², direct computation shows that 12Δ𝑄=𝑢²_𝑖𝑗+𝑢_𝑗𝑢_𝑖𝑖𝑗+𝑅_𝑖𝑗𝑢_𝑖𝑢_𝑗+𝜆₁(𝑀)𝑛|∇𝑢|²+𝜆₁(𝑀)𝑛≥𝑢Δ𝑢1−𝑛𝑛𝜆₁(𝑀)|∇𝑢|²+𝑅_𝑖𝑗𝑢_𝑖𝑢_𝑗. (33)
Integrating both sides on 𝑀^𝑛, we have 0≥1−𝑛𝑛𝜆₁(𝑀)_𝑀|∇𝑢|²𝑑vol+_𝑀𝑅_𝑖𝑗𝑢_𝑖𝑢_𝑗≥𝑑vol1−𝑛𝑛𝜆₁(𝑀)_𝑀|∇𝑢|²𝑑vol+_𝑀Ric₋|∇𝑢|²=𝑑vol1−𝑛𝑛𝜆₁(𝑀)_𝑀|∇𝑢|²𝑑vol+(𝑛−1)_𝑀|∇𝑢|²−𝑑vol_𝑀(𝑛−1)−Ric₋|∇𝑢|²𝑑vol; (34) therefore, 𝜆₁(𝑀)(𝑀)≥𝑛−Vol𝜆₁(𝑀)∫_𝑀|𝑢|²1𝑑vol(𝑀)×Vol_𝑀(𝑛−1)−Ric₋₊|∇𝑢|²𝑑vol (35)
if we suppose that 1Vol(𝑀)_𝑀(𝑛−1)−Ric₋₊𝑑vol≤𝜖₁(𝑛). (36) If 𝜖₁ is the one obtained in Lemma 7, then one has: 𝜆₁(𝑀)𝐶≥𝑛−²𝐶₄𝐶₃𝐶₅1Vol(𝑀)_𝑀(𝑛−1)−Ric₋₊𝑑vol. (37)
Finally, if one chooses 0<𝜖=𝜖(𝑛,𝑖,𝛿)≤min𝜖₁𝐶,𝛿³𝐶₅𝐶₂𝐶₄, (38) then 𝜆₁(𝑀)≥𝑛−𝛿 as long as (1/Vol(𝑀))∫_𝑀((𝑛−1)−Ric₋)₊𝑑vol≤𝜖, 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.