DSpace at VNU: p-harmonic l-forms on Riemannian manifolds with a weighted Poincare inequality

Nonlinear Analysis 150 (2017) 138–150

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Nonlinear Analysis
www.elsevier.com/locate/na

p-harmonic ℓ-forms on Riemannian manifolds with a weighted
Poincar´e inequality
Nguyen Thac Dung
Department of Mathematics, Mechanics, and Informatics (MIM), Hanoi University of Sciences
(HUS-VNU), Vietnam National University, 334 Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam

article

info

Article history:
Received 7 June 2016
Accepted 10 November 2016
Communicated by Enzo Mitidieri
MSC:
53C24
53C21
Keywords:
Flat normal bundle
p-harmonic ℓ-forms
The second fundamental form
Weighted Poincar´
e inequality
Weitzenb¨

ock curvature operator

abstract
Given a Riemannian manifold with a weighted Poincar´
e inequality, in this paper, we
will show some vanishing type theorems for p-harmonic ℓ-forms on such a manifold.
We also prove a vanishing result on submanifolds in Euclidean space with flat
normal bundle. Our results can be considered as generalizations of the work of
Lam, Li–Wang, Lin, and Vieira (see Lam (2008), Li and Wang (2001), Lin (2015),
Vieira (2016)). Moreover, we also prove a vanishing and splitting type theorem for
p-harmonic functions on manifolds with Spin (9) holonomy provided a (p, p, λ)Sobolev type inequality which can be considered as a general Poincar´
e inequality
holds true.
© 2016 Elsevier Ltd. All rights reserved.

1. Introduction
Let (M n , g) be a Riemannian manifold of dimension n and ρ ∈ C(M ) be a positive function on M . We
say that M has a weighted Poincar´e inequality, if


ρϕ2 ≤
|∇ϕ|2
(1.1)
M

C0∞ (M )

M

holds true for any smooth function ϕ ∈

with compact support in M . The positive function ρ is
called the weighted function. It is easy to see that if the bottom of the spectrum of Laplacian λ1 (M ) is
positive then M satisfies a weighted Poincar´e inequality with ρ ≡ λ1 . Here λ1 (M ) can be characterized by
variational principle


2
| ∇ϕ |
∞
M
λ1 (M ) = inf
: ϕ ∈ C0 (M ) .
ϕ2
M
E-mail address:
/>0362-546X/© 2016 Elsevier Ltd. All rights reserved.


N.T. Dung / Nonlinear Analysis 150 (2017) 138–150

139

When M satisfies a weighted Poincar´e inequality then M has many interesting properties concerning
topology and geometry. It is worth to notice that weighted Poincar´e inequalities not only generalize the first
eigenvalue of the Laplacian, but also appear naturally in other PDE and geometric problems. For example,
λ1 (M ) is related to the problem of finding the best constant in the inequality
∥u∥L2 ≤ C∥∇u∥L2
obtained by the continuous embedding W01,2 → L2 (M ). It is also well known that a stable minimal
hypersurface satisfies a weighted Poincar´e inequality with the weight function
2

ρ = |A| + Ric(ν, ν)
where A is the second fundamental form and Ric(ν, ν) is the Ricci curvature of the ambient space in the
normal direction. For further discussion on this topic, we refer to [12,15,18,20,23] and the references there
in.
On the other hand, suppose that M is a complete noncompact oriented Riemannian manifold of dimension
n. At a point x ∈ M , let {ω1 , . . . , ωn } be a positively oriented orthonormal coframe on Tx∗ (M ), for ℓ ≥ 1,
the Hodge star operator is given by
∗(ωi1 ∧ · · · ∧ ωiℓ ) = ωj1 ∧ · · · ∧ ωjn−ℓ ,
where j1 , . . . , jn−ℓ is selected such that {ωi1 , . . . , ωiℓ , ωj1 , . . . , ωjn−ℓ } gives a positive orientation. Let d is the
exterior differential operator, so its dual operator δ is defined by
δ = ∗d ∗ .
Then the Hodge–Laplace–Beltrami operator ∆ acting on the space of smooth ℓ-forms Ω ℓ (M ) is of form
∆ = −(δd + dδ).
In [15], Li studied Sobolev inequality on spaces of harmonic ℓ-forms. Then he gave estimates of the bottom
of ℓ-spectrum and proved that the space of harmonic ℓ-forms is of finite dimension provided the Ricci
curvature bounded from below. When M is compact, it is well-known that the space of harmonic ℓ-forms is
isomorphic to its ℓ-th de Rham cohomology group. This is not true when M is non-compact but the theory
of L2 harmonic forms still has some interesting applications. For further results, one can refer to [4,5]. Later,
in [20], the authors investigated spaces of L2 harmonic ℓ-forms H ℓ (L2 (M )) on submanifolds in Euclidean
space with flat normal bundle. Assuming that the submanifolds are of finite total curvature, Lin showed
that the space H ℓ (L2 (M )) has finite dimension. Recently, in [23], Vieira obtained vanishing theorems for L2
harmonic 1 forms on complete Riemannian manifolds satisfying a weighted Poincar´e inequality and having
a certain lower bound of the curvature. His theorems improve Li–Wang’s and Lam’s results. Moreover,
some applications to study geometric structures of minimal hypersurfaces are also given. Therefore, it
is very natural for us to study p-harmonic ℓ-forms on Riemannian manifolds with a weighted Poincar´e
inequality.
Recall that an ℓ-form ω on M is said to be p-harmonic (p > 1) if ω satisfies dω = 0 and δ(|ω|p−2 ω) = 0.
When p = 2, a p-harmonic ℓ-form is exactly a harmonic ℓ-form. Some properties of the space of p-harmonic
ℓ-forms are given by X. Zhang and Chang–Guo–Sung (see [24,7]). In particular, in [6], Chang–Chen–Wei

studied p-harmonic functions with finite Lq energy and proved some vanishing type theorems on Riemannian
manifold satisfying a weighted Poincar´e inequality, recently. Moreover, Sung–Wang, Dat and the author used
theory of p-harmonic functions to show some interesting rigidity properties of Riemannian manifolds with
maximal p-spectrum. (See also [8,22]).


N.T. Dung / Nonlinear Analysis 150 (2017) 138–150

140

In this paper, we will prove the following theorem.
Theorem 1.1. Suppose that M is a Riemannian manifold satisfying the weighted Poincar´e inequality (1.1) with the positive weighted function ρ. If the Weitzenb¨
ock curvature operator Kℓ > −aρ, and
4(p−1)
a < p2 then every p-harmonic ℓ-form ( 2 ≤ ℓ ≤ n − 2) with finite Lp norm on M is trivial.
This theorem can be considered as a generalization of the work of Li–Wang, Lam, and Vieira (see
[17,13,23]). The second vanishing property of this paper is an extension of Lin’s result. Our theorem is
stated as follows.
Theorem 1.2. Let M n , n ≥ 3 be a complete non-compact immersed submanifold of Rn+k with flat normal
bundle. Denote by H the mean curvature vector of M . If one of the following conditions
2

2

n |H|
,
1. |A|2 ≤ max{l,n−ℓ}
2. the total curvature ∥A∥n is bounded by

∥A∥2n < min



4(p − 1)
n2
,
2
max {ℓ, n − ℓ} p CS 2 max {ℓ, n − ℓ} CS


,

3. supM |A| is bounded and the fundamental tone satisfies
sup |A|2 p2
λ1 (M ) > max {ℓ, n − ℓ}

M

4(p − 1)

holds true then every p-harmonic ℓ-form on M is trivial.
Here the submanifold M is said to have flat normal bundle if the normal connection of M is flat, namely
the components of the normal curvature tensor of M are zero.
On the other hand, it is worth to mention that in [2] (see also [21]), the author considered complete
manifolds M with some (p, q, λ)-Sobolev inequality

λ

ϕq

 pq


≤

|∇ϕ|p

(1.2)

for some constant λ and for every ϕ ∈ C0∞ (M ). Here p, q are real numbers satisfying 1 < p ≤ q < ∞. Defining
1,p
(M ) by
the p-Laplacian of a function u ∈ Wloc
∆p u = div(|∇u|p−2 ∇u).
Hence if u ∈ C ∞ (M ) is a p-harmonic function, namely ∆p u = 0 then du is a p-harmonic 1-form. Buckley
and Koskela noted that when p = q and M is a bounded Euclidean domain then −∆p u = λ|u|p−2 u has a
solution u ∈ C01 (M ). The variational principle tells us that


|∇ϕ|p
∞
M

λ ≤ inf
(M
)
:
ϕ
∈
C

.
0
ϕp
M
In the case p = 2, λ is the least eigenvalue for the Laplacian Dirichlet problem. Therefore, a (p, pλ)Sobolev inequality can be considered as a generalization of the Poincar´e inequality. When M is a complete
noncompact with Spin(9) holonomy, we will show that if a (p, p, λ)-Sobolev inequality holds true then either
M has no p-parabolic end; or M is splitting. Let us recall the definition of p-parabolic ends.


N.T. Dung / Nonlinear Analysis 150 (2017) 138–150

141

Definition 1.3. An end E of the Riemannian manifold M is called p-parabolic if for every compact subset
K⊂E

capp (K, E) := inf
|∇f |p = 0,
E

where the infimum is taken among all f ∈
p-nonparabolic.

Cc∞ (E)

such that f ≥ 1 on K. Otherwise, the end E is called

The paper has three sections. In Section 2, we will recall some auxiliary lemmas then give proofs of
Theorems 1.1 and 1.2. Then we will give an application to study p-harmonic ℓ-forms on submanifolds with
a stable condition. Finally, in Section 3, we will show that complete noncompact manifolds with Spin(9)

holonomy are splitting via the theory of p-harmonic functions.
2. p-harmonic ℓ-forms on Riemannian manifolds with a weighted Poincar´e inequality
Suppose that M is a complete noncompact Riemannian manifold satisfying a weighted Poincar´e inequality
(1.1). Let {e1 , . . . , en } be a local orthonormal frame on M on M with dual coframe {ω1 , . . . , ωn }. Given a
ℓ-form ω on M , the Weitzenb¨
ock curvature operator Kℓ acting on ω is defined by
Kℓ (ω, ω) =

n


ωk ∧ iej R(ek , ej )ω.

j.k=1

Using the Weitzenb¨
ock curvature operator, we have the following Bochner type formula for ℓ-forms.
Lemma 2.1 ([16,23]). Let ω =



I

aI ωI be a ℓ-form on M . Then

∆|ω|2 = 2 ⟨∆ω, ω⟩ + 2|∇ω|2 + 2 ⟨Eℓ (ω), ω⟩
= 2 ⟨∆ω, ω⟩ + 2|∇ω|2 + 2Kℓ (ω, ω)
where Eℓ (ω) =

n

j,k=1

ωk ∧ iej R(ek , ej )ω.

Apply the above Bochner formula to the form |ω|p−2 ω we obtain


1
∆|ω|2(p−1) = |∇(|ω|p−2 ω)|2 − (δd + dδ)(|ω|p−2 ω), |ω|p−2 ω + Kℓ (|ω|p−2 ω, |ω|p−2 ω)
2


= |∇(|ω|p−2 ω)|2 − δd(|ω|p−2 ω), |ω|p−2 ω + |ω|2(p−2) Kℓ (ω, ω)
where we used ω is p-harmonic in the second equality. This can be read as




|ω|p−1 ∆|ω|p−1 = |∇(|ω|p−2 ω)|2 − |∇|ω|p−1 |2 − |ω|p−2 δd(|ω|p−2 ω), ω + |ω|2(p−2) Kℓ (ω, ω).
By Kato type inequality |∇(|ω|p−2 ω)|2 ≥ |∇|ω|p−1 |2 (see [3]) and Kℓ ≥ −aρ, this implies


|ω|∆|ω|p−1 ≥ − δd(|ω|p−2 ω), ω − aρ|ω|p .

(2.3)

Now we will give a proof of Theorem 1.1.
Proof of Theorem 1.1. Let ϕ ∈ C0∞ (M ), then multiply both sides of (2.3) by ϕ2 then integrate the obtained
results, we have






ϕ2 |ω|∆|ω|p−1 ≥ −
δd(|ω|p−2 ω), ϕ2 ω − a
ρϕ2 |ω|p .
M

M

M


142

N.T. Dung / Nonlinear Analysis 150 (2017) 138–150

Integration by parts implies







∇(ϕ2 |ω|), ∇|ω|p−1 ≤
d(|ω|p−2 ω), d(ϕ2 ω) + a
ρϕ2 |ω|p

M
M
M



p−2
2
=
d(|ω| ) ∧ ω, d(ϕ ) ∧ ω + a
ρϕ2 |ω|p
M
M



 

p−2
2




≤
d(|ω| ) ∧ ω . d(ϕ ) ∧ ω + a
ρϕ2 |ω|p .
M

(2.4)

M

Here we used d(|ω|p−2 ω) = d(|ω|p−2 ) ∧ ω since dω = 0 in the second equality. Moreover, we used Schwartz
inequality in the last equality.
Observe that for any ℓ-form α and m-form β, it is proved in [10] that



 ℓ+m
.
|α ∧ β| ≤ c|α| · |β|, where c =
ℓ
Therefore the first term in the right hand side of (2.4) is estimated by





 

∇(|ω|p−2 ) |ω| · |d(ϕ2 )||ω|
d(|ω|p−2 ) ∧ ω  . d(ϕ2 ) ∧ ω  ≤ c2
M
M

2
= 2c |p − 2|
ϕ|ω|p−1 |∇ϕ| · |∇(|ω|)|
M



c2 |p − 2|
2
p−2
2 2
≤ c |p − 2|ε
|ω| |∇(|ω|)| ϕ +
|ω|p |∇ϕ|2
ε
M
M

(2.5)

for any ε > 0. Here we used the elementary inequality 2AB ≤ εA2 + ε−1 B 2 for any A, B ∈ R in the last
inequality.
Since M satisfies the weighted Poincar´e inequality, we can estimate the second term in the right hand
side of (2.4) by


  


2
p 2
p




ρϕ2 |ω|2 =
ρ ϕ|ω| 2  ≤
∇ ϕ|ω| 2 
M
M
M




2
1
2
≤ (1 + ε)
ϕ ∇|ω|p/2  + 1 +
|ω|p |∇ϕ|2
ε
M
M




2
1
(1 + ε)p2
2
p−2 
ϕ |ω|
∇|ω| + 1 +

|ω|p |∇ϕ|2
(2.6)
=
4
ε
M
M
for any ε > 0.
On the other hand, we compute the left hand side of (2.4) as follows.





∇(ϕ2 |ω|), ∇|ω|p−1 ≥ (p − 1)
|ω|p−2 |∇|ω||2 ϕ2 − 2(p − 1)
ϕ|ω|p−1 |∇ϕ||∇|ω||
M
M
M


p−2
2 2
≥ (p − 1)
|ω| |∇|ω|| ϕ − (p − 1)ε
|ω|p−2 |∇|ω||2 ϕ2
M
M


p−1
−
|ω|p |∇ϕ|2 .
ε
M
By (2.5)–(2.7), it turns out that there exist A, B ∈ R such that


A
|ω|p−2 |∇|ω|| 2 ϕ2 ≤ B
|ω|p |∇ϕ|2
M

M

(2.7)


N.T. Dung / Nonlinear Analysis 150 (2017) 138–150

143

where
a(1 + ε)p2
A = (1 − ε)(p − 1) −
− c2 |p − 2|ε
4


c2 |p − 2| p − 1

1
+
+
.
B =a 1+
ε
ε
ε
Since a <
that

4(p−1)
p2 ,

choosing ε small enough, this implies there exists a positive constant C = C(p, ε) > 0 such


|ω|p−2 |∇|ω||2 ϕ2 ≤ C



M

|ω|p |∇ϕ|2 .

M

Now, we choose ϕ ∈ C0∞ (M ) satisfying 0 ≤ ϕ ≤ 1, |∇ϕ| ≤ R2 and

ϕ = 1 on B(o, R)

ϕ = 0 on M \ B(o, 2R)
where o ∈ M is a fixed point and B(o, R) is the geodesic ball centered at o with radius R > 0. Then the
above inequality implies



2 
4C
4

p/2 
p−2
2
=
|ω|
|∇|ω||
≤
|ω|p .
∇|ω|

2
p2 B(o,R)
R
B(o,R)
M
Letting R → ∞, we conclude that |∇|ω|p/2 | = 0. Hence |ω| is constant, but since |ω| ∈ Lp (M ), it forces
|ω| = 0. Therefore ω = 0. The proof is complete.
Remark 2.1. If we assume p ≥ 2, ℓ = 1 then the refined Kato type inequality in [9] reads



κ
|∇(|ω|p−2 ω)|2 ≥ 1 +
|∇|ω|p−1 |2 ,
(p − 1)2
where


(p − 1)2
.
κ = min 1,
n−1
Using this refined Kato inequality we can show that if
a<

4(p − 1 + κ)
p2

then any p-harmonic 1-form with finite Lp norm is trivial. Hence, our result is a generalization of Li–Wang’s,
Lam’s, Vieira’s and Chang–Chen–Wei’s results [6,17,12,23].
Recall that let M n be an n-dimensional submanifold in the (n + k)-dimensional Euclidean space Rn+k .
M is said to be satisfied a δ-super stability condition for 0 < δ ≤ 1 if


δ
|A|2 ϕ2 ≤
|∇ϕ|2 ,
M

M

for any compactly supported Lipschitz function ϕ on M .
In particular, when δ = 1, M is said to be super-stable. Note that if k = 1 and δ = 1 then the concept of
super stability is the same as the usual definition of stability.
Corollary 2.2. Let M be an n-dimensional complete immersed minimal submanifold in Rn+k . If M is superstable then there is no non-trivial p-harmonic 1-form on M with finite Lp norm provided that
√
√
2(n + n)
2(n − n)
.
n−1
n−1
Consequently, for p = 2, every harmonic 1-form on M is trivial.


N.T. Dung / Nonlinear Analysis 150 (2017) 138–150

144

Proof. By Leung’s estimate for Ric M (see [14]), we have that
n−1 2
|A| .
n
Since M satisfies the stable condition, this implies that a weighted Poincar´e inequality holds true on M , with
2
the weighted function ρ = |A| . Therefore, by Theorem 1.1, M does not admit any non-trivial p-harmonic
ℓ-form if
n−1
4(p − 1)
<

n
p2
Ric M ≥ −

or equivalently,
√
√
2(n − n)
2(n + n)
.
n−1
n−1
The proof is complete.
Now, we consider p-harmonic forms on submanifolds in Euclidean space with flat normal bundle. To begin
with, let us recall a well-known Sobolev inequality.
Lemma 2.3 ([11]). Let M n (n ≥ 3) be an n-dimensional complete submanifold in a complete simply-connected
manifold with nonpositive sectional curvature. Then for any f ∈ W01,2 (M ) we have

 n−2

2n
n
n−2
≤ CS
dv
|∇f |2 + (|H|f )2 dv,
(2.8)
|f |
M

M

where CS is the Sobolev constant which depends only on n.
Let us recall Theorem 1.2.
Theorem 2.4. Let M n , n ≥ 3 be a complete non-compact immersed submanifold of Rn+k with flat normal
bundle. If one of the following conditions
2

2

n |H|
1. |A|2 ≤ max{ℓ,n−ℓ}
,
2. the total curvature ∥A∥n is bounded by

∥A∥2n < min



n2
4(p − 1)
,
2
max {ℓ, n − ℓ} p CS 2 max {ℓ, n − ℓ} CS


,

3. supM |A| is bounded and the fundamental tone satisfies

sup |A|2 p2
λ1 > max {ℓ, n − ℓ}

M

4(p − 1)

holds true then every p-harmonic ℓ-form on M is trivial.
Proof. Let ω be any p-harmonic ℓ-form with finite Lp norm. By Bochner formula and Kato inequality, we
have


|ω|∆|ω|p−1 ≥ − δd(|ω|p−2 ω), ω + Kℓ |ω|p .
It is proved in [20] that
Kℓ ≥


1 2
n |H|2 − max {ℓ, n − ℓ} |A|2 .
2


N.T. Dung / Nonlinear Analysis 150 (2017) 138–150

145

Therefore,

 1 2


|ω|∆|ω|p−1 ≥ − δd(|ω|p−2 ω), ω +
n |H|2 − max {ℓ, n − ℓ} |A|2 |ω|p .
2

1. If |A|2 ≤

n2 |H|2
max{ℓ,n−ℓ}

(2.9)

then


|ω|∆|ω|p−1 ≥ − δd(|ω|p−2 ω), ω .

Choosing ϕ ∈ C0∞ (M ) such that 0 ≤ ϕ ≤ 1 on M, ϕ = 1 on B(o, R) and ϕ = 0 outside the geodesic ball
B(o, 2R), where o ∈ M is a fixed point. Moreover, |∇ϕ| ≤ R2 for R > 0. Multiplying both sides of the
above inequality by ϕ2 then integrating the result, we obtain


ϕ2 |ω|∆|ω|p−1 ≥ −



M





δd(|ω|p−2 ω), ϕ2 ω .

M

Consequently,




∇(ϕ2 |ω|), ∇|ω|p−1 ≤

M






d(|ω|p−2 ω), d(ϕ2 ω) .

M

Using (2.5) and (2.7), we infer for any ε > 0

((p − 1)(1 − ε) − c(p − 2)ε)

2

p−2

ϕ |ω|

2



|∇|ω|| ≤

M

c(p − 2) p − 1
+
ε
ε



|ω|p |∇ϕ|2 .

M

Choosing ε > 0 small enough, this implies there exist a constant C > 0 such that
4
p2





2

C


|ω|p .
∇|ω|p/2  ≤ 2
R M
B(o,R)

Let R → ∞, we conclude that |ω| is constant. Since |ω| ∈ Lp (M ), it turns out that ω is zero.
2. By the previous part, we may assume |A|2 >

n2 |H|2
max{ℓ,n−ℓ}

then by (2.9), we have


 n2
∇(ϕ2 |ω|), ∇|ω|p−1 +
ϕ2 |H|2 |ω|p
2
M
M




p−2
2
≤

δd(|ω| ω), ϕ ω + max {ℓ, n − ℓ}





M

|A|2 ϕ2 |ω|p .

(2.10)

M

Now, we will estimate the last term of the right hand side of (2.10). By Lemma 2.3 and H¨older inequality,
we have
n−2
 

2n  n
 n−2
p
2 2
p
2
|ω| 2 ϕ
|A| ϕ |ω| ≤ ∥A∥n
M
M
  


2 
p


≤ CS ∥A∥2n
|H|2 |ω|p ϕ2 ,
∇ |ω| 2 ϕ  +
M

M


N.T. Dung / Nonlinear Analysis 150 (2017) 138–150

146

where CS is the Sobolev constant depending only on n. Since
  
2
2  
p
p
p




ϕ∇|ω| 2 + |ω| 2 ∇ϕ
∇ |ω| 2 ϕ  =

M
M





p 2
1
2
2
≤ (1 + ε)
ϕ ∇|ω|  + 1 +
|ω|p |∇ϕ|2
ε
Bx (R)
M



(1 + ε)p2
1
≤
ϕ2 |ω|p−2 |∇|ω||2 + 1 +
|ω|p |∇ϕ|2 ,
4
ε
M
M
we get


+ ε)p2
|A| |ω| ϕ ≤
ϕ2 |ω|p−2 |∇|ω||2
4
M
M



1
|H|2 |ω|p ϕ2 .
|ω|p |∇ϕ|2 + CS ∥A∥2n
+ CS ∥A∥2n 1 +
ε
M
M



p

2

(1
CS ∥A∥2n

2

Combining the above inequality, (2.5), (2.7) and (2.10), we have


A1

|ω|p−2 |∇|ω||2 +



M

n2
− max {ℓ, n − ℓ} CS ∥A∥2n
2



|H|2 |ω|p ϕ2 ≤ B1

M



|ω|p |∇ϕ|2 ,

M

where
(1 + ε)p2
A1 := (p − 1)(1 − ε) − c|p − 2|ε − max {ℓ, n − ℓ} CS ∥A∥2n
,
4



1
p − 1 c(p − 2)
+
+
.
B1 := max {ℓ, n − ℓ} CS ∥A∥2n 1 +
ε
ε
ε


4(p−1)
n2
Since ∥A∥2n < min max{ℓ,n−ℓ}p
, choosing ε > 0 small enough, we conclude that there
2 C , 2 max{ℓ,n−ℓ}C
S
S
exist two positive constants C1 , C2 > 0 such that


|ω|p−2 |∇|ω||2 + C1



B(o,R)

|H|2 |ω|p ≤

B(o,R)

C2
R2



|ω|p .

M

Let R → ∞, we infer |ω| is constant and |H||ω| = 0. Since |ω| ∈ Lp (M ), this implies ω is trivial.
3. Suppose that supM |A|2 < ∞, the last term of the right hand side can be estimated as follows.


2

2

p

sup |A|2 
M

|∇(|ω|p/2 ϕ)|2
λ1
M





sup |A|2 
(1 + ε)p2
1
M
2
p−2
2
p
2
≤
ϕ |ω| |∇|ω|| + 1 +
|ω| |∇ϕ| .
λ1
4
ε
M
M

|A| ϕ |ω| ≤
M

Combining this inequality, (2.5), (2.7) and (2.10), we infer

A2

|ω|p−2 |∇|ω||2 +

M

n2
2



|H|2 |ω|p ϕ2 ≤ B2

M



|ω|p |∇ϕ|2 ,

M

where
sup |A|2
A2 := (p − 1)(1 − ε) − c|p − 2|ε − max {ℓ, n − ℓ}
2

B2 := max {ℓ, n − ℓ}

M

λ1

(1 + ε)p2
,
4


sup |A| 
1
p − 1 c(p − 2)
M
1+
+
+
.
λ1
ε
ε
ε


N.T. Dung / Nonlinear Analysis 150 (2017) 138–150

147

2 2

M |A| p
Since λ1 > max {ℓ, n − ℓ} sup4(p−1)
, choose ε > 0 small enough, we conclude that there exist two positive
constants D1 , D2 > 0 such that


|ω|

p−2

2

|∇|ω|| + D1

B(o,R)



D2
|H| |ω| ≤ 2
R
B(o,R)
2

p



|ω|p .

M

Let R → ∞, we infer |ω| is constant and |H||ω| = 0. Since |ω| ∈ Lp (M ), this implies ω is trivial.
The proof is complete.
It is also worth to note that whenever a refined Kato inequality for p-harmonic ℓ-forms holds true then
we can improve the bound of ∥A∥ and λ1 (M ) as in Theorem 1.2. When p = 2, a refined Kato type inequality
for harmonic ℓ-forms was given in [3]. Let us finish this section by the following remark.
Remark 2.2. Recall that a Riemannian manifold N is said to have nonnegative isotropic curvature if

R1313 + R1414 + R2323 + R2424 − 2R1234 ≥ 0.
Furthermore, assume that N has pure curvature tensor, namely for every p ∈ N there is an orthonormal basis
{e1 , . . . , en } of the tangent space Tp N such that Rijkl := ⟨R(ei , ej )ek , el ⟩ = 0 whenever the set {i, j, k, l}
contains more than two elements. Here Rijkl denote the curvature tensors of N . It is worth to notice that
all 3-manifolds and conformally flat manifolds have pure curvature tensor.
By [20], we know that if M n be a compact immersed submanifold in N n+k with flat normal bundle, N
has pure curvature tensor and nonnegative isotropic curvature then

1 2
n |H|2 − max {ℓ, n − ℓ} |A|2 .
Kℓ ≥
2
Therefore, the results in Theorem 1.2 are valid provided the above conditions on ambient spaces N hold.
3. Rigidity of manifolds with spin(9) holonomy
Let M be a complete noncompact Riemannian manifold with holonomy group Spin(9). It was proved
in [1] that a manifold with holonomy group Spin(9) must be locally symmetric and its universal covering is
either the Cayley projective plane or the Cayley hyperbolic space H2O . Since M is noncompact, its universal
covering is H2O . The following theorem is proved in [12].
Theorem 3.1. Let M be a locally symmetric space with universal covering H2O then
∆M r ≤ 14 coth 2r + 8 coth r
in the sense of distributions. Here r(q) is the distance function between q ∈ M and a fixed point o ∈ M .
Moreover, V (B(R)) ≤ Ce22R , where B(R) stands for the geodesic ball centered at the point o ∈ M with
radius R.
1,p
Recall that for any function u ∈ Wloc
(M ) and p > 1, the p-Laplacian operator is defined by

∆p := div(|∇u|p−2 ∇u).
1,p
If u is a positive function u ∈ Wloc

(M ) such that for any ϕ ∈ W01,p (M ) we have




|∇u|p−2 ∇u, ∇ϕ = λ1,p
ϕup−1
M

M


N.T. Dung / Nonlinear Analysis 150 (2017) 138–150

148

then u is said to be p-eigenfunction with respect to the first eigenvalue λ1,p . By Theorem 3.1, we can estimate
λ1,p as follows.
Corollary 3.2. Let M be a locally symmetric space with universal covering H2O . If λ1,p is the first eigenvalue
with respect to the p-Laplacian then
 p
22
.
λ1,p ≤
p
Proof. Without loss of generality, we may assume that λ1,p is positive. The variational characterization of
λ1,p implies that M has infinite volume. Hence, by Theorem 0.1 in [2], M is p-nonparabolic, moreover
1/p

V (B(R)) ≥ C1 epλ1,p R .

Note that by Theorem 3.1, the growth of V (B(R)) is of at most e22R . Therefore,
 p
22
λ1,p ≤
.
p
The proof is complete.
Now, we will show a splitting property of complete noncompact manifolds with holonomy Spin(9).
Theorem 3.3. Let M be a complete noncompact 16-dimensional manifold with holonomy group Spin(9).
p
Suppose that the first eigenvalue λ1,p achieves the maximal value, namely λ1,p = 22
pp . Then either M has no
p-parabolic end; or M is a warped product M = R × N , where N is a compact manifold given by a compact
quotient of the horosphere of the universal cover of M .
Proof. Assume that M has a p-parabolic end E. Let β be the Busemann function associated with a geodesic
ray γ contained in E, namely
β(q) = lim (t − dist(q, γ(t))).
t→∞

The Laplacian comparison theorem in Theorem 3.1 implies that
∆β ≥ −22.
Let a =

22
p

and define g = eaβ , we compute
∆p g = ∆p (eaβ ) = div(ap−2 ea(p−2)β ∇(eaβ ))
= ap−1 div(ea(p−1)β ∇β)
= ap−1 (ea(p−1)β ∆β + b(p − 1)ea(p−1)β |∇β|2 )

≥ ap−1 ea(p−1)β (−ap + a(p − 1)) = −ap g p−1 = −λ1,p g p−1 .

For any nonnegative compactly supported smooth function ϕ on M , the variational characterization of λ1,p
implies


p
λ1,p
(ϕg) ≤
|∇(ϕg)|p .
M

M

On the other hand, integration by parts infers



ϕp g∆p g = −
ϕp |∇g|p − p
M

M

M

ϕp−1 g ⟨∇ϕ, ∇g⟩ |∇g|p−2 .


N.T. Dung / Nonlinear Analysis 150 (2017) 138–150

149

As in [22], we have that

p
|∇(ϕg)|p = |∇ϕ|2 g 2 + 2gϕ ⟨∇ϕ, ∇g⟩ + ϕ2 |∇g|2 2
≤ ϕp |∇g|p + pϕg ⟨∇ϕ, ∇g⟩ ϕp−2 |∇g|p−2 + c|∇ϕ|2 g p
for some constant c depending on p. Therefore, we obtain




ϕp g(∆p g + λ1,p g p−1 ) = λ1,p
(ϕg)p −
ϕp |∇g|p − p
ϕp−1 g ⟨∇ϕ, ∇h⟩ |∇h|p−2
M
M
M
M



p
p
p
≤
|∇(ϕg)| −
ϕ |∇g| − p

ϕp−1 g ⟨∇ϕ, ∇h⟩ |∇h|p−2
M
M
M

≤c
|∇ϕ|2 g p .

(3.1)

M

Now, for R > 0, we choose ϕ ∈ C0∞ (M ), 0 ≤ ϕ ≤ 1 such that

1, on B(R)
ϕ=
0 on M \ B(2R)
and |∇ϕ| ≤

2
R.

Then the right hand side of (3.1) can be estimated by



4
2 p
2 22β
e22β

|∇ϕ| g =
|∇ϕ| e
≤ 2
R M
M
M


4
4
≤ 2
e22β + 2
e22β .
R M ∩(B(2R)\B(R))
R (M \E)∩(B(2R)\B(R))

(3.2)

p

−22R
. Hence, the first term in the right
Since λ1,p = 22
pp , Theorem 0.1 in [2] implies that V (E \ B(R)) ≤ Ce
hand side of (3.2) tends to zero when R goes to ∞. It is well known that (for example, [19,12])

β(q) ≤ −r(q) + C
on M \ E. However, by Theorem 3.1, we have V (B(R)) ≤ Ce22R . This yields that the second term in the
right hand side of (3.2) approaches 0 when R converges to ∞. Letting R → ∞ in (3.1), we conclude that
∆p g + λ1,p g p−1 ≡ 0.

This means
∆β ≡ −22.
The proof is followed by using the argument in [12, Theorem 14]. We omit the details. The proof is
complete.
Acknowledgments
The author would like to express his deep thanks to the referees for their useful and constructive
comments/suggestions to improve the manuscript. In particular, the author thanks an anonymous referee
for pointing out many typos in the manuscript and correcting them.
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