ESTIMATES ON EIGENVALUES OF LAPLACIAN

Qing-Ming Cheng* and Hongcang Yang**

Abstract. In this paper, we study eigenvalues of Laplacian on either a bounded connected domain in an n-dimensional unit sphere S^{n}(1), or a compact homogeneous Riemannian manifold, or an n-dimensional compact minimal submanifold in an N-dimensional unit sphere S^{N} (1). We estimate the k+1-th eigenvalue by the frist k eigenvalues. As a corollary, we obtain an estimate of difference between consecutive eigenvlaues. Our results are sharper than ones of P. C. Yang and Yau [25], Leung [19], Li [20] and Harrel II and Michel [12], respectively. From Weyl’s asymptotical formula, we know that our estimates are optimal in the sense of the order of k for eigenvalues of Laplacian on a bounded connected domain in an n-dimensional unit sphere S^{n}(1).

1. Introduction

Let M be a compact manifold with or without boundary. The spectrum of Laplacian on M is an important analytic invariant and has important geometric meanings (cf. Chavel [7]). There are many works denoted to estimating eigenvalues in term of other geometric qunatities of M. When M = Ω̄, where Ω is a connected bounded domain in a Euclidean space ℝ^{n}, an eigenvalue problem of Laplacian on a bounded domain Ω in ℝ^{n} is the following:

(1.1)  \{ \begin{matrix}Δu = −λu, in Ω, \\ u|_{∂Ω} = 0, \end{matrix}

where Δ is the Lapalacian in ℝ^{n}.

It is well known that this problem has a real and purely discrete spectrum

(1.2)  0 < λ_{1} < λ_{2} ≤ λ_{3} ≤ ⋯ → ∞.

Key words and phrases. eigenvalues of Laplacian, inequalitites of eigenvalues, comapct minimal hypersurface and a unit sphere.

2000 Mathematics Subject Classification: 35P15, 58G25, 53C42

* Research was partially supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science.

** Research was partially Supported by SF of CAS, Chinese NSF and NSF of USA.

Typeset by AMS-TEX

Here each eigenvalue is repeated from its multiplicity. When n = 2, in 1955 and 1956, Payne, Pólya and Weinberger proved that, in [21] and [22],

\frac{λ_{2}}{λ_{1}} ≤ 3 for Ω ⊂ ℝ^{2}

and they conjectured

(1.3)  \frac{λ_{2}}{λ_{1}} ≤ \frac{λ_{2}}{λ_{1}} |_{disk} ≈ 2.5387

with equality if and only if Ω is a disk. For general n ≥ 2, an analogous statement is

\frac{λ_{2}}{λ_{1}} ≤ 1 + \frac{4}{n} for Ω ⊂ ℝ^{n}

and the conjecture of Payne, Pólya and Weinberger is

(1.4)  \frac{λ_{2}}{λ_{1}} ≤ \frac{λ_{2}}{λ_{1}} |_{n-ball}

with equality if and only if Ω is an n-ball. In their excellent papers [3], [4] and [5], Ashbaugh and Benguria solved this important cconjecture of Payne, Pólya and Weinberger.

On the other hand, for estimates of higher eigenvalues, Payne, Pólya and Weinberger in [22] proved

(1.5)  λ_{k+1} − λ_{k} ≤ \frac{2}{k} ∑_{i=1}^{k} λ_{i}, k = 1, 2, ⋯

for Ω ⊂ ℝ^{2}. For general n ≥ 2, we have

(1.6)  λ_{k+1} − λ_{k} ≤ \frac{4}{kn} ∑_{i=1}^{k} λ_{i}, k = 1, 2, ⋯

for Ω ⊂ ℝ^{n}. Although these results introduced by Payne, Pólya and Weinberger have been extended by many authors, a result of Hile and Protter in [14] and a result of the second author in [23] are two main developments. Namely, in 1980, Hile and Protter [12] proved

(1.7)  ∑_{i=1}^{k} \frac{λ_{i}}{λ_{k+1} − λ_{i}} ≥ \frac{kn}{4}, for k = 1, 2, · · · .

It is not hard to check the inequality (1.7) of Hile and Protter is better than the inequality (1.5) of Payne, Pólya and Weinberger. In 1991, Yang [23] obtained very sharp inequalities, that is, he derived

(1.8)  ∑_{i=1}^{k}(λ_{k+1} − λ_{i})(λ_{k+1} − (1 + \frac{4}{n}λ_{i}) ≤ 0, for k = 1, 2, · · · .

According to the inequality, we can infer

(1.9)  λ_{k+1} ≤ \frac{1}{k} (1 + \frac{4}{n}) ∑_{i=1}^{k} λ_{i}, for k = 1, 2, · · · .

It is easy to prove that Yang’s inequalities (1.8) and (1.9) are sharper than the inequality (1.7) of Hile and Protter (see [1] and [2] in detail). For a recent development on estimates of eigenvalues for a clamped plate problem, see our manuscript [8].

When M is an n-dimensional compact manifold without boundary, the eigenvalue comparison theorem introduced by S. Y. Cheng [10] gives a computable sharp upper bound of the first eigenvalue λ_{1} in terms of the diameter and the lower bound of the Ricci curvature of M. In particular, If M is a Riemannian surface of genus g with area A, then P. C. Yang and Yau [25] proved λ_{1} ≤ 8π(1 + g)A^{−1}.

On the other hand, it is natural to extend the results on higher eigenvalues of Laplacian on a connected bounded domain in a Euclidean space ℝ^{n} introduced by Payne, Pólya and Weinberger [22], Hile and Protter [14] and Yang [23] to Riemannian manifolds. When M is a homogeneous Riemannian manifold, Li [20] proved

(1.10)  λ_{k+1} − λ_{k} ≤ \frac{2}{k + 1} (\sqrt{(∑_{i=1}^{k} λ_{i})^{2} + (k + 1) ∑_{i=1}^{k} λ_{i}λ_{1}} + ∑_{i=1}^{k} λ_{i}) + λ_{1}.

For compact minimal submanifolds in a unit sphere, P. C. Yang and Yau [25] studied a similar problem and made an estimate of difference between consecutive eigenvalues of Laplacian on such submanifolds. That is, they proved that if M is an n-dimensional compact minimal submanifold in a unit sphere S^{N}(1) and let λ_{i} denote the i-th eigenvalue of Laplacian on M, then

(1.11)  λ_{k+1} − λ_{k} ≤ n + \frac{2}{n(k + 1)} (\sqrt{(∑_{i=1}^{k} λ_{i})^{2} + n^{2}(k + 1) ∑_{i=1}^{k} λ_{i}} + ∑_{i=1}^{k} λ_{i}) .

Here we must remark that Leung [19] corrected a mistake in P. C. Yang and Yau [25] and improved their result. But it is very complicated looking version of (1.11). Furthermore, Harrel II and Michel ( see [12] and [13]) obtained an abstract algebraic inequality. As its applications, they made signaficant improvement to earlier estimates of differences of consecutive eigenvalues of Laplacian introduced by P. C. Yang and Yau [25], Leung [19], Li [20] and Harrel II [11]. Namely, they obtained that if M is an n-dimensional compact minimal submanifold in a unit sphere S^{N}(1) and let λ_{i} denote the i-th eigenvalue of Laplacian on M, then

(1.12)  λ_{k+1} − λ_{k} ≤ n + \frac{4}{n(k + 1)} ∑_{i=1}^{k} λ_{i}

and if M is an n-dimensional compact homogeneous Riemannian manifold and let λ_{i} denote the i-th eigenvalue of Laplacian on M, then

(1.13)  λ_{k+1} − λ_{k} ≤ \frac{4}{k + 1} ∑_{i=1}^{k} λ_{i} + λ_{1}.

In this paper, we shall study eigenvalue problems of Laplacian on domains of a unit sphere, on compact minimal submanifolds in a unit sphere and on compact homogeneous Riemannian manfolds. We shall give an estimate of the k + 1-th eigenvalue in terms of the first k eigenvalues. As a consequence, we give explicit estimates on the difference of any two consecutive eigenvalues of Laplacian on such objects, which are much sharper than ones of P. C. Yang and Yau [25], Leung [19], Li [20] and Harrel II and Michel [12], respectively.

Theorem 1. Let Ω be a bounded connected domain in a unit sphere S^{n}(1). We consider the following eigenvalue problem:

(1.14)  \{ \begin{matrix}Δu = −λu, in Ω, \\ u|_{∂Ω} = 0. \end{matrix}

Assume that λ_{i}, i = 1, 2, ⋯, is the i-th eigenvalue of the above eigenvalue problem. Then

(1.15)  \begin{aligned}λ_{k+1} ≤ & (1 + \frac{2}{n}) \frac{1}{k} ∑_{i=1}^{k} λ_{i} + \frac{n}{2} \\ & + [(\frac{2}{n} \frac{1}{k} ∑_{i=1}^{k} λ_{i} + \frac{n}{2})^{2} − (1 + \frac{4}{n}) \frac{1}{k} ∑_{j=1}^{k} (λ_{j} − \frac{1}{k} ∑_{i=1}^{k} λ_{i})^{2}]^{1/2} . \end{aligned}

Corollary 1. Under the assumptions of Theorem 1, we have

(1.16) λ_{k+1} − λ_{k} ≤2 [(\frac{2}{n} \frac{1}{k} ∑_{i=1}^{k} λ_{i} + \frac{n}{2})^{2} − (1 + \frac{4}{n}) \frac{1}{k} ∑_{j=1}^{k} (λ_{j} − \frac{1}{k} ∑_{i=1}^{k} λ_{i})^{2}]^{1/2} .

Remark 1. According to Theorem 1, we can infer, by a straightforward calculation,

(1.17)  λ_{k+1} ≤ C(n)k^{\frac{2}{n}} (λ_{1} + \frac{n^{2}}{4} ) ,

where C(n) ≤ 1 + \frac{4}{n} is a constant only depended on n. From the Weyl’s asymptotic formula of eigenvalues (see. [7] in details), we know that our estimate is optimal in the sense of the order on k.

For eigenvalues of Laplacian on compact homogeneous Riemannian manifolds without boundary, we have Theorem 2. Let M^{n} be an n-dimensional compact homogeneous Riemannian manifold without boundary. If λ_{i}, i = 0, 1, 2, ⋯, is the i-th eigenvalue of Δu = −λu on M^{n}, then

(1.18)  \begin{aligned}λ_{k+1} & ≤ \frac{3}{k + 1} ∑_{i=1}^{k} λ_{i} + \frac{λ_{1}}{2} \\ & + \frac{1}{2}\sqrt{(\frac{4}{k + 1} ∑_{i=1}^{k} λ_{i} + λ_{1})^{2} − \frac{20}{k + 1} ∑_{i=0}^{k} (λ_{i} − \frac{1}{k + 1} ∑_{j=1}^{k} λ_{j})^{2}}, \end{aligned}

where λ_{0} = 0.

Corollary 2. Under the assumptions of Theorem 2, we have

(1.19)  λ_{k+1} − λ_{k} ≤ \sqrt{(\frac{4}{k + 1} ∑_{i=1}^{k} λ_{i} + λ_{1})^{2} − \frac{20}{k + 1} ∑_{i=0}^{k} (λ_{i} − \frac{1}{k + 1} ∑_{j=1}^{k} λ_{j})^{2}},

where λ_{0} = 0.

Remark 2. Since λ_{k} ≥ \frac{1}{k} ∑_{i=1}^{k} λ_{i} holds, we have, from Theorem 2,

λ_{k+1} − λ_{k} ≤ \frac{2k − 1}{k(k + 1)} ∑_{i=1}^{k} λ_{i} + \frac{λ_{1}}{2}

+\frac{1}{2}\sqrt{(\frac{4}{k + 1} ∑_{i=1}^{k} λ_{i} + λ_{1})^{2} − \frac{20}{k + 1} ∑_{i=0}^{k} (λ_{i} − \frac{1}{k + 1} ∑_{j=1}^{k} λ_{j})^{2}}.

From (^{\frac{∑_{i=1}^{k} λ_{i}}{k+1}} )^{2} ≤ \frac{k}{k+1} ^{\frac{∑_{i=1}^{k} λ_{i}^{2}}{k+1}} and the inequality above, we infer

(1.20)  λ_{k+1} − λ_{k} < \frac{4k − 1}{k(k + 1)} ∑_{i=1}^{k} λ_{i} + λ_{1} < \frac{4}{k + 1} ∑_{i=1}^{k} λ_{i} + λ_{1}.

It is obvious that our estimate is sharper than ones of Li [20] and Harrel II and Stubbe [12]. In particular, when k = 1, from Theorem 2, we have

(1.21)  λ_{2} ≤ 3λ_{1}.

But the corresponding result of Harrel II and Michel [12] is λ_{2} ≤ (2 + \sqrt{3})λ_{1} and the estimate of Li [20] is λ_{2} ≤ 4λ_{1}. Hence, our estimate is much better than ones of Li in [20] and Harrel II and Michel [12]. Furthermore, when k = 1, our result in Theorem 2 can also be seen as a generalization of the estimate for eigenvalues in a domain in ℝ^{2} introduced by Payne, Pólya and Weinberger to compact homogeneous Riemannian manifolds without boundary.

Next, we consider the eigenvalue problem of Laplacian on an n-dimensional compact minimal submanifold without boundary in S^{n+1}(1). We have the following:

Theorem 3. Let M^{n} be an n-dimensional compact minimal submanifold without boundary in a unit sphere S^{N}(1). If λ_{i}, i = 1, 2, ⋯, is the i-th eigenvalue of Δu = −λu on M^{n}, then

(1.22) \begin{matrix}λ_{k+1} & ≤ (1 + \frac{2}{n}) \frac{1}{k + 1} ∑_{i=1}^{k} λ_{i} + \frac{n}{2} \\ + & [(\frac{n}{2} + \frac{2}{n} \frac{1}{k + 1} ∑_{i=1}^{k} λ_{i})^{2} − (1 + \frac{4}{n}) \frac{1}{k + 1} ∑_{j=0}^{k} (λ_{j} − \frac{1}{k + 1} ∑_{i=1}^{k} λ_{i})^{2}]^{1/2} , \end{matrix}

where λ_{0} = 0.

Corollary 3. Under the assumptions of Theorem 3, we have

\begin{matrix}λ_{k+1} − λ_{k} \\ ≤ 2 \{[\frac{2}{n} \frac{1}{k + 1} ∑_{i=0}^{k} λ_{i} + \frac{n}{2}]^{2} − (1 + \frac{4}{n}) \frac{1}{k + 1} ∑_{i=0}^{k} (λ_{i} − \frac{1}{k + 1} ∑_{j=0}^{k} λ_{j})^{2}\}^{1/2} . \end{matrix}

Remark 3. Making use of the same assertion as in Remark 2, we know that our result in Theorem 3 is much sharper than those results of P. C. Yang and Yau [25], Leung [19] and Harrel II and Michel [12]. From Theorem 3, we can also obtain the following estimate:

(1.23)  λ_{k+1} ≤ C(n)k^{\frac{2}{n}} λ_{1},

where C(n) ≤ n(1 + \frac{4}{n}) is a constant only depended on n.

Remark 4. We should notice that a part of our results was announced in [24]. But we did not publish them yet. As one knows, the importance of these results was emphasized by Asbaugh in [1] and [2], and by Harrell II and Stubbe in [13], we shall take the opportunity to publish them together .

2. Proofs of theorems

In this section, we shall prove our theorems.

Let Ω ⊂ S^{n}(1) be a connected domain in an n-dimensional unit sphere S^{n}(1). We consider the following eigenvalue problem:

(2.1)  \{ \begin{matrix}Δu = −λu, in Ω, \\ u|_{∂Ω} = 0. \end{matrix}

Proof of Theorem 1. Assume that u_{k} is an orthonormal eigenfunction corresponding to the k-th eigenvalue λ_{k}, i.e. u_{k} satisfies

(2.2)  \{ \begin{matrix}Δu_{k} = −λ_{k}u_{k}, in Ω, \\ u_{k}|_{∂Ω} = 0, \\ ∫_{Ω} u_{k}u_{j} = δ_{kj}, \end{matrix}

where Δ is the Laplacian of S^{n}(1). We have

0 < λ_{1} < λ_{2} ≤ ⋯ ≤ λ_{k+1} ≤ ….

Let x^{1}, x^{2}, ⋯, x^{n+1} be the standard coordinate functions of the Euclidean space ℝ^{n+1}. We know that S^{n}(1) = {(x^{1}, x^{2}, ⋯, x^{n+1}) ∈ ℝ^{n+1}; ∑_{j=1}^{n+1}(x^{j})^{2} = 1}. It is well known that x^{p} (p = 1, ⋯, n + 1) satisfy

(2.3)  Δx^{p} = −nx^{p}.

For a fixed p, 1 ≤ p ≤ n + 1, we put g = x^{p} and define φ_{i}, a_{ij}, and b_{ij} for i, j = 1, 2, ⋯, k by

(2.4)  \{ \begin{matrix}a_{ij} = ∫_{Ω} gu_{i}u_{j}, \\ φ_{i} = gu_{i} − ∑_{j=1}^{k} a_{ij}u_{j}, \\ b_{ij} = ∫_{Ω} u_{j}(∇u_{i} · ∇g + \frac{1}{2}u_{i}Δg), \end{matrix}

where ∇ denotes the gradient operator. Then, we have, from (2.2),

(2.5)  a_{ij} = a_{ji}, ∫_{Ω} φ_{i}u_{j} = 0, for j = 1, 2, ⋯, k.

Thus, we have

(2.6)  λ_{k+1} ≤ \frac{∫_{Ω} |∇φ_{i}|^{2}}{∫_{Ω} φ_{i}^{2}} .

According to Green’s formula, we infer

λ_{j}a_{ij} = ∫_{Ω} g(−Δu_{j})u_{i}
= ∫_{Ω}(−2u_{j}∇u_{i} · ∇g − Δgu_{i}u_{j} − gu_{j}Δu_{i})
= −2b_{ij} + λ_{i}a_{ij}.

Hence, we obtain

(2.7)  2b_{ij} = −2b_{ji} = (λ_{i} − λ_{j})a_{ij}.

From the definition of φ_{i}, (2.5) and a simple calculation, we have

\begin{aligned}Δφ_{i} & = gΔu_{i} + 2∇g · ∇u_{i} + Δgu_{i} − ∑_{j=1}^{k} a_{ij}Δu_{j} \\ & = −λ_{i}gu_{i} + 2∇g · ∇u_{i} − ngu_{i} + ∑_{j=1}^{k} a_{ij}λ_{j}u_{j}. \end{aligned}

Therefore, from (2.5) and the above equality, we have

(2.8)  \begin{aligned}∫_{Ω} |∇φ_{i}|^{2} & = − ∫_{Ω} φ_{i}(Δφ_{i}) \\ & = λ_{i} ∫_{Ω} φ_{i}^{2} − ∫_{M} φ_{i}(2∇g · ∇u_{i} − ngu_{i}). \end{aligned}

From (2.6) and (2.8), we infer

(2.9)  (λ_{k+1} − λ_{i})∥φ_{i}∥^{2} ≤ − ∫_{Ω} φ_{i}(2∇g · ∇u_{i} − ngu_{i}) ≡ w_{i},

where

∥v∥^{2} = ∫_{Ω} v^{2}.

Since

(2.10)  2 ∫_{Ω} u_{j}(∇u_{i} · ∇g + \frac{1}{2}u_{i}Δg) = 2b_{ij} = −2b_{ji} = (λ_{i} − λ_{j})a_{ij},

from φ_{i} = gu_{i} − ∑_{j=1}^{k} a_{ij}u_{j} and the definition of a_{ij}, we have

w_{i} = − ∫_{Ω}(gu_{i} − ∑_{j=1}^{k} a_{ij}u_{j})(2∇g · ∇u_{i} − ngu_{i}) (2.11)

= ∫_{Ω} \frac{1}{2}u_{i}^{2}Δg^{2} + n ∫_{Ω} g^{2}u_{i}^{2} + ∑_{j=1}^{k}(λ_{i} − λ_{j})a_{ij}^{2}.

On the other hand, from the definition of w_{i} and (2.5), we have

w_{i} = − ∫_{Ω} φ_{i}(2∇g · ∇u_{i} − ngu_{i} − 2 ∑_{j=1}^{k} b_{ij}u_{j}).

From Schwarz inequality, we obtain

(2.12)  (λ_{k+1} − λ_{i})w_{i}^{2} ≤ (λ_{k+1} − λ_{i})∥φ_{i}∥^{2}∥2∇g · ∇u_{i} − ngu_{i} − 2 ∑_{j=1}^{k} b_{ij}u_{j}∥^{2}.

From (2.9), (2.12) and the definition of b_{ij}, we infer

(λ_{k+1} − λ_{i})w_{i}^{2} ≤ w_{i} (4∥∇g · ∇u_{i}∥^{2} + n ∫_{Ω} ∇g^{2} · ∇u_{i}^{2} + n^{2}∥gu_{i}∥^{2} − 4 ∑_{j=1}^{k} b_{ij}^{2}) .

Hence,

(2.13)  (λ_{k+1} − λ_{i})w_{i} ≤ 4∥∇g · ∇u_{i}∥^{2} + n ∫_{Ω} ∇g^{2} · ∇u_{i}^{2} + n^{2}∥gu_{i}∥^{2} − 4 ∑_{j=1}^{k} b_{ij}^{2}.

Multiplying (2.13) by (λ_{k+1} − λ_{i}) and taking sum on i from 1 to k, we have

∑_{i=1}^{k}(λ_{k+1} − λ_{i})^{2}w_{i} ≤ −4 ∑_{i,j=1}^{k}(λ_{k+1} − λ_{i})b_{ij}^{2} (2.14)

+ ∑_{i=1}^{k}(λ_{k+1} − λ_{i})(4∥∇g · ∇u_{i}∥^{2} + n ∫_{Ω} ∇g^{2} · ∇u_{i}^{2} + n^{2}∥gu_{i}∥^{2}).

Since b_{ij} = −b_{ji}, we have b_{ii} = 0 and

4∥∇g · ∇u_{i}∥^{2} + n ∫_{Ω} ∇g^{2} · ∇u_{i}^{2} + n^{2}∥gu_{i}∥^{2} = ∥2∇g · ∇u_{i} − ngu_{i}∥^{2}.

In terms of the symmetry of a_{ij} and 2b_{ij} = (λ_{i} − λ_{j})a_{ij}, we have

(2.15)  \begin{matrix}∑_{i=1}^{k}(λ_{k+1} − λ_{i})^{2}w_{i} ≤ & ∑_{i=1}^{k}(λ_{k+1} − λ_{i}) (∥2∇g · ∇u_{i} − ngu_{i}∥^{2} \\ − & ∑_{i,j=1}^{k}(λ_{k+1} − λ_{i})(λ_{i} − λ_{j})^{2}a_{ij}^{2}. \end{matrix}

Further, from (2.11), the symmetry of a_{ij} and

∥u_{i}∇g∥^{2} = ∫_{Ω} \frac{1}{2}u_{i}^{2}Δg^{2} + n ∫_{Ω} g^{2}u_{i}^{2},

we have

∑_{i=1}^{k}(λ_{k+1} − λ_{i})^{2}w_{i} = ∑_{i=1}^{k}(λ_{k+1} − λ_{i})^{2}∥u_{i}∇g∥^{2}

+ ∑_{i,j=1} \frac{1}{2} [(λ_{k+1} − λ_{i})^{2} − (λ_{k+1} − λ_{j})^{2}] (λ_{i} − λ_{j})a_{ij}^{2}

= ∑_{j=1}^{k}(λ_{k+1} − λ_{i})^{2}∥u_{i}∇g∥^{2} − ∑_{i,j=1}(λ_{k+1} − λ_{i})(λ_{i} − λ_{j})^{2}a_{ij}^{2}.

Therefore, we infer, from (2.15) and the above equality,

(2.16)  ∑_{i=1}^{k}(λ_{k+1} − λ_{i})^{2}∥u_{i}∇g∥^{2} ≤ ∑_{i=1}^{k}(λ_{k+1} − λ_{i}) (∥2∇g · ∇u_{i} − ngu_{i}∥^{2})

For any fixed point x_{0} ∈ Ω, we can find a coordinate system (x̄^{1}, …, x̄^{n+1}) of S^{n}(1) such that at the point x_{0}

(2.17)  \begin{matrix}x̄^{1} = ⋯ = x̄^{n} = 0, x̄^{n+1} = 1, \\ ∇x̄^{n+1} = 0; ∇_{p}x̄^{q} = δ_{p}^{q}. (p, q = 1, …, n). \end{matrix}

In fact, we can choose a constant (n + 1) × (n + 1) orthonormal matrix (a_{j}^{i}) satisfying

∑_{α=1}^{n+1} a_{p}^{α}a_{q}^{α} = δ_{pq},

such that

x^{p} = ∑_{α=1}^{n+1} a_{α}^{p}x̄^{α}

and (2.17) is satisfied at the point x_{0}. Thus, at the point x_{0}, we have

∑_{p=1}^{n+1}(∇x^{p} · ∇u_{i})^{2} = ∑_{p,q,α=1}^{n+1} a_{p}^{α}a_{q}^{α}(∇x̄^{p} · ∇u_{i})(∇x̄^{q} · ∇u_{i})

= ∑_{p=1}^{n+1}(∇x̄^{p} · ∇u_{i})^{2} = ∑_{p=1}^{n}(∇_{p}u_{i})^{2}

= |∇u_{i}|^{2}.

Since x_{0} is an arbitrary point, we know that for any point x ∈ Ω,

(2.18)  ∑_{p=1}^{n+1}(∇x^{p} · ∇u_{i})^{2} = |∇u_{i}|^{2}.

On the other hand, since ∑_{p=1}^{n+1}(x^{p})^{2} = 1, Δx^{p} = −nx^{p}, we have

\begin{matrix}∑_{p=1}^{n+1} ∇(x^{p})^{2} = 0, \\ ∑_{p=1}^{n+1} |∇x^{p}|^{2} = − ∑_{p=1}^{n+1} x^{p}Δx^{p} = n. \end{matrix}

Since, for a fixed p, g = x^{p} and (2.16) is satisfied, taking sum on p from 1 to n + 1 for formula (2.16), we obtain

(2.19)  n ∑_{i=1}^{k}(λ_{k+1} − λ_{i})^{2} ≤ ∑_{i=1}^{k}(λ_{k+1} − λ_{i})(4λ_{i} + n^{2}).

Since the formula (2.19) is a quadratic inequality of λ_{k+1}, we obtain

(2.20)  \begin{matrix}λ_{k+1} ≤ (1 + \frac{2}{n}) \frac{1}{k} ∑_{i=1}^{k} λ_{i} + \frac{n}{2} \\ + \{[\frac{2}{n} \frac{1}{k} ∑_{i=1}^{k} λ_{i} + \frac{n}{2}]^{2} − (1 + \frac{4}{n}) \frac{1}{k} ∑_{i=1}^{k} (λ_{i} − \frac{1}{k} ∑_{j=1}^{k} λ_{j})^{2}\}^{1/2} . \end{matrix}

This completes the proof of Theorem 1.

Proof of Corollary 1. Since k is an any integer, we know that (2.19) is also true if we replace k + 1 with k, that is, we can obtain

n ∑_{i=1}^{k−1}(λ_{k} − λ_{i})^{2} ≤ ∑_{i=1}^{k−1}(λ_{k} − λ_{i})(4λ_{i} + n^{2}).

Hence, we have

(2.21)  n ∑_{i=1}^{k}(λ_{k} − λ_{i})^{2} ≤ ∑_{i=1}^{k}(λ_{k} − λ_{i})(4λ_{i} + n^{2}).

Namely, λ_{k} also satisfies the same quadratic inequality. Therefore, we infer

(2.22)  \begin{matrix}λ_{k} ≥ (1 + \frac{2}{n}) \frac{1}{k} ∑_{i=1}^{k} λ_{i} + \frac{n}{2} \\ − \{[\frac{2}{n} \frac{1}{k} ∑_{i=1}^{k} λ_{i} + \frac{n}{2}]^{2} − (1 + \frac{4}{n}) \frac{1}{k} ∑_{i=1}^{k} (λ_{i} − \frac{1}{k} ∑_{j=1}^{k} λ_{j})^{2}\}^{1/2} . \end{matrix}

Thus, we obtain an estimate of λ_{k+1} − λ_{k}:

(2.23) λ_{k+1} − λ_{k} ≤2 [(\frac{2}{n} \frac{1}{k} ∑_{i=1}^{k} λ_{i} + \frac{n}{2})^{2} − (1 + \frac{4}{n}) \frac{1}{k} ∑_{j=1}^{k} (λ_{j} − \frac{1}{k} ∑_{i=1}^{k} λ_{i})^{2}]^{1/2} .

This means that Corollary 1 is true.

Proof of Theorem 2. Let {g_{p}}_{p=1}^{l} be an orthonormal basis of the first eigenspace E_{λ_{1}} of the eigenvlaue problem

Δf = −λf, on M,

that is, {g_{p}}_{p=1}^{l} satisfies

Δg_{p} = −λ_{1}g_{p}, on M.

Since M is an n-dimensional compact homogeneous Riemannian manifold without boundary, we know that ∑_{p=1}^{l} g_{p}^{2} = α^{2} is constant (cf. Proposition 1 of Li [20]). Let {u_{i}}_{i=0}^{k} be the set of the first k + 1-th orthonormal eigenfunctions. Then, u_{i} satisfies

(2.24)  \{ \begin{matrix}Δu_{i} = −λ_{i}u_{i}, on M, \\ ∫_{Ω} u_{i}u_{j} = δ_{ij}, for 0 ≤ i, j ≤ k, \end{matrix}

where Δ is the Laplacian of M. For a fixed p, 1 ≤ p ≤ l, we define φ_{i}, a_{ij}, and b_{ij} for i, j = 0, 2, ⋯, k by

(2.25)  \{ \begin{matrix}a_{ij} = ∫_{Ω} g_{p}u_{i}u_{j}, \\ φ_{i} = g_{p}u_{i} − ∑_{j=0}^{k} a_{ij}u_{j}, \\ b_{ij} = ∫_{Ω} u_{j}(∇u_{i} · ∇g_{p} + \frac{1}{2}u_{i}Δg_{p}), \end{matrix}

where ∇ denotes the gradient operator. Making use of the same assertions as in the proof of Theorem 1, we can obtain

(2.26)  ∑_{i=0}^{k}(λ_{k+1} − λ_{i})^{2}∥u_{i}∇g_{p}∥^{2} ≤ ∑_{i=0}^{k}(λ_{k+1} − λ_{i}) (∥2∇g_{p} · ∇u_{i} − λ_{1}g_{p}u_{i}∥^{2}) .

Taking sum on p from 1 to k, we have

(2.27) ∑_{i=0}^{k}(λ_{k+1} − λ_{i})^{2} ∑_{p=1}^{l} ∥u_{i}∇g_{p}∥^{2} ≤ ∑_{i=0}^{k}(λ_{k+1} − λ_{i}) ∑_{p=1}^{l} (∥2∇g_{p} · ∇u_{i} − λ_{1}g_{p}u_{i}∥^{2}) .

Since ∑_{p=1}^{l} g_{p}^{2} = α^{2} is constant, we infer

(2.28) ∑_{p=1}^{l} g_{p}∇g_{p} = 0, ∑_{p=1}^{l} ∇g_{p} · ∇g_{p} = − ∑_{p=1}^{l} g_{p}Δg_{p} = λ_{1}α^{2}.

Hence, we infer

(2.29) ∑_{p=1}^{l} ∥u_{i}∇g_{p}∥^{2} = λ_{1}α^{2}

and

∑_{p=1}^{l} ∥2∇g_{p} · ∇u_{i} − λ_{1}g_{p}u_{i}∥^{2} (2.30)

= ∑_{p=1}^{l} ∫_{M}{4(∇g_{p} · ∇u_{i})^{2} + λ_{1}^{2}g_{p}^{2}u_{i}^{2}}
= ∫_{M} ∑_{p=1}^{l} 4(∇g_{p} · ∇u_{i})^{2} + λ_{1}^{2}α^{2}.

Since (∇g_{p} · ∇u_{i})^{2} ≤ |∇g_{p}|^{2}|∇u_{i}|^{2} and ∑_{p=1}^{l} |∇g_{p}|^{2} = λ_{1}α^{2} hold, we infer, from (2.29) and (2.30)

∑_{p=1}^{l} ∥2∇g_{p} · ∇u_{i} − λ_{1}g_{p}u_{i}∥^{2} (2.31)

≤ ∫_{M} 4λ_{1}α^{2}|∇u_{i}|^{2} + λ_{1}^{2}α^{2}
=4λ_{1}α^{2}λ_{i} + λ_{1}^{2}α^{2}.

From λ_{1}α^{2} > 0, (2.27), (2.29) and (2.31), we have

(2.32) ∑_{i=0}^{k}(λ_{k+1} − λ_{i})^{2} ≤ ∑_{i=0}^{k}(λ_{k+1} − λ_{i})(4λ_{i} + λ_{1}).

This is a quadratic inequality of λ_{k+1}. Therefore, we infer

λ_{k+1} ≤ \frac{3}{k + 1} ∑_{i=0}^{k} λ_{i} + \frac{λ_{1}}{2} + \frac{1}{2}\sqrt{36(\frac{∑_{i=0}^{k} λ_{i}}{k + 1} )^{2} + 8λ_{1} \frac{∑_{i=0}^{k} λ_{i}}{k + 1} − 20\frac{∑_{i=0}^{k} λ_{i}^{2}}{k + 1} + λ_{1}^{2}}.

From λ_{0} = 0, we know that Theorem 2 is true.

Proof of Corollary 2. By using the same proof as in the Corollary 1, Corollary 2 is obvious.

From now, we shall assume that M is an n-dimensional compact minimal submanifold without boundary in S^{N}(1). We consider an eigenvalue problem of Laplacian on M, that is, we consider

Δu = −λu.

We know that this problem has a real and purely discrete spectrum

(2.33)  0 = λ_{0} < λ_{1} ≤ λ_{2} ≤ λ_{3} ≤ ⋯ → ∞.

Here each eigenvalue is repeated from its multiplicity.

Proof of Theorem 3. Let x^{1}, x^{2}, ⋯, x^{N+1} be the standard coordinate functions of ℝ^{N+1}. Thus, we have S^{N}(1) = {(x^{1}, x^{2}, ⋯, x^{N+1}) ∈ ℝ^{N+1}; ∑_{j=1}^{N+1}(x^{j})^{2} = 1}. It is well known that x^{1}, x^{2}, ⋯, x^{N+1} are eigenfunctions of Laplacian on M correspanding to eigenvalue n, namely,

(2.34)  Δx^{p} = −nx^{p}, p = 1, 2, ⋯, n, ⋯, N, N + 1.

Since M is a compact minimal submanifold without boundary and the above formulas hold, making use of the same assertion as in the proof of Theorem 1, we can infer

(2.35)  ∑_{i=0}^{k}(λ_{k+1} − λ_{i})^{2}∥u_{i}∇g∥^{2} ≤ ∑_{i=0}^{k}(λ_{k+1} − λ_{i}) (2∥∇g · ∇u_{i} − ngu_{i}∥^{2})

where, for a fixed p, 1 ≤ p ≤ N + 1, g = x^{p}. We should notice that in this case, we must consider the eigenfunction corresponding to the eigenvalue λ_{0} = 0. Taking the same assertion as in the proof of Theorem 1, we can conclude

(2.36)  ∑_{p=1}^{N+1}(∇x^{p} · ∇u_{i})^{2} = |∇u_{i}|^{2}

Since ∑_{p=1}^{N+1}(x^{p})^{2} = 1, Δx^{p} = −nx^{p}, we have

\begin{matrix}∑_{p=1}^{N+1} ∇(x^{p})^{2} = 0, \\ ∑_{p=1}^{N+1} |∇x^{p}|^{2} = − ∑_{p=1}^{N+1} x^{p}Δx^{p} = n. \end{matrix}

Since, for a fixed p, g = x^{p} and (2.36) is satisfied, taking sum on p from 1 to N + 1 for formula (2.35), we obtain

(2.37)  n ∑_{i=0}^{k}(λ_{k+1} − λ_{i})^{2} ≤ ∑_{i=0}^{k}(λ_{k+1} − λ_{i})(4λ_{i} + n^{2}).

Thus, we obtain

(2.38) \begin{matrix}λ_{k+1} ≤ (1 + \frac{2}{n}) \frac{1}{k + 1} ∑_{i=0}^{k} λ_{i} + \frac{n}{2} \\ + \{[\frac{2}{n} \frac{1}{k + 1} ∑_{i=0}^{k} λ_{i} + \frac{n}{2}]^{2} − (1 + \frac{4}{n}) \frac{1}{k + 1} ∑_{i=0}^{k} (λ_{i} − \frac{1}{k + 1} ∑_{j=0}^{k} λ_{j})^{2}\}^{1/2} . \end{matrix}

This completes the proof of Theorem 3.

Remark 5. Let k = 0 in (2.38), we obtain a well known fact

λ_{1} ≤ n.

Proof of Corollary 3. From the same assertion, we know that Corollary 3 is true.

References

1. M. S. Ashbaugh, Isoperimetric and universal inequalities for eigenvalues, in Spectral theory and geometry (Edinburgh, 1998), E B Davies and Yu Safalov eds., London Math. Soc. Lecture Notes, vol. 273 (1999), Cambridge Univ. Press, Cambridge, 95-139.

2. M. S. Ashbaugh, Universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Prottter and H.C. Yang, Proc. Indian Acad. Sci. Math. Sci. 112 (2002), 3-30.

3. M. S. Ashbaugh and R. D. Benguria, Proof of the Payne-Pólya-Weinberger conjecture, Bull. Amer. Math. Soc. 25 (1991), 19-29.

4. M. S. Ashbaugh and R. D. Benguria, A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions, Ann. of Math. 135 (1992), 601-628.

5. M. S. Ashbaugh and R. D. Benguria, a second proof of the Payne-Pólya-Weinberger conjecture, Commun. Math. Phys. 147 (1992), 181-190.

6. J. J. A. M. Brands, Bounds for the ratios of the first three membrane eigenvalues, Arch. Rational Mech. Anal. 16 (1964), 265-268.

7. I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, New York, 1984.

8. Q. -M. Cheng and H. C. Yang, Inequalities of eigenvalues for a clamped plate problem, Preprint.

9. Q. -M. Cheng and H. C. Yang, Inequalities of eigenvalues on Laplacian on domains and compact hypersurfaces in complex projective spaces, Preprint.

10. S. Y. Cheng, Eigenfunctions and eigenvalues of Laplacian, Proc. Symp. Pure Math. vol. 27 part 2, Differntial Geometry, S. S. Chern and R. Osserman, editors, Amer. Math. Soc., Providence, Rhode Island, 1975, 185-193.

11. E. M. Harrell II, Some geometric bounds on eigenvalue gaps, Comm. in Part. Diff. Eqns 18 (1993), 179-198.

12. E. M. Harrell II and P. L. Michel, Commutator bounds for eigenvalues with applications to spectral geometry, Comm. in Part. Diff. Eqns 19 (1994), 2037-2055.

13. E. M. Harrell II and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997), 1797-1809.

14. G. N. Hill and M. H. Protter, Inequalities for eigenvalues of Laplacian, Indiana Univ. Math. 29 (1980), 523–538.

15. G. N. Hile and R. Z. Yeh, Inequalities for eigenvalues of the biharmonic operator, Pac. J. Math. 112 (1984), 115-133.

16. S. M. Hook, Domain independent upper bounds for eigenvalues of elliptic operator, Trans. Amer. Math. Soc. 318 (1990), 615-642.

17. V. Ya. Ivrii, Second term of the spectrual asymptotic expansion of the Laplace-Beltrami operator on manifolds with boundary, Funct. Analy. Appl. 14(2) (1980), 98-105.

18. J. M. Lee, The gaps in the spectrum of the Laplace-Beltrami operator, Houston J. Math. 17 (1991), 1-24.

19. P. -F. Leung, On the consecutive eigenvalues of the Laplacain of a compact minimal submanifold in a sphere, J. Austral. Math. Soc. 50 (1991), 409-426.

20. P. Li, Eigenvalue estimates on homogeneous manifolds, Comment. Math. Helvetici 55 (1980), 347-363.

21. L. E. Payne, G. Polya and H. F. Weinberger, Sur le quotient de deux fréquences propres consécutives, Comptes Rendus Acad. Sci. Paris 241 (1955), 917-919.

22. L. E. Payne, G. Polya and H. F. Weinberger, On the ratio of consecutive eigenvalues, J. Math. and Phys. 35 (1956), 289 -298.

23. H. C. Yang, An estimate of the differance between consecutive eigenvalues, Preprint IC/91/60 of ICTP, Trieste, 1991.

24. H. C. Yang, Estimates of the differance between consecutive eigenvalues, Preprint of ICTP, Trieste, 1995.

25. P. C. Yang and S. T. Yau, Eigenvalues of the Laplacian of compact Riemannian surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa CI. Sci. 7 (1980), 55-63.

26. S. T. Yau and R. Schoen, Differential Geometry, Science Press, Beijing, 1988, pp. 142– 145.

Qing-Ming Cheng

Department of Mathematics

Faculty of Science and Engineering

Saga University, Saga 840-8502, Japan

cheng@math.saga-u.ac.jp

and

Hongcang Yang

Academy of Mathematics and Systematical Sciences CAS, Beijing 100080, China e-mail: yanghc@math03.math.ac.cn