Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 31, pp. 1–19.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.31

EIGENVALUE BOUNDS FOR THE CLAMPED PLATE PROBLEM OF L2
ξ

OPERATOR

LINGZHONG ZENG, ZIYI ZHOU

Abstract. The operator LII is an important extrinsic differential operator, which is elliptic of

divergence type and plays significant roles in the study of translating solitons. In this article,
we extend LII to a more general elliptic differential operator Lξ, for studying the clamped plate

problem of the bi-Lξ operator, denoted by L2
ξ , on the complete Riemannian manifolds. By

establishing a general formula of eigenvalues for L2
ξ , we give a new estimate for the eigenvalues

of bi-Lξ operator. Some further applications of this result includes obtaining some universal

inequalities for bi-LII operator on translators, and studying the eigenvalues on the submanifolds

of the Euclidean spaces, unit spheres, and projective spaces.

1. Introduction

Let D be a bounded domain with piecewise smooth boundary ∂D on Mn, where (Mn, g) is an
n-dimensional complete Riemannian submanifold isometrically immersed into the N -dimensional
Euclidean space RN , with smooth induced metric g. Throughout this paper, we assume that ξ
is a constant vector field defined on Mn and use ⟨·, ·⟩g, | · |2g, div, ∆, ∇ and ξ⊤ to denote the
Riemannian inner product with respect to the induced metric g, norm associated with the inner
product ⟨·, ·⟩g, divergence, Laplacian, the gradient operator on Mn and the projective of the vector
ξ on the tangent bundle TMn, respectively. In addition, we assume that {e1, . . . , en} is a local
orthonormal basis of Mn with respect to the induced Riemannian metric g, and {en+1, . . . , eN}
is the corresponding local unit orthonormal normal vector fields. Assume that

H =
1

n

N∑
α=n+1

Hαeα =
1

n

N∑
α=n+1

( n∑
i=1

hαii

)
eα,

is the mean curvature vector field, and

H = |H| = 1

n

( N∑
α=n+1

( n∑
i=1

hαii

)2)1/2

,

is the mean curvature of Mn. Assume that Π is a set defined as the following form:

Π =: {σ : Mn → RN : σ is a isometric immersion}.
We define an elliptic differential operator on Mn as follows

Lξ = ∆+ ⟨ξ,∇(·)⟩g0 = e−⟨ξ,X⟩g0 div(e⟨ξ,X⟩g0∇(·)), (1.1)

where ⟨·, ·⟩g0 stands for the standard inner product of RN . We remark that the elliptic differential

operator Lξ is a self-adjoint operator with respect to the weighted measure e⟨ξ,X⟩g0dv. Namely,
for any u, ū ∈ C2

0 (D), the following Stokes’ formula holds:

−
∫
D

⟨∇u,∇ū⟩ge⟨ξ,X⟩g0dv =

∫
D

(Lξū)ue
⟨ξ,X⟩g0dv =

∫
D

(Lξu)ūe
⟨ξ,X⟩g0dv. (1.2)

2020 Mathematics Subject Classification. 35P15, 53C40.

Key words and phrases. L2
ξ operator; clamped plate problem; eigenvalues; submanifolds; translating solitons.

©2025. This work is licensed under a CC BY 4.0 license.

Submitted December 3, 2024. Published March 31, 2025.

1



2 L. ZENG, Z. ZHOU EJDE-2025/31

Accordingly, we use | · |g0 to denote the norm on RN associated with the standard inner product
⟨·, ·⟩g0 . In particular, we assume that ξ is a unit constant vector defined on a translating soliton in
the sense of the means curvature flows (5.1) and denote it by ξ0. For this special case, the above
differential operator will be denoted by LII , which is introduced by Xin in [33] and of important
geometric meaning. We refer the readers to section 5 for details. Just like the other weighted
Laplacian, for example, L operator and Witten-Laplacian, Lξ operator is also very important in
geometric analysis. Next, let us consider an eigenvalue problem of L2

ξ operator on the bounded
domain D ⊂ Mn with Dirichlet boundary condition:

L2
ξu = Γu, in D,

u =
∂u

∂n
= 0, on ∂D,

(1.3)

where n denotes the normal vector to the boundary ∂D. Let Γk denote the kth eigenvalue, and
then the spectrum of the eigenvalue problem (1.3) is discrete and satisfies

0 < Γ1 ≤ Γ2 ≤ · · · ≤ Γk ≤ · · · → +∞,

where each eigenvalue is repeated according to its multiplicity. Furthermore, we assume that
|ξ|

g0
= 0, and then the Lξ operator exactly is the classical Laplacian defined on Riemannian

manifold Mn. For this case, eigenvalue problem (1.3) correspondingly becomes the following
Dirichlet problem of biharmonic operator associated with Riemannian manifold Mn:

∆2u = Γu, in D,

u =
∂u

∂n
= 0, on ∂D.

(1.4)

In particular, when Mn is an n-dimensional Euclidean space Rn

, eigenvalue problem (1.4) is
called a clamped plate problem, which is used to describe vibrations of a clamped plate in elastic
mechanics. In 1956, Payne, Pólya and Weinberger [28] investigated the above eigenvalue problem
with respect to the Euclidean space and obtained a universal bound for eigenvalue problem (1.4)
as follows:

Γk+1 − Γk ≤ 8(n+ 2)

n2
1

k

k∑
i=1

Γi. (1.5)

In 1984, by means of improved method due to Hile and Protter in [21], Hile and Yeh [22] obtained
the universal inequality

k∑
i=1

Γ
1/2
i

Γk+1 − Γi
≥ n2k3/2

8(n+ 2)

( k∑
i=1

Γi

)−1/2

, (1.6)

which generalizes universal inequality (1.5). In 1990, Hook [23] proved the inequality:

n2k2

8(n+ 2)
≤

[∑
i=1

Γ
1/2
i

Γk+1 − Γi

] k∑
i=1

Γ
1/2
i . (1.7)

In 2006, Cheng and Yang [13] gave an affirmative answer to an interesting problem proposed by
Ashbaugh in his survey paper [4]. Specifically, they obtained the following universal bound of
Yang type:

Γk+1 −
1

k

k∑
i=1

Γi ≤
[8(n+ 2)

n2

]1/2 1
k

k∑
i=1

[
Γi (Γk+1 − Γi)

]1/2
, (1.8)

which is sharper than

Γk+1 ≤
[
1 +

8(n+ 2)

n2

]1
k

k∑
i=1

Γi. (1.9)



EJDE-2025/31 EIGENVALUE BOUNDS FOR THE CLAMPED PLATE PROBLEM 3

We note that, in fact, inequality (1.9) is better than inequality (1.5) given by Payne, Pólya and
Weinberger. In 2011, Wang and Xia [32] investigated the eigenvalues with higher order of bi-
harmonic operator on the complete Riemannian manifolds and proved the inequality

k∑
i=1

(Γk+1 − Γi)
2 ≤ 4

n

{ k∑
i=1

(Γk+1 − Γi)
2
[ (n

2
+ 1

)
Γ
1/2
i + C0

]}1/2

×
{ k∑

i=1

(Γk+1 − Γi)
(
Γ
1/2
i + C0

)}1/2

,

(1.10)

where

C0 =
1

4
inf
σ∈Π

max
D

(
n2H2

)
.

For more progresses on the clamped plate eigenvalue problem of bi-harmonic operators, we refer
the reader to [11] and references therein. We remark that Wang and Xia’s result is extended by Du
et al. [16] to the setting of bi-drifting Laplacian on the smooth metric measure spaces. Further-
more, in [19, 20], He and Pu investigated the clamped plate problem of the drifting Laplacian in
several cases, and established some eigenvalue inequalities that are different from those obtained
previously in [16]. In this paper, we consider the clamped plate problem (1.3) with respect to the
bi-Lξ operator L2

ξ on the complete Riemannian manifold Mn and obtain an eigenvalue inequality.
Specially, we prove the following theorem.

Theorem 1.1. Let (Mn, g) be an n-dimensional complete Riemannian manifold isometrically
embedded into the Euclidean space RN with mean curvature H, then the eigenvalues Γi of the
clamped plate problem (1.3) of the L2

ξ operator satisfy

k∑
i=1

(Γk+1 − Γi) ≤
4

n

{ k∑
i=1

(Γk+1 − Γi)
((n

2
+ 1

)
Γ
1/2
i + 4C̃1Γ

1/4
1 + 4C̃2

1 + C1

)}1/2

×
{ k∑

i=1

(
Γ
1/2
i + 4C̃1Γ

1/4
1 + 4C̃2

1 + C1

)}1/2

,

(1.11)

where

C1 =
1

4
inf
σ∈Π

max
D

(
n2H2

)
, C̃1 =

1

4
max
D

|ξ⊤|g0 .

Remark 1.2. We recall that, the first author established the following eigenvalue inequality [36,
Theorem 1.1],

k∑
i=1

(Γk+1 − Γi)
2 ≤ 4

n

{ k∑
i=1

(Γk+1 − Γi)
2
((n

2
+ 1

)
Γ
1/2
i + 4C̃1Γ

1/4
1 + 4C̃2

1 + C1

)}1/2

×
{ k∑

i=1

(Γk+1 − Γi)
(
Γ
1/2
i + 4C̃1Γ

1/4
1 + 4C̃2

1 + C1

)}1/2

,

(1.12)

By a similar argument as in [20, Remark 1.2], we can show that inequality (1.11) is better than
inequality (1.12) in some sense. In addition, by weighted Chebyshev inequality (see citeHLP), we
know that inequality (1.11) can deduce to upper bound of the (k + 1)-th eigenvalue via the first
k eigenvalues more quickly and directly than inequality (1.12). Here, we left the details to the
reader. Also, see [20, Remark 1.2].

As an application of Theorem 1.1, we obtain a universal bound of the eigenvalues of L2
II operator

on the translating solitons (see the definition (5.2)), which occurs as Type-II singularity of the
mean curvature flow (MCF for short). In other words, we prove the following domain independent
bound.



4 L. ZENG, Z. ZHOU EJDE-2025/31

Theorem 1.3. Let (Mn, g) be an n-dimensional complete translating soliton isometrically em-
bedded into an N -dimensional Euclidean space RN , then eigenvalues of the clamped plate problem
(1.3) of the L2

II operator satisfy

k∑
i=1

(Γk+1 − Γi) ≤
4

n

{ k∑
i=1

(Γk+1 − Γi)
((n

2
+ 1

)
Γ
1/2
i + Γ

1/4
i +

n2

4

)}1/2

×
{ k∑

i=1

(
Γ
1/2
i + Γ

1/4
i +

n2

4

)}1/2

.

(1.13)

Remark 1.4. Since inequality (1.13) does not depend on the domainD, it is a universal inequality.

2. General formula and its proof

In this section, we establish a general formula, which will play an important role in the proof
of Theorem 1.1. Toward this end, we need to prove some auxiliary lemmas.

Lemma 2.1. Let Γi, i = 1, 2, . . . , be the i-th eigenvalue of the clamped plate problem (1.3) and
ui be the orthonormal eigenfunction corresponding to Γi, that is,

L2
ξui = Γiui, in D,

ui =
∂ui
∂n

= 0, on ∂D,∫
D

uiuje
⟨ξ,X⟩g0dv = δij , ∀i, j = 1, 2, . . . .

(2.1)

Let us use ⟨·, ·⟩ to denote the inner product of two vector fields. For any function ψ ∈ C4(D) ∩
C3(∂D), we define

Φi := 2⟨∇ψ,∇ (Lξui)⟩+ LξψLξui + 2Lξ (⟨∇ψ,∇ui⟩) + Lξ (uiLξψ) , (2.2)

sij :=

∫
D

ujΦie
⟨ξ,X⟩g0dv, (2.3)

aij :=

∫
D

ψuiuje
⟨ξ,X⟩g0dv. (2.4)

Then for each positive integer k, we have

(Γj − Γi) aij = sij . (2.5)

Proof. From the definitions of sij and Φi, we have

sij =

∫
D

uj

[
2⟨∇ψ,∇ (Lξui)⟩+ LξψLξui + 2Lξ (⟨∇ψ,∇ui⟩)

+ Lξ (uiLξψ)
]
e⟨ξ,X⟩g0dv.

(2.6)

Multiplying both sides of L2
ξui = Γiui by ψuj , we obtain

ψujL
2
ξui = Γiψuiuj . (2.7)

Exchanging the order of the subscripts i and j yields

ψuiL
2
ξuj = Γjψuiuj . (2.8)

Subtracting (2.7) from (2.8) and integrating over the bounded domain D, we have

(Γj − Γi) aij =

∫
D

(
ψuiL

2
ξuj − ψujL

2
ξui

)
e⟨ξ,X⟩g0dv. (2.9)

A straightforward calculation yields

Lξ (ψui) = ψLξui + 2⟨∇ψ,∇ui⟩+ uiLξψ, (2.10)



EJDE-2025/31 EIGENVALUE BOUNDS FOR THE CLAMPED PLATE PROBLEM 5

Furthermore, applying Stokes’ formula (1.2), (2.6), (2.9) and (2.10), we infer that

(Γj − Γi) aij

=

∫
D

{[uiLξψ + 2⟨∇ψ,∇ui⟩]Lξuj − [ujLξψ + 2⟨∇ψ,∇uj⟩]Lξui} e⟨ξ,X⟩g0dv

=

∫
D

{
uj [Lξ (uiLξψ) + 2Lξ (⟨∇ψ,∇ui⟩)]− ujLξψLξui

+ 2uje
−⟨ξ,X⟩g0 div

(
e⟨ξ,X⟩g0Lξui∇ψ

)}
e⟨ξ,X⟩g0dv

=

∫
D

uj [Lξ (uiLξψ) + 2Lξ (⟨∇ψ,∇ui⟩) + LξψLξui + 2⟨∇Lξui,∇ψ⟩] e⟨ξ,X⟩g0dv

= sij .

(2.11)

The proof is complete. □

Lemma 2.2. Under the same convention as Lemma 2.1, we define

tij :=

∫
D

uj

(
⟨∇ψ,∇ui⟩+

uiLξψ

2

)
e⟨ξ,X⟩g0dv.

Then, tij is antisymmetric with respect to the subscripts, i.e., it holds

tij = −tji. (2.12)

Proof. Utilizing Stokes’ formula (1.2), by the definition of tij , we have

tij + tji =

∫
D

uj

(
⟨∇ψ,∇ui⟩+

uiLξψ

2

)
e⟨ξ,X⟩g0dv

+

∫
D

ui

(
⟨∇ψ,∇uj⟩+

ujLξψ

2

)
e⟨ξ,X⟩g0dv

=

∫
D

[⟨∇ψ, uj∇ui + ui∇uj⟩+ uiujLξψ] e
⟨ξ,X⟩g0dv

=

∫
D

[⟨∇ψ,∇ (uiuj)⟩+ uiujLξψ] e
⟨ξ,X⟩g0dv

=

∫
D

(−uiujLξψ + uiujLξψ) e
⟨ξ,X⟩g0dv = 0.

The proof is complete. □

Lemma 2.3. We define

G :=

k∑
i,j=1

(Γk+1 − Γi) aijtij , and K :=

k∑
i,j=1

(Γk+1 − Γi) aijsij .

Then we have

G =
1

2

k∑
i,j=1

sijtij , (2.13)

K =
1

2

k∑
i,j=1

s2ij . (2.14)



6 L. ZENG, Z. ZHOU EJDE-2025/31

Proof. By exchanging the summation order of i and j in the definition of G, and noticing (2.4),
(2.11) and (2.12), one can carry out the following calculations:

G =

k∑
i,j=1

(Γk+1 − Γj) aijtij +

k∑
i,j=1

(Γj − Γi) aijtij

=

k∑
j,i=1

(Γk+1 − Γi) aijtji +

k∑
i,j=1

sijtij

= −
k∑

j,i=1

(Γk+1 − Γi) aijtij +

k∑
i,j=1

sijtij

= −G+

k∑
i,j=1

sijtij ,

(2.15)

which implies (2.13). By the same method as in the proof of (2.13), one can infer that

K =

k∑
i,j=1

[(Γk+1 − Γj) + (Γj − Γi)] aijsij

=

k∑
i,j=1

(Γk+1 − Γj) aijsij +

k∑
i,j=1

s2ij

=

k∑
j,i=1

(Γk+1 − Γi) ajisji +

k∑
i,j=1

s2ij

= −
k∑

i,j=1

(Γk+1 − Γi) aijsij +

k∑
i,j=1

s2ij

= −K +

k∑
i,j=1

s2ij .

(2.16)

In view of (2.15), we derive (2.14). □

Lemma 2.4. Under the same convention as in Lemma 2.1, we have∫
D

ψuiΦie
⟨ξ,X⟩g0dv =

∫
D

[
u2i (Lξψ)

2
+ 4

(
⟨∇ψ,∇ui⟩2 + uiLξψ⟨∇ψ,∇ui⟩

)
− 2|∇ψ|2uiLξui

]
e⟨ξ,X⟩g0dv.

(2.17)

Proof. By direct calculations, we obtain∫
D

ψuiΦie
⟨ξ,X⟩g0dv

=

∫
D

ψui

[
Lξ (uiLξψ) + 2Lξ (⟨∇ψ,∇ui⟩)

+ 2⟨∇ψ,∇ (Lξui)⟩+ LξψLξui

]
e⟨ξ,X⟩g0dv

=

∫
D

{
Lξ (ψui)uiLξψ + 2Lξ (ψui) ⟨∇ψ,∇ui⟩

− 2Lξui

[
e−⟨ξ,X⟩g0 div

(
e⟨ξ,X⟩g0ψui∇ψ

)]
+ ψuiLξψLξui

}
e⟨ξ,X⟩g0dv.

(2.18)



EJDE-2025/31 EIGENVALUE BOUNDS FOR THE CLAMPED PLATE PROBLEM 7

A straightforward calculation yields the following equalities:∫
D

Lξ (ψui)uiLξψe
⟨ξ,X⟩g0dv

=

∫
D

(uiLξψ + 2⟨∇ψ,∇ui⟩+ ψLξui)uiLξψe
⟨ξ,X⟩g0dv

=

∫
D

[
u2i (Lξψ)

2
+ 2uiLξψ⟨∇ψ,∇ui⟩+ ψuiLξuiLξψ

]
e⟨ξ,X⟩g0dv,

(2.19)

∫
D

Lξ (ψui) ⟨∇ψ,∇ui⟩e⟨ξ,X⟩g0dv

=

∫
D

(uiLξψ + 2⟨∇ψ,∇ui⟩+ ψLξui) ⟨∇ψ,∇ui⟩e⟨ξ,X⟩g0dv

=

∫
D

(
uiLξψ⟨∇ψ,∇ui⟩+ 2⟨∇ψ,∇ui⟩2

+ ⟨∇ψ,∇ui⟩ψLξui

)
e⟨ξ,X⟩g0dv,

(2.20)

and ∫
D

Lξui
[
e−⟨ξ,X⟩g0 div

(
e⟨ξ,X⟩g0ψui∇ψ

)]
e⟨ξ,X⟩g0dv

=

∫
D

Lξui
[
⟨∇ (ψui) ,∇ψ⟩+ ψuiLξψ

]
e⟨ξ,X⟩g0dv

=

∫
D

Lξui
(
|∇ψ|2ui + ψ⟨∇ui,∇ψ⟩+ ψuiLξψ

)
e⟨ξ,X⟩g0dv

=

∫
D

(
|∇ψ|2uiLξui + ψLξui⟨∇ui,∇ψ⟩+ ψuiLξuiLξψ

)
e⟨ξ,X⟩g0dv

(2.21)

Combining (2.18)-(2.21) yields∫
D

ψui [Lξ (uiLξψ) + 2Lξ⟨∇ψ,∇ui⟩+ 2⟨∇ψ,∇ (Lξui)⟩+ LξψLξui] e
⟨ξ,X⟩g0dv

=

∫
D

[
u2i (Lξψ)

2
+ 4

(
⟨∇ψ,∇ui⟩2 + uiLξψ⟨∇ψ,∇ui⟩

)
− 2|∇ψ|2uiLξui

]
e⟨ξ,X⟩g0dv,

which implies (2.17). The proof is complete. □

Combining the strategies in [16, 19, 20, 31, 32], and applying Lemmas 2.1, 2.2, 2.3 and 2.4, we
can establish the following general formula.

Lemma 2.5. Under the same convention of Lemma 2.1, Then for each function ψ ∈ C4(D) ∩
C3(∂D) and each positive integer k, we have

k∑
i=1

(Γk+1 − Γi)

∫
D

u2i |∇ψ|2e⟨ξ,X⟩g0dv

≤ ε

k∑
i=1

(Γk+1 − Γi)

∫
D

[Ψi (ψ)−Θi (ψ)] e
⟨ξ,X⟩g0dv +

1

4ε

k∑
i=1

∫
D

Ψi (ψ) e
⟨ξ,X⟩g0dv,

(2.22)

where ε is a positive constant,

Ψi (ψ) = (uiLξψ + 2⟨∇ψ,∇ui⟩g)2 , (2.23)

Θi (ψ) = |∇ψ|2g uiLξui. (2.24)

Proof. For each i = 1, . . . , k, we define a function φi : D → R on the bounded domain D as
follows:

φi = ψui −
k∑

j=1

aijuj , (2.25)



8 L. ZENG, Z. ZHOU EJDE-2025/31

where aij is given by (2.1). Clearly, such functions satisfy the variation condition of eigenvalue
problem (1.3). This is to say that

φi

∣∣
∂D

=
∂φi

∂n

∣∣
∂D

= 0 and

∫
D

ujφie
⟨ξ,X⟩g0dv = 0, ∀i, j = 1, . . . , k. (2.26)

Therefore, the min-max principle (Rayleigh-Ritz inequality) implies that

Γk+1

∫
D

φ2
i e

⟨ξ,X⟩g0dv ≤
∫
D

φiL
2
ξφie

⟨ξ,X⟩g0dv. (2.27)

Equation (2.10) implies that

L2
ξ (ψui) = Lξ (ψLξui + 2⟨∇ψ,∇ui⟩+ uiLξψ)

= ψL2
ξui + 2⟨∇ψ,∇ (Lξui)⟩+ LξψLξui + 2Lξ (⟨∇ψ,∇ui⟩) + Lξ (uiLξψ)

= Γiψui +Φi,

(2.28)

where Φi is given by (2.2). By (2.3), (2.25), (2.26) and (2.28), we infer that

∫
D

φiL
2
ξφie

⟨ξ,X⟩g0dv

=

∫
D

φi

[
L2
ξ (ψui)−

k∑
j=1

aijΓjuj

]
e⟨ξ,X⟩g0dv

=

∫
D

φi

[
Φi + Γiψui −

k∑
j=1

aijΓjuj

]
e⟨ξ,X⟩g0dv

=

∫
D

φi

[
Φi + Γi

(
ψui −

k∑
j=1

aijuj

)]
e⟨ξ,X⟩g0dv

=

∫
D

φiΦie
⟨ξ,X⟩g0dv + Γi∥φi∥2

=

∫
D

ψuiΦie
⟨ξ,X⟩g0dv −

k∑
j=1

aij

∫
D

ujΦie
⟨ξ,X⟩g0dv + Γi∥φi∥2

=

∫
D

ψuiΦie
⟨ξ,X⟩g0dv −

k∑
j=1

aijsij + Γi∥φi∥2,

(2.29)

where

∥φi∥2 =

∫
D

φ2
i e

⟨ξ,X⟩g0dv.

It follows from (2.5), (2.17) (2.27) and (2.29), that

(Γk+1 − Γi) ∥φi∥2 ≤
∫
D

[
u2i (Lξψ)

2
+ 4

(
⟨∇ψ,∇ui⟩2 + uiLξψ⟨∇ψ,∇ui⟩

)
− 2|∇ψ|2uiLξui

]
e⟨ξ,X⟩g0dv −

k∑
j=1

aijsij .

(2.30)



EJDE-2025/31 EIGENVALUE BOUNDS FOR THE CLAMPED PLATE PROBLEM 9

By a direct computation, we have

− 2

∫
D

φi

(
⟨∇ψ,∇ui⟩+

uiLξψ

2

)
e⟨ξ,X⟩g0dv

=

∫
D

[
− 2ψiui⟨∇ψ,∇ui⟩ − u2iψLξψ

]
e⟨ξ,X⟩g0dv + 2

k∑
j=1

aijtij

=

∫
D

[
− 1

2
⟨∇

(
ψ2

)
,∇

(
u2i

)
⟩ − u2iψLξψ

]
e⟨ξ,X⟩g0dv + 2

k∑
j=1

aijtij

=

∫
D

[
1

2
u2iLξ

(
ψ2

)
− u2iψLξψ

]
e⟨ξ,X⟩g0dv + 2

k∑
j=1

aijtij

=

∫
D

[
u2i

(
ψLξψ + |∇ψ|2

)
− u2iψLξψ

]
e⟨ξ,X⟩g0dv + 2aijtij

=

∫
D

u2i |∇ψ|2e⟨ξ,X⟩g0dv + 2

k∑
j=1

aijtij .

(2.31)

Multiplying (2.31) by (Γk+1 − Γi), we have

(Γk+1 − Γi)
(∫

D

u2i |∇ψ|2e⟨ξ,X⟩g0dv + 2

k∑
j=1

aijtij

)
= −2 (Γk+1 − Γi)

∫
D

φi

(
⟨∇ψ,∇ui⟩+

uiLξψ

2

)
e⟨ξ,X⟩g0dv

= −2 (Γk+1 − Γi)

∫
D

φi

(
⟨∇ψ,∇ui⟩+

uiLξψ

2
−

k∑
j=1

tijuj

)
e⟨ξ,X⟩g0dv.

Utilizing Cauchy-Schwarz inequality, one can conclude from the above equation that

(Γk+1 − Γi)
(∫

D

u2i |∇ψ|2e⟨ξ,X⟩g0dv + 2

k∑
j=1

aijtij

)

≤ ε (Γk+1 − Γi)
2 ∥φi∥2 +

1

ε

∫
D

(
⟨∇ψ,∇ui⟩+

uiLξψ

2
−

k∑
j=1

tijuj

)2

e⟨ξ,X⟩g0dv.

(2.32)

From (2.30) and (2.32), we infer that

(Γk+1 − Γi)
(∫

D

u2i |∇ψ|2e⟨ξ,X⟩g0dv + 2

k∑
j=1

aijtij

)

≤ ε (Γk+1 − Γi)
2 ∥φi∥2 +

1

ε

[ ∫
D

(
⟨∇ψ,∇ui⟩+

uiLξψ

2

)2

e⟨ξ,X⟩g0dv −
k∑

j=1

t2ij

]
≤ ε (Γk+1 − Γi)

{∫
D

[
u2i (Lξψ)

2 + 4
(
⟨∇ψ,∇ui⟩2 + uiLξψ⟨∇ψ,∇ui⟩

)
− 2|∇ψ|2uiLξui

]
e⟨ξ,X⟩g0dv −

k∑
j=1

aijsij

}

+
1

ε

[ ∫
D

(
⟨∇ψ,∇ui⟩+

uiLξψ

2

)2

e⟨ξ,X⟩g0dv −
k∑

j=1

t2ij

]
.

(2.33)



10 L. ZENG, Z. ZHOU EJDE-2025/31

Summing over i from 1 to k for (2.33), we obtain

k∑
i=1

(Γk+1 − Γi)
(∫

D

u2i |∇ψ|2 + 2

k∑
j=1

aijtij

)
e⟨ξ,X⟩g0dv

≤ ε

k∑
i=1

(Γk+1 − Γi)
{∫

D

[
u2i (Lξψ)

2
+ 4

(
⟨∇ψ,∇ui⟩2 + uiLξψ⟨∇ψ,∇ui⟩

)
− 2|∇ψ|2uiLξui

]
e⟨ξ,X⟩g0dv −

k∑
j=1

aijsij

}

+
1

ε

k∑
i=1

[ ∫
D

(
⟨∇ψ,∇ui⟩+

uiLξψ

2

)2

e⟨ξ,X⟩g0dv −
k∑

j=1

t2ij

]
,

which implies that

k∑
i=1

(Γk+1 − Γi)
(∫

D

u2i |∇ψ|2 + 2

k∑
j=1

aijtij

)
e⟨ξ,X⟩g0dv

≤ ε

k∑
i=1

(Γk+1 − Γi)

∫
D

[
u2i

(
Lξψ

)2
+ 4

(
⟨∇ψ,∇ui⟩2 + uiLξψ⟨∇ψ,∇ui⟩

)
− 2|∇ψ|2uiLξui

]
e⟨ξ,X⟩g0dv − ε

k∑
i,j=1

(Γk+1 − Γi) aijsij

+
1

ε

k∑
i=1

∫
D

(
⟨∇ψ,∇ui⟩+

uiLξψ

2

)2

e⟨ξ,X⟩g0dv − 1

ε

k∑
i,j=1

t2ij .

(2.34)

We can rewrite the left-hand side of (2.34) as

k∑
i=1

(Γk+1 − Γi)
(∫

D

u2i |∇ψ|2e⟨ξ,X⟩g0dv + 2

k∑
j=1

aijtij

)

=

k∑
i=1

(Γk+1 − Γi)

∫
D

u2i |∇ψ|2e⟨ξ,X⟩g0dv + 2

k∑
i,j=1

(Γk+1 − Γi) aijtij .

(2.35)

It follows from (2.13), (2.14), (2.34) and (2.35) that

k∑
i=1

(Γk+1 − Γi)

∫
D

u2i |∇ψ|2e⟨ξ,X⟩g0dv +

k∑
i,j=1

sijtij

≤ ε

k∑
i=1

(Γk+1 − Γi)

∫
D

[
u2i (Lξψ)

2
+ 4

(
⟨∇ψ,∇ui⟩2 + uiLξψ⟨∇ψ,∇ui⟩

)
− 2|∇ψ|2uiLξui

]
e⟨ξ,X⟩g0dv − ε

2

k∑
i,j=1

s2ij

+
1

ε

k∑
i=1

∫
D

(
⟨∇ψ,∇ui⟩+

uiLξψ

2

)2

e⟨ξ,X⟩g0dv − 1

ε

k∑
i,j=1

t2ij .



EJDE-2025/31 EIGENVALUE BOUNDS FOR THE CLAMPED PLATE PROBLEM 11

We infer from the above inequality that

k∑
i=1

(Γk+1 − Γi)

∫
Q

u2i |∇ψ|2e⟨ξ,X⟩g0dv +
(ε
2

k∑
i,j=1

s2ij +

k∑
i,j=1

sijtij +
1

ε

k∑
i,j=1

s2ij

)

≤ ε

k∑
i=1

(Γk+1 − Γi)

∫
Q

[
u2i (Lξψ)

2
+ 4

(
⟨∇ψ,∇ui⟩2 + uiLξψ⟨∇ψ,∇ui⟩

)
− 2|∇ψ|2uiLξui

]
e⟨ξ,X⟩g0dv +

1

ε

k∑
i=1

∫
D

(
⟨∇ψ,∇ui⟩+

uiLξψ

2

)2

e⟨ξ,X⟩g0dv.

(2.36)

Noticing that

ε

2

k∑
i,j=1

s2ij +

k∑
i,j=1

sijtij +
1

ε

k∑
i,j=1

t2ij =
1

2ε

k∑
i,j=1

[
(εsij + tij)

2
+ t2ij

]
≥ 0, (2.37)

and ∫
D

(
⟨∇ψ,∇ui⟩+

uiLξψ

2

)2

e⟨ξ,X⟩g0dv

=
1

4

∫
D

[
u2i (Lξψ)

2
+ 4

(
⟨∇ψ,∇ui⟩2 + uiLξψ⟨∇ψ,∇ui⟩

)]
e⟨ξ,X⟩g0dv,

(2.38)

from (2.36)-(2.38), we abtain inequality (2.22). The proof is complete. □

3. Some extrinsic formulas of Chen-Cheng type

From now on, we set the following convention on the ranges of indices:

1 ≤ i, j, . . . ,≤ n; 1 ≤ α, β, . . . ,≤ N.

Suppose that
(
x1, . . . , xn

)
is an arbitrary coordinate system in a neighborhood U of P in Mn.

Assume that x with components xα defined by xα = xα
(
x1, . . . , xn

)
, 1 ≤ α ≤ N , is the position

vector of P in RN . To prove our main results, the following auxiliary lemmas will play very
important roles, and their proofs can be found in [8, 11, 35, 36].

Lemma 3.1. For an n-dimensional submanifold Mn in the Euclidean space RN , let x = (x1, x2, . . . , xN )
be the position vector of a point P ∈ Mn with xα = xα(x1, . . . , xn), 1 ≤ α ≤ N , where (x1, . . . , xn)
denotes a local coordinate system of Mn. Then, we have

N∑
α=1

⟨∇xα,∇xα⟩g = n, (3.1)

N∑
α=1

⟨∇xα,∇u⟩g⟨∇xα,∇ū⟩g = ⟨∇u,∇ū⟩g, (3.2)

for any functions u, ū ∈ C1(Mn),

N∑
α=1

(∆xα)2 = n2H2, (3.3)

N∑
α=1

∆xα∇xα = 0, (3.4)

where H is the mean curvature of Mn.

From (3.1), we have ∫
D

u2i

N∑
α=1

|∇xα|2ge⟨ξ,X⟩g0dv = n. (3.5)



12 L. ZENG, Z. ZHOU EJDE-2025/31

From (3.2), it is easy to check that

N∑
α=1

⟨∇xα,∇ui⟩2g = |∇ui|2g. (3.6)

From (3.4), we can verify that

N∑
α=1

∆xα⟨∇xα,∇ui⟩g =

N∑
α=1

⟨∆xα∇xα,∇ui⟩g = 0, (3.7)

N∑
α=1

∆xα⟨∇xα, ξ⟩g0 =

N∑
α=1

⟨∆xα∇xα, ξ⟩g0 = 0. (3.8)

Straightforward calculations show that

N∑
α=1

⟨∇xα, ξ⟩2g0 = |ξ⊤|2g0 . (3.9)

By the Cauchy-Schwarz inequality and (3.9), we have

N∑
α=1

⟨∇xα,∇ui⟩g⟨∇xα, ξ⟩g0 ≤ |∇ui|g|ξ⊤|g0 . (3.10)

Combining (3.10) with (3.7), we conclude that

N∑
α=1

Lξxα⟨∇xα,∇ui⟩g =

N∑
α=1

(∆xα + ⟨∇xα, ξ⟩g0) ⟨∇xα,∇ui⟩g ≤ |∇ui|g|ξ⊤|g0 . (3.11)

Lemma 3.2. Let
(
x1, . . . , xn

)
be an arbitrary coordinate system in a neighborhood U of P ∈ Mn.

Assume that x with components xα defined by xα = xα
(
x1, . . . , xn

)
, 1 ≤ α ≤ N , is the position

vector of P in RN . Then
N∑

α=1

⟨∇xα, ξ⟩2g0 = |ξ⊤|2g0 , (3.12)

where ∇ is the gradient operator on Mn.

Lemma 3.3. Let
(
x1, . . . , xn

)
be an arbitrary coordinate system in a neighborhood U of P ∈ Mn.

Assume that x with components xα defined by xα = xα
(
x1, . . . , xn

)
, 1 ≤ α ≤ N , is the position

vector of P ∈ RN . Then

N∑
α=1

⟨∇xα,∇u⟩g⟨∇xα, ξ⟩g0 ≤ |∇u|g|ξ⊤|g0 , (3.13)

where ∇ is the gradient operator on Mn.

4. Proof of Theorem 1.1

Based on the arguments from the previous section, we can establish the following lemma (see
[35, 36]).

Lemma 4.1. Let x1, x2, . . . , xN be the standard coordinate functions of RN . For any i = 1, 2, . . . k
and α = 1, 2, . . . , N , let

Ψi,α :=
1

4

∫
D

Ψi(xα)e
⟨ξ,X⟩g0dv, Θi,α :=

1

4

∫
D

Θi(xα)e
⟨ξ,X⟩g0dv,

where the functions Ψi and Θi are given by (2.23) and (2.24), respectively. Then, we have

N∑
α=1

Ψi,α ≤
∫
D

[
|∇ui|2g +

1

4
u2i

(
n2H2 + |ξ⊤|2g0

) ]
e⟨ξ,X⟩g0dv

+ Γ
1/4
i

[ ∫
D

(ui|ξ⊤|g0)2e⟨ξ,X⟩g0dv
]1/2

,

(4.1)



EJDE-2025/31 EIGENVALUE BOUNDS FOR THE CLAMPED PLATE PROBLEM 13

and

4

N∑
α=1

(Ψi,α −Θi,α) ≤
∫
D

[
− 2nuiLξui + 4|∇ui|2g + u2i

(
n2H2 + |ξ⊤|2g0

) ]
e⟨ξ,X⟩g0dv

+ 4Γ
1/4
i

(∫
D

u2i |ξ⊤|2g0e
⟨ξ,X⟩g0dv

)1/2

.

(4.2)

To prove Theorem 1.1, we need the following embedding theorem due to Nash [27].

Theorem 4.2. Each complete Riemannian manifold Mn can be isometrically immersed into a
Euclidian space RN .

Proof of Theorem 1.1. Since Mn is a complete Riemannian manifold, Nash’s embedding Theorem
4.2 implies that there exists an isometric embedding from Mn into a Euclidean space RN . Thus,
Mn can be considered as an n-dimensional complete isometrically embedded submanifold in RN .
Taking ψ = xα in Lemma 2.5, by the definitions of Ψ̂i,α and Θ̂i,α in Lemma 4.1, we have

k∑
i=1

(Γk+1 − Γi)

∫
D

u2i |∇xα|2e⟨ξ,X⟩g0dv

≤ ε

k∑
i=1

(Γk+1 − Γi)

∫
D

[Ψi (xα)−Θi (xα)] e
⟨ξ,X⟩g0dv +

1

4ε

k∑
i=1

∫
D

Ψi (xα) e
⟨ξ,X⟩g0dv

= 4ε

k∑
i=1

(Γk+1 − Γi)

∫
D

(Ψi,α −Θi,α) e
⟨ξ,X⟩g0dv +

1

ε

k∑
i=1

∫
D

Ψi,αe
⟨ξ,X⟩g0dv.

(4.3)

By (3.1), we have
N∑

α=1

∫
D

u2i |∇xα|2ge⟨ξ,X⟩g0dv = n. (4.4)

Using (4.4), and summing over α from 1 to N for (4.3), one has

n

k∑
i=1

(Γk+1 − Γi) ≤
k∑

i=1

N∑
α=1

4ε (Γk+1 − Γi) (Ψi,α −Θi,α) +

k∑
i=1

N∑
α=1

1

ε
Ψi,α

= 4ε

k∑
i=1

(Γk+1 − Γi)
[ N∑
α=1

(Ψi,α −Θi,α)
]
+

1

ε

k∑
i=1

( N∑
α=1

Ψi,α

)
.

(4.5)

Next, we give the upper bounds for Ψi,α and Ψi,α −Θi,α. Clearly, eigenvalues are some invariants
in the sense of isometries, therefore, we can set

C1 =
1

4
inf
σ∈Π

max
D

(
n2H2

)
, C̃1 =

1

4
max
D

|ξ⊤|g0 ,

where Π stands for the set of all isometric immersions from Mn into a Euclidean space. By the
divergence theorem and Cauchy-Schwarz inequality, we conclude that∫

D

|∇ui|2ge⟨ξ,X⟩g0dv = −
∫
D

uiLξuie
⟨ξ,X⟩g0dv

≤
{∫

D

u2i e
⟨ξ,X⟩g0dv

}1/2{∫
D

(Lξui)
2
e⟨ξ,X⟩g0dv

}1/2

= Γ
1/2
i .

(4.6)

It follows from (4.2), (4.6) and (4.1) that

4

N∑
α=1

(Ψi,α −Θi,α) ≤ (2n+ 4)Γ
1/2
i + 4

(
4C̃1Γ

1/4
1 + 4C̃2

1 + C1

)
, (4.7)

and
N∑

α=1

Ψi,α ≤ Γ
1/2
i + 4C̃1Γ

1/4
1 + 4C̃2

1 + C1. (4.8)



14 L. ZENG, Z. ZHOU EJDE-2025/31

Substituting (4.7) and (4.8) into (4.5), we have

n

k∑
i=1

(Γk+1 − Γi) ≤ ε

k∑
i=1

(Γk+1 − Γi)
[
(2n+ 4)Γ

1/2
i + 4C1

]
+

1

ε

k∑
i=1

(
Γ
1/2
i + C1

)
,

where

C1 = 4C̃1Γ
1/4
1 + 4C̃2

1 + C1.

In above inequality, taking

ε =

[∑k
i=1

(
Γ
1/2
i + C1

)]1/2[∑k
i=1(Γk+1 − Γi)

(
(2n+ 4)Γ

1/2
i + 4C1

)]1/2 ,
we obtain

n

k∑
i=1

(Γk+1 − Γi)

≤ 2
{ k∑

i=1

(Γk+1 − Γi)
[
(2n+ 4)Γ

1/2
i + 4C1

]}1/2{ k∑
i=1

(
Γ
1/2
i + C1

)}1/2
,

which is equivalent to (1.11). The proof is complete. □

Observing the proof of Theorem 1.1, one can establish the following result.

Corollary 4.3. Let (Mn, g) be an n-dimensional complete Riemannian manifold isometrically
embedded into the Euclidean space RN with mean curvature H, then eigenvalues Γi of the clamped
plate problem (1.3) of the L2

ξ operator satisfy

k∑
i=1

(Γk+1 − Γi)

≤ 4

n

{ k∑
i=1

(Γk+1 − Γi)
(
(
n

2
+ 1)Γ

1/2
i + 4C̃1Γ

1/4
1 + 4C̃2

1 +

∫
D

n2H2u2i e
⟨ξ,X⟩g0dv

)}1/2

×
{ k∑

i=1

(
Γ
1/2
i + 4C̃1Γ

1/4
1 + 4C̃2

1 +

∫
D

n2H2u2i e
⟨ξ,X⟩g0dv

)}1/2

,

(4.9)

where C̃1 is a constant given by

C̃1 =
1

4
max
D

|ξ⊤|g0 .

5. Applications of Theorem 1.1

5.1. Eigenvalue inequalities on the translating solitons. In this subsectionwe discuss the
eigenvalues of L2

II on the complete translating solitons. Firstly, let us consider a smooth family of
immersions Xt(·) = X(·, t) : Mn → RN with corresponding images Mn

t = Xt(M
n) such that the

mean curvature equation system [24]

d

dt
Xt(x) = Ht(x), x ∈ Mn,

X(·, 0) = X(·) := Mn
0 ,

(5.1)

is satisfied, where Ht(x) := H(x, t) is the mean curvature vector of Mn
t at Xt(x) in RN and Mn

0

denotes the initial submanifold associated with the MCF (5.1). We assume that ξ0 is a constant
vector with unit length and denote by ξN0 the normal projection of ξ0 onto the normal bundle of
Mn in RN . A immersed submanifold X : Mn → RN is said to be a translating soliton of the MCF
(5.1), if it satisfies the system

H(x) = ξN0 (x), x ∈ Mn. (5.2)



EJDE-2025/31 EIGENVALUE BOUNDS FOR THE CLAMPED PLATE PROBLEM 15

We remark that translating soliton is a special solutions of the MCF equations (5.1) and occurs
as Type-II singularity of the MCF equations (5.1), which play an important role in the study of
the MCF [5]. In 2015, Xin [33] studied some basic properties of translating solitons: the volume
growth, generalized maximum principle, Gauss maps and certain functions related to the Gauss
maps. In addition, by estimating the point-wise estimates and integral estimates for |A|2, Xin
proved some rigidity theorems for translating solitons in the Euclidean space in higher codimension.
Here, |A|2 denotes the squared norm of the second fundamental form. For more details, we refer the
reader to the excellent survey paper [34] and references therein. In 2016, using a new Omori-Yau
maximal principle, Chen and Qiu [10] proved the nonexistence of spacelike translating solitons.

Suppose that ξ0 is a unit vector field satisfying (5.2). Then Lξ0 exactly is the LII operator
introduced by Xin in [33] and similar to the L operator introduced by Colding and Minicozzi in
[15]. Therefore, Lξ operator can be regarded as a extension of LII operator. As an application
of Theorem 1.1, we investigate the eigenvalues of the L2

II operator on the complete translating
solitons. In other words, we prove the following theorem.

Theorem 5.1 (see Theorem 1.3). Let Mn be an n-dimensional complete translating soliton iso-
metrically embedded into the Euclidean space RN with mean curvature H. Then, eigenvalues of
clamped plate problem (1.3) of the L2

II operator satisfy

k∑
i=1

(Γk+1 − Γi) ≤
4

n

{ k∑
i=1

(Γk+1 − Γi)

((n
2
+ 1

)
Γ
1/2
i + Γ

1/4
i +

n2

4

)}1/2

×
{ k∑

i=1

(
Γ
1/2
i + Γ

1/4
i +

n2

4

)}1/2

.

(5.3)

Proof. Since Mn is an n-dimensional complete translator isometrically embedded into the Eu-
clidean space RN , we have

H = ξ⊥0 , (5.4)

and

|ξ⊤0 |2g0 ≤ |ξ0|2g0 = 1, (5.5)

which implies that

n2H2 + |ξ⊤0 |2g0 = n2|ξ⊥0 |2g0 + |ξ⊤0 |2g0 ≤ n2. (5.6)

combining (5.4), (5.5) and (5.6) yields

1

4

∫
D

u2i
(
n2H2 + |ξ⊤0 |2g0

)
e⟨ξ0,X⟩g0dv ≤ n2

4
. (5.7)

Substituting (5.7) into (4.9), we obtain

k∑
i=1

(Γk+1 − Γi) ≤
4

n

{ k∑
i=1

(Γk+1 − Γi)
((n

2
+ 1

)
Γ
1/2
i + Γ

1/4
i +

n2

4

)}1/2

×
{ k∑

i=1

(
Γ
1/2
i + Γ

1/4
i +

n2

4

)}1/2

.

The proof is complete. □

5.2. Submanifolds in unit sphere and projective spaces. In this subsection, we investigate
the eigenvalues on the setting of the submanifolds in unit sphere and projective spaces. To this
end, let us recall some fundamental facts for the submanifolds on the projective spaces and refer
the reader to [7] for more information. Suppose that F is the field R of real numbers, the field
C of complex numbers or the field Q of quaternions. In what follows, we use the notations from
[35, 36]:

dF = dimR F =


1, if F = R;
2, if F = C;

4, if F = Q.

(5.8)



16 L. ZENG, Z. ZHOU EJDE-2025/31

Denote by FPm the real projective space with dimension m if F = R, the complex projective
space with real dimension 2m if F = C, and the quaternionic projective space with real dimension
4m if F = Q, respectively. It is well known that the manifold FPm carries a natural metric such
that the Hopf fibration η : SdF·(m+1)−1 ⊂ Fm+1 → FPm is a Riemannian fibration. Let

Hm+1(F) =
{
A ∈ Mm+1(F) : A

∗ := tA = A
}

be the vector space consisting of (m + 1) × (m + 1) Hermitian matrices with coefficients in the
field F. Now, let us endow Hm+1(F) with an inner product of the form

⟨A,B⟩ = 1

2
tr(AB),

where tr(·) represents the trace for the given matrix of (m+1)× (m+1) type. It is clear that the
map η : SdF·(m+1)−1 ⊂ Fm+1 → Hm+1(F) given by

η(ζ) = ζζ∗ =


|ζ0|2 ζ0ζ1 . . . ζ0ζm
ζ1ζ0 |ζ1|2 . . . ζ1ζm
. . . . . . . . . . . .

ζmζ0 ζmζ1 . . . |ζm|2


induces through the Hopf fibration: an isometric embedding η from FPm intoHm+1(F), where ζ =
(ζ0, ζ1, . . . , ζm) ∈ SdF·(m+1)−1. In addition, η

(
FPm

)
is a minimal submanifold of the hypersphere

S
(

I
m+1 ,

√
m

2(m+1)

)
of Hm+1(F) with radius

√
m

2(m+1) and center I
m+1 , where I stands for the

identity matrix. In accordance with the above notations, one can show the following lemma (see
[7, Lemma 6.3 in Chapter 4]):

Lemma 5.2. Let ρ : Mn → FPm be an isometric immersion, and let Ĥ and H be the mean
curvature vector fields of the immersions ρ and η ◦ ρ, respectively (here η is the induced isometric
embedding η from FPm into Hm+1(F) explained above). Then

|H|2 = |Ĥ|2 + 4(n+ 2)

3n
+

2

3n2

∑
i ̸=j

K (ei, ej) ,

where {ei}ni=1 is a local orthonormal basis of Γ(TMn) and K is the sectional curvature of FPm

expressed by

K (ei, ej) =


1, if F = R;
1 + 3 (ei · Jej)2 , if F = C;

1 +
∑3

r=1 3 (ei · Jrej)
2
, if F = Q,

where J is the complex structure of CPm and Jr is the quaternionic structure of QPm.

One can infer from Lemma 5.2 that

|H|2 =


|Ĥ|2 + 2(n+1)

2n , for RPm;

|Ĥ|2 + 2(n+1)
2n + 2

n2

∑n
i,j=1 (ei · Jej)

2 ≤ |Ĥ|2 + 2(n+2)
n , for CPm;

|Ĥ|2 + 2(n+1)
2n + 2

n2

∑n
i,j=1

∑3
r=1 (ei · Jrej)

2 ≤ |Ĥ|2 + 2(n+4)
n , for QPm.

From this equality it follows that

|H|2 ≤ |Ĥ|2 + 2 (n+ dF)

n
. (5.9)

In (5.9), the equality holds iff Mn is a complex submanifold of CPm (for the case CPm) while
n ≡ 0(mod4) and Mn is an invariant submanifold of QPm for the case QPm). Let X : Mn −→
M̄m be the standard embeddings from the submanifold Mn to ambient space M̄m, where M̄m

denotes the Euclidean space Rm, unit sphere Sm or projective spaces FPm by the coordinate
functions, respectively. In addition, Ĥ, H̄ and H̃ are used to denote the mean curvature vector



EJDE-2025/31 EIGENVALUE BOUNDS FOR THE CLAMPED PLATE PROBLEM 17

fields of the embeddings from Mn to Rm, Sm and FPm, respectively. For convenience, we define
the nonnegative integers c(n) and the set Π̂ as follows:

c(n) =


1
4

∫
D
n2|H|2u2i e⟨ξ,X⟩g0dv, if M̄m = Rm;

1
4

∫
D

(
n2|H̄|2 + n2

)
u2i e

⟨ξ,X⟩g0dv, if M̄m = Sm;
1
4

∫
D

(
n2|H̃|2 + 2n(n+ dF)

)
u2i e

⟨ξ,X⟩g0dv, if M̄m = FPm,

where

dF = dimR F =


1, if F = R;
2, if F = C;

4, if F = Q,

(5.10)

and Π̂ =: {σ : Mn → FPm|σ is a isometric immersion}. Then, by the same arguments as in [35,
Corollaries 6.1, 6.2, 6.3, 6.5] or [36, Corollaries 4.1, 4.2, 4.3, 4.5], and applying Corollary 4.3 and
Lemma 5.2, we can prove the following theorem.

Theorem 5.3. Let M̄m be Rm, Sm or FPm and X : Mn −→ M̄m be an isometric immersion
with mean curvature vector fields H, H̄ or H̃. For any bounded potential q on Mn, the spectrum
of L2

ν must satisfy

k∑
i=1

(Γk+1 − Γi) ≤
4

n

{ k∑
i=1

(Γk+1 − Γi)
((n

2
+ 1

)
Γ
1/2
i + 4C̃1Γ

1/4
1 + 4C̃2

1 + c(n)
)}1/2

×
{ k∑

i=1

(
Γ
1/2
i + 4C̃1Γ

1/4
1 + 4C̃2

1 + c(n)
)}1/2

, .

where C̃1 is a constant given by

C̃1 =
1

4
max
D

|ξ⊤|g0 .

We define a constant

c̄(n) =


0, if M̄m = Rm;
n2

4 , if M̄m = Sm;
n(n+dF)

2 , if M̄m = FPm.

As a consequence of Theorem 5.3, we can establish the following corollary.

Corollary 5.4. Under the hypotheses of Theorem 5.3, if the immersion are minimal, then

k∑
i=1

(Γk+1 − Γi) ≤
4

n

{ k∑
i=1

(Γk+1 − Γi)
((n

2
+ 1

)
Γ
1/2
i + 4C̃1Γ

1/4
1 + 4C̃2

1 + c̄(n)
)}1/2

×
{ k∑

i=1

(
Γ
1/2
i + 4C̃1Γ

1/4
1 + 4C̃2

1 + c̄(n)
)}1/2

,

(5.11)

where C̃1 is a constant given by

C̃1 =
1

4
max
D

|ξ⊤|g0 .

Acknowledgments. This research was partially supported by the National Natural Science Foun-
dation of China (Grant No. 12461007), and by the Natural Science Foundation of Jiangxi Province
(Grant No. 20224BAB201002). The author wants ot express his sincere gratitude to the anony-
mous referees for their helpful comments and suggestions.



18 L. ZENG, Z. ZHOU EJDE-2025/31

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Lingzhong Zeng
College of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022, China

Email address: Lingzhongzeng@yeah.net

Ziyi Zhou

College of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022, China

Email address: zyzhouu@163.com