INEQUALITIES FOR EIGENVALUES OF LAPLACIAN

ON DOMAINS AND COMPACT COMPLEX

HYPERSURFACES IN COMPLEX PROJECTIVE SPACES

Qing-Ming Cheng* and Hongcang Yang**

Abstract. It is well known that the spectrum of Laplacian on a compact Riemannian manifold M is an important analytic invariant and has important geometric meanings. There are many mathematicians to investigate properties of the spectrum of Laplacian and to estimate the spectrum in term of the other geometric quantities of M. When M is a bounded domain in Euclidean spaces, a compact homogeneous Riemannian manifold, a bounded domain in the standard unit sphere or a compact minimal submanifold in the standard unit sphere, the estimates of the k + 1-th eigenvalue was given by the first k eigenvalues (see [9], [12], [19], [20], [22], [23], [24] and [25]). In this paper, we shall consider the eigenvalue problem of the Laplacian on compact Riemannian manifolds. First of all, we shall give a general inequality of eigenvalues. As its applications, we study the eigenvalue problem of the Laplacian on a bounded domain in the standard complex projective space ℂℙ^{n}(4) and on a compact complex hypersurface without boundary in ℂℙ^{n}(4). We shall give an explicit estimate of the k + 1-th eigenvalue of Laplacian on such objects by its first k eigenvalues.

1. Introduction

In this paper, we consider the eigenvalue problem of the Laplacian on a compact Riemannian manifold M with boundary (possible empty):

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

where Δ is the Lapalacian on M.

Key words and phrases. Eigenvalues and eigenfunctions, Laplacian, complex projective space and complex hypersurface.

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

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

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

Typeset by AMS-TEX

This problem has a real and purely discrete spectrum

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

We put λ_{0} = 0 if ∂M = ∅. Here each eigenvalue is repeated from its multiplicity.

It is well known that the spectrum of Laplacian on M is an important analytic invariant and has important geometric meanings (cf. Chavel [7]). There are many mathematicians to investigate properties of spectrum of Laplacian and to estimate the spectrum in term of the other geometric quantities of M. First of all, we will review several important results on estimates of eigenvalues.

(1). When M = Ω̄, where Ω is a connected bounded domain in the Euclidean space ℝ^{n}, the main contributions on estimates of higher eigenvalues are obtained by Payne, Pólya and Weinberger [22] and Thompson [23], Hile and Protter [14] and Yang [24]. The following very sharper inequality was obtained by Yang [24]

(1.3)  ∑_{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, he also obtained (cf. [24])

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

We must remark that when k = 1, Ashbaugh and Benguria in [3], [4] and [5] gave an optimal estimate of λ_{2} by λ_{1}.

(2). When M is an n-dimensional compact homogeneous Riemannian manifold, Li [20] estimated the difference of any two consecutive eigenvalues of Laplacian on M. He proved

(1.5)  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}.

Furthermore, Harrel II and Michel [12] improved the estimates introduced by Li [20]. Namely, they obtained that

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

Recently, the authors [9] obtained

λ_{k+1} ≤ \frac{3}{k + 1} ∑_{i=1}^{k} λ_{i} + \frac{λ_{1}}{2} (1.7)

+ \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}},

where λ_{0} = 0. Since λ_{k} ≥ \frac{1}{k} ∑_{i=1}^{k} λ_{i} holds, we infered, from the above inequality,

λ_{k+1} − λ_{k} ≤ \frac{2k − 1}{k(k + 1)} ∑_{i=1}^{k} λ_{i} + \frac{λ_{1}}{2} (1.8)

+\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 obtained, in [9],

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}.

(3). When M is a compact minimal submanifold in the standard unit sphere S^{N}(1) , P. C. Yang and Yau [25] obtained

(1.9)  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}) .

(cf. Leung [19] ). Furthermore, Harrel II and Michel [12] proved

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

In [9], we gave an estimate of the k + 1-th eigenvalue in terms of the first k eigenvalues:

λ_{k+1} ≤ (1 + \frac{2}{n}) \frac{1}{k + 1} ∑_{i=0}^{k} λ_{i} + \frac{n}{2}

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

where λ_{0} = 0. From λ_{k} ≥ \frac{1}{k} ∑_{i=1}^{k} λ_{i}, it is obvious that our inequality above is much sharper than these inequalities (1.9) and (1.10) of P. C. Yang and Yau [25] and Harrel II and Michel [12].

(4). When M is a connected bounded domain of S^{n}(1), in [9], we obtained

\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} ∑_{j=1}^{k} (λ_{j} − \frac{1}{k} ∑_{i=1}^{k} λ_{i})^{2}]^{1/2} \end{matrix}

and

\begin{matrix}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} . \end{matrix}

In this paper, we shall consider the eigenvalue problem of the Laplacian on compact Riemannian manifolds. In section 2, we shall give a general inequality of eigenvalues for the eigenvalue problem (1.1). Namely, when M is a compact Riemannian manifold with boundary ∂M, for any function g ∈ C^{3}(M) ∩ C^{2}(∂M) and any integer k, we obtained (see Proposition 1)

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

where u_{i} is the orthonormal eigenfunction corresponding to eigenvalue λ_{i} of (1.1). As applications of the above inequality, in section 3, we study the eigenvalue problem of the Laplacian on a connected bounded domain of the standard complex projective space ℂℙ^{n}(4) with holomorphic sectional curvature 4. We proved (see Theorem 1)

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

In section 4, for a compact complex hypersurface without boundary in ℂℙ^{n+1}(4), we derived (see Theorem 2)

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

Acknowledgement. We like to express our gratitude to the referee for the valuable suggestions and comments.

2. A general Inequality of Eigenvalues

Let M be an n-dimensional compact Riemannian manifold with boundary ∂M (possible empty). We consider the following eigenvalue problem:

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

where Δ is the Laplacian of M. We shall prove a general inequality of eigenvalues for the above eigenvalue problem which will be applied to prove Theorems 1 and 2.

Proposition 1. Let λ_{i} be the i-th eigenvalue of the above eigenvalue problem (2.1) and u_{i} be the orthonormal eigenfunction corresponding to λ_{i}, that is, u_{i} satisfies

Δu_{i} = −λ_{i}u_{i}, in M,
u_{i}|_{∂M} = 0,
∫_{M} u_{i}u_{j} = δ_{ij}, for any i, j = 1, 2, · · · .

Then, for any function g ∈ C^{3}(M) ∩ C^{2}(∂M) and any integer k, we have

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

where

∥f∥^{2} = ∫_{M} f^{2}.

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

(2.2)  \{ \begin{matrix}Δu_{i} = −λ_{i}u_{i}, in M, \\ u_{i}|_{∂M} = 0, \\ ∫_{M} u_{i}u_{j} = δ_{ij}. \end{matrix}

We have

(2.3)  0 < λ_{1} ≤ λ_{2} ≤ ⋯ ≤ λ_{k+1} ≤ · · · .

Define φ_{i}, a_{ij} and b_{ij}, for i, j = 1, ⋯, k, by

(2.4)  \{ \begin{matrix}a_{ij} = ∫_{M} gu_{i}u_{j}, \\ φ_{i} = gu_{i} − ∑_{j=1}^{k} a_{ij}u_{j}, \\ b_{ij} = ∫_{M} 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}, ∫_{M} φ_{i}u_{j} = 0, for j = 1, 2, ⋯, k.

Thus, we have

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

According to Stokes formula, we infer

λ_{j}a_{ij} = ∫_{M} g(−Δu_{j})u_{i}
= ∫_{M}(−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} and a simple calculation, we have

\begin{aligned}Δφ_{i} & = gΔu_{i} + 2∇g · ∇u_{i} + u_{i}Δg − ∑_{j=1}^{k} a_{ij}Δu_{j} \\ & = −λ_{i}gu_{i} + 2∇g · ∇u_{i} + u_{i}Δg + ∑_{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}∫_{M} |∇φ_{i}|^{2} & = − ∫_{M} φ_{i}(Δφ_{i}) \\ & = λ_{i} ∫_{M} φ_{i}^{2} − ∫_{M} φ_{i}(2∇g · ∇u_{i} + u_{i}Δg). \end{aligned}

From (2.6) and (2.8), we infer

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

Since

(2.10)  2 ∫_{M} 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} = − ∫_{M}(gu_{i} − ∑_{j=1}^{k} a_{ij}u_{j})(2∇g · ∇u_{i} + u_{i}Δg) (2.11)

= ∥u_{i}∇g∥^{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} = − ∫_{M} φ_{i}(2∇g · ∇u_{i} + u_{i}Δg − 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} + u_{i}Δg − 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} (∥2∇g · ∇u_{i} + u_{i}Δg∥^{2} − 4 ∑_{j=1}^{k} b_{ij}^{2}) .

Hence,

(2.13)  (λ_{k+1} − λ_{i})w_{i} ≤ ∥2∇g · ∇u_{i} + u_{i}Δg∥^{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})(∥2∇g · ∇u_{i} + u_{i}Δg∥^{2}.

In terms of 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} + u_{i}Δg∥^{2} \\ − & ∑_{i,j=1}^{k}(λ_{k+1} − λ_{i})(λ_{i} − λ_{j})^{2}a_{ij}^{2}. \end{matrix}

Further, from (2.11) and the symmetry of a_{ij}, 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}^{k} \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}^{k}(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} + u_{i}Δg∥^{2}.

This completes the proof of Proposition 1.

3. Eigenvalues of Laplacian on a domain in ℂℙ^{n}(4)

In this section, we shall consider the eigenvalue problem of the Laplacian on a domain in ℂℙ^{n}(4), that is, let M be a connected domain in ℂℙ^{n}(4) with boundary ∂M, we consider the following:

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

where Δ is the Laplacian of ℂℙ^{n}(4).

Theorem 1. Let M be a connected bounded domain in the standard n-dimensional complex projective space ℂℙ^{n}(4). Assume that λ_{i} is the i-th eigenvalue of the above eigenvalue problem (3.1). Then, we have

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

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

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

Proof of Theorem 1. Let Z = (Z^{0}, Z^{1}, …, Z^{n}) be a homogeneous coordinate system of ℂℙ^{n}(4), (Z^{p} ∈ ℂ). Defining f_{pq̄}, for p, q = 0, 1, ⋯, n, by

(3.3)  f_{pq̄} = \frac{Z^{p}\overline{Z^{q}}}{∑_{r=0}^{n} Z^{r}\overline{Z^{r}}} ,

we have

(3.4)  f_{p\overline{q}} = \overline{f_{q\overline{p}}}, ∑_{p,q=0}^{n} f_{p\overline{q}}\overline{f_{p\overline{q}}} = 1.

For any fixed point P ∈ M, we can choose a new homogeneous coordinate system of ℂℙ^{n}(4) such that, at P

(3.5)  Z̃^{0} ≠ 0, Z̃^{1} = ⋯ = Z̃^{n} = 0

and

(3.6)  Z^{p} = ∑_{r=0}^{n} C_{pr}Z̃^{r},

where C = (C_{pq}) ∈ U(n + 1) is an (n + 1) × (n + 1)-unitary matrix, that is, C_{pq} satisfies

(3.7)  ∑_{s=0}^{n} C_{sp}\overline{C}_{sq} = ∑_{s=0}^{n} C_{ps}\overline{C}_{qs} = δ_{pq}.

Then, we know that z = (z^{1}, …, z^{n}), z^{p} = Z̃^{p}/Z̃^{0}, is a local holomorphic coordinate system of ℂℙ^{n}(4) in a neighborhood U of the point P ∈ M and satisfies, at P,

(3.8)  z^{1} = ⋯ = z^{n} = 0.

Hence, we infer, for p, q = 0, 1, ⋯, n,

(3.9)  \begin{matrix}f̃_{p\overline{q}} = \begin{matrix}Z̃^{p} & \overline{Z̃^{q}} \\ ∑ & {n \atop r=0} & Z̃^{r} & \overline{Z̃^{r}} \end{matrix} = \frac{z^{p}\overline{z^{q}}}{1 + ∑_{r=1}^{n} z^{r}\overline{z}^{r}} , \\ f_{p\overline{q}} = ∑_{r,s=0}^{n} C_{pr}\overline{C_{qs}}f̃_{r\overline{s}}, z^{0} ≡ 1. \end{matrix}

Putting G_{p\overline{q}} = Re(f_{p\overline{q}}) and F_{p\overline{q}} = Im(f_{p\overline{q}}), for p, q = 0, 1, …, n, then, we have

(3.10)  \begin{matrix}∑_{p,q=0}^{n}(G_{p\overline{q}}^{2} + F_{p\overline{q}}^{2}) = ∑_{p,q=0}^{n} f_{p\overline{q}}\overline{f_{p\overline{q}}} = ∑_{p,q=0}^{n} f̃_{p\overline{q}}\overline{f̃_{p\overline{q}}} = 1, \\ ∑_{p,q=0}^{n}(G_{p\overline{q}}∇G_{p\overline{q}} + F_{p\overline{q}}∇F_{p\overline{q}}) = 0. \end{matrix}

Let ds^{2} = ∑_{p,q=1}^{n} g_{p\overline{q}}dz^{p}d\overline{z^{q}} is the Fubini-Study metric of ℂℙ^{n}(4). Then,

(3.11)  g_{p\overline{q}} = \frac{δ_{p\overline{q}}}{1 + ∑_{r=1}^{k} |z^{r}|^{2}} − \frac{z^{q}\overline{z^{p}}}{(1 + ∑_{r=1}^{n} |z^{r}|^{2})^{2}} .

(3.12)  \begin{matrix}(g_{p\overline{q}})^{−1} = (g^{p\overline{q}}), \\ g^{p\overline{q}} = (1 + ∑_{r=1}^{n} |z^{r}|^{2}) (δ^{p\overline{q}} + z^{q}\overline{z^{p}}). \end{matrix}

Under the local coordinate system, for any smooth function f, we have

(3.13)  Δf = ∑_{p,q=1}^{n} 4g^{p\overline{q}} \frac{∂^{2}}{∂z^{p}\overline{∂z^{q}}} f.

And, at P,

Δ = 4 ∑_{r=1}^{n} \frac{∂^{2}}{∂z^{r}\overline{∂z^{r}}} ,

(3.14)

∇f̃_{p\overline{q}} = 0, if p ≠ 0 and q ≠ 0,
∇f̃_{p\overline{p}} = 0,
Δf̃_{p\overline{q}} = 0, if p ≠ q
Δf̃_{0¯0} = −4n, Δf̃_{r\overline{r}} = 4, r = 1, ⋯, n.

10

Thus, we obtain from (3.10), at P, (3.15)

∑_{p,q=0}^{n}(∇G_{p\overline{q}} · ∇G_{p\overline{q}} + ∇F_{p\overline{q}} · ∇F_{p\overline{q}})

= − ∑_{p,q=0}^{n}(G_{p\overline{q}}ΔG_{p\overline{q}} + F_{p\overline{q}}ΔF_{p\overline{q}})

= −Re ∑_{p,q=0}^{n} \overline{f_{p\overline{q}}}Δf_{p\overline{q}}

= −Re ∑_{p,q=0}^{n} ∑_{r,s=0}^{n} \overline{C_{pr}\overline{C_{qs}}}\overline{f̃_{r\overline{s}}} ∑_{u,v=0}^{n} C_{pu}\overline{C_{qv}}Δf̃_{u\overline{v}}

= − ∑_{p,q=0}^{n} Re\overline{f̃_{p\overline{q}}}Δf̃_{p\overline{q}} = −f̃_{0\overline{0}}Δf̃_{0\overline{0}} = 4n.

By a similar calculation, we have, at P,

(3.16) ∑_{p,q=0}^{n}(∇G_{p\overline{q}}ΔG_{p\overline{q}} + ∇F_{p\overline{q}}ΔF_{p\overline{q}}) = Re ∑_{p,q=0}^{n} ∇\overline{f_{p\overline{q}}}Δf_{p\overline{q}} = 0,

(3.17)

∑_{p,q=0}^{n}(ΔG_{p\overline{q}}ΔG_{p\overline{q}} + ΔF_{p\overline{q}}ΔF_{p\overline{q}})

=Re ∑_{p,q=0}^{n} \overline{Δf_{p\overline{q}}}Δf_{p\overline{q}} = Re ∑_{p,q=0}^{n} \overline{Δf̃_{p\overline{q}}}Δf̃_{p\overline{q}}
=(−4n) × (−4n) + 4 × 4 × n = 16n(n + 1)

and

(3.18)  \begin{matrix}∑_{p,q=0}^{n}\{(∇G_{p\overline{q}} · ∇u_{i})^{2} + (∇F_{p\overline{q}} · ∇u_{i})^{2}\} \\ = & Re ∑_{p,q=0}^{n} \overline{(∇f_{p\overline{q}} · ∇u_{i})}(∇f_{p\overline{q}} · ∇u_{i}) \\ = & Re ∑_{p,q=0}^{n} \overline{(∇f̃_{p\overline{q}} · ∇u_{i})}(∇f̃_{p\overline{q}} · ∇u_{i}) \\ = & 2|∇u_{i}|^{2}. \end{matrix}

Since P is arbitrary, we have at any point x ∈ M,

(3.19)  \{ \begin{matrix}∑_{p,q=0}^{n}(∇G_{p\overline{q}} · ∇G_{p\overline{q}} + ∇F_{p\overline{q}} · ∇F_{p\overline{q}}) = 4n, \\ ∑_{p,q=0}^{n}(ΔG_{p\overline{q}}ΔG_{p\overline{q}} + ΔF_{p\overline{q}}ΔF_{p\overline{q}}) = 16n(n + 1), \\ ∑_{p,q=0}^{n}(∇G_{p\overline{q}}ΔG_{p\overline{q}} + ∇F_{p\overline{q}}ΔF_{p\overline{q}}) = 0, \\ ∑_{p,q=0}^{n}\{(∇G_{p\overline{q}} · ∇u_{i})^{2} + (∇F_{p\overline{q}} · ∇u_{i})^{2}\} = 2|∇u_{i}|^{2}. \end{matrix}

By applying the Proposition 1 to the functions G_{p\overline{q}} and F_{p\overline{q}} and taking sum on p and q from 0 to n, we infer

(3.20)  \begin{matrix}∑_{i=1}^{k}(λ_{k+1} − λ_{i})^{2} ∫_{M} ∑_{p,q=0}^{n}(∥u_{i}∇G_{p\overline{q}}∥^{2} + ∥u_{i}∇F_{p\overline{q}}∥^{2}) \\ ≤ & ∑_{i=1}^{k}(λ_{k+1} − λ_{i})\{4 ∫_{M} ∑_{p,q=0}^{n}\{(∇G_{p\overline{q}} · ∇u_{i})^{2} + (∇F_{p\overline{q}} · ∇u_{i})^{2}\} \\ + & 4 ∫_{M} ∑_{p,q=0}^{n}(∇G_{p\overline{q}}ΔG_{p\overline{q}} · u_{i}∇u_{i} + ∇F_{p\overline{q}}ΔF_{p\overline{q}} · u_{i}∇u_{i}) \\ + & ∫_{M} ∑_{p,q=0}^{n}(ΔG_{p\overline{q}}ΔG_{p\overline{q}} + ΔF_{p\overline{q}}ΔF_{p\overline{q}})u_{i}^{2}\}. \end{matrix}

From (2.2), (3.19) and (3.20), we obtain

4n ∑_{i=1}^{k}(λ_{k+1} − λ_{i})^{2} ≤ ∑_{i=1}^{k}(λ_{k+1} − λ_{i})[8λ_{i} + 16n(n + 1)],

namely,

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

Letting μ_{i} = λ_{i} + 2n(n + 1), we have

(3.22)  ∑_{i=1}^{k}(μ_{k+1} − μ_{i})^{2} ≤ \frac{2}{n} ∑_{i=1}^{k}(μ_{k+1} − μ_{i})μ_{i},

Since (3.22) is a quadratic inequality of μ_{k+1}, we have

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

This finishes the proof of Theorem 1. Proof of Corollary 1. In the proof of Theorem 1, since k is any integer, we know that (3.22) is also true if we replace k + 1 with k, namely, we have

∑_{i=1}^{k−1}(μ_{k} − μ_{i})^{2} ≤ \frac{2}{n} ∑_{i=1}^{k−1}(μ_{k} − μ_{i})μ_{i}.

Hence, we infer

(3.24)  ∑_{i=1}^{k}(μ_{k} − μ_{i})^{2} ≤ \frac{2}{n} ∑_{i=1}^{k}(μ_{k} − μ_{i})μ_{i}.

Thus, μ_{k} also satisfies the same quadratic inequality. Hence, we have

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

Therefore, we have

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

4. Eigenvalues of Laplacain on a complex hypersurface in ℂℙ^{n+1}(4)

In this section, we shall consider the eigenvalue problem of the Laplacian on a compact complex hypersurface M without boundary in ℂℙ^{n+1}(4):

(4.1)  Δu = −λu, in M,

where Δ is the Laplacian of M. We know that this eigenvalue problem has a discrete spectrum: 0 = λ_{0} < λ_{1} ≤ λ_{2} ≤ ⋯ → ∞. As an application of the Proposition 1, we obtain the following Theorem 2. In this case, we must consider the eigenfunction u_{0} corresponding to eigenvalue λ_{0} = 0.

Theorem 2. Let M be a compact complex hypersurface without boundary in ℂℙ^{n+1}(4), Assume that λ_{i} is the i-th eigenvalue of the above eigenvalue problem (4.1). Then, we have

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

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

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

Proof of Theorem 2. Since the method of proof is the same as in the proof of Theorem 1, we shall only give its outline. Let Z = (Z^{0}, Z^{1}, …, Z^{n+1}) be a homogeneous coordinate system of ℂℙ^{n+1}(4). Defining f_{pq̄}, for p, q = 0, 1, ⋯, n, n + 1, by

(4.3)  f_{pq̄} = \frac{Z^{p}\overline{Z^{q}}}{∑_{r=0}^{n+1} Z^{r}\overline{Z^{r}}} ,

we have

(4.4)  f_{p\overline{q}} = \overline{f_{q\overline{p}}}, ∑_{p,q=0}^{n+1} f_{p\overline{q}}\overline{f_{p\overline{q}}} = 1.

For any fixed point P ∈ M, we can choose a new homogeneous coordinate system of ℂℙ^{n+1}(4) such that, at P

(4.5)  Z̃^{0} ≠ 0, Z̃^{1} = ⋯ = Z̃^{n+1} = 0

and

(4.6)  Z^{p} = ∑_{r=0}^{n+1} C_{pr}Z̃^{r},

where C = (C_{pq}) ∈ U(n + 2) is an (n + 2) × (n + 2)-unitary matrix, that is, C_{pr} satisfies

(4.7)  ∑_{s=0}^{n+1} C_{sp}\overline{C}_{sq} = ∑_{s=0}^{n+1} C_{ps}\overline{C}_{qs} = δ^{pq}.

Let

z^{p} = Z̃^{p}/Z̃^{0}, for p = 0, 1, ⋯, n, n + 1.

Then, we know that z = (z^{1}, …, z^{n}) is a local holomorphic coordinate system of M in a neighborhood U of the point P ∈ M and z^{n+1} = h(z^{1}, …, z^{n}) ∈ O(U) is a holomorphic function of z^{1}, …, z^{n} satisfying

(4.8)  \frac{∂h}{∂z^{p}} |_{P} = 0, for p = 1, ⋯, n,

and, at P,

(4.9)  z^{1} = ⋯ = z^{n} = z^{n+1} = 0.

Hence, we infer, for p, q = 0, 1, ⋯, n, n + 1,

(4.10)  \begin{matrix}f̃_{p\overline{q}} = \begin{matrix}Z̃^{p} & \overline{Z̃^{q}} \\ ∑ & {n+1 \atop r=0} & Z̃^{r} & \overline{Z̃^{r}} \end{matrix} = \frac{z^{p}\overline{z^{q}}}{1 + ∑_{r=1}^{n+1} z^{r}\overline{z}^{r}} , \\ f_{p\overline{q}} = ∑_{r,s=0}^{n+1} C_{pr}\overline{C_{qs}}f̃_{r\overline{s}}, z^{0} ≡ 1. \end{matrix}

Putting G_{p\overline{q}} = Re(f_{p\overline{q}}) and F_{p\overline{q}} = Im(f_{p\overline{q}}), for p, q = 0, 1, ⋯, n, n + 1, then, for the 2(n + 2)^{2} functions G_{p\overline{q}} and F_{p\overline{q}}, we have

(4.11)  \begin{matrix}∑_{p,q=0}^{n+1}(G_{p\overline{q}}^{2} + F_{p\overline{q}}^{2}) = ∑_{p,q=0}^{n+1} f_{p\overline{q}}\overline{f_{p\overline{q}}} = ∑_{p,q=0}^{n+1} f̃_{p\overline{q}}\overline{f̃_{p\overline{q}}} = 1, \\ ∑_{p,q=0}^{n+1}(G_{p\overline{q}}∇G_{p\overline{q}} + F_{p\overline{q}}∇F_{p\overline{q}}) = 0. \end{matrix}

Since, under the local coordinate system, we have, for z ∈ U,

ds_{M}^{2} = ∑_{p,q=1}^{n}(1 + O(|z|^{2})dz^{p}d\overline{z^{q}},

we obtain, at P,

Δ = 4 ∑_{r=1}^{n} \frac{∂^{2}}{∂z^{r}\overline{∂z^{r}}} ,

(4.12)

∇f̃_{p\overline{q}} = 0, if p ≠ 0 and q ≠ 0,
∇f̃_{p\overline{p}} = 0,
∇f̃_{p\overline{(}n+1)} = ∇f̃_{(n+1)\overline{p}} = ∇f̃_{(n+1)\overline{(n+1)}} = 0, for p = 1, ⋯, n,
Δf̃_{p\overline{q}} = 0, if p ≠ q, Δf̃_{n+1\overline{n+1}} = 0,
Δf̃_{0¯0} = −4n, Δf̃_{r\overline{r}} = 4, r = 1, ⋯, n,

Thus, making use of the same arguments as in the proof of Theorem 1, we obtain, at any point x ∈ M,

(4.13)  \{ \begin{matrix}∑_{p,q=0}^{n+1}(∇G_{p\overline{q}} · ∇G_{p\overline{q}} + ∇F_{p\overline{q}} · ∇F_{p\overline{q}}) = 4n, \\ ∑_{p,q=0}^{n+1}(ΔG_{p\overline{q}}ΔG_{p\overline{q}} + ΔF_{p\overline{q}}ΔF_{p\overline{q}}) = 16n(n + 1), \\ ∑_{p,q=0}^{n+1}(∇G_{p\overline{q}}ΔG_{p\overline{q}} + ∇F_{p\overline{q}}ΔF_{p\overline{q}}) = 0, \\ ∑_{p,q=0}^{n+1}\{(∇G_{p\overline{q}} · ∇u_{i})^{2} + (∇F_{p\overline{q}} · ∇u_{i})^{2}\} = 2|∇u_{i}|^{2}. \end{matrix}

By applying the Proposition 1 to the functions G_{p\overline{q}} and F_{p\overline{q}} and taking sum on p and q from 0 to n + 1, we infer

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

This finishes the proof of Theorem 2.

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

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Qing-Ming Cheng Department of Mathematics Faculty of Science and Engineering Saga University, Saga 840-8502, Japan cheng@ms.saga-u.ac.jp and Hongcang Yang Academy of Mathematics and Systematical Sciences CAS, Beijing 100080, China e-mail: yanghc@netease.com