DSpace at VNU: p-harmonic l-forms on Riemannian manifolds with a weighted Poincare inequality tài liệu, giáo án, bài giả...
Trang 1Contents lists available at ScienceDirect 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
a r t i c l e i n f o
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
a b s t r a c t
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
M
ρϕ2≤
M
holds true for any smooth function ϕ ∈ C0∞(M ) 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
λ1(M ) = inf
M | ∇ϕ |2
M ϕ2 : ϕ ∈ C0∞(M )
.
E-mail address:dungmath@gmail.com
http://dx.doi.org/10.1016/j.na.2016.11.008
0362-546X/© 2016 Elsevier Ltd All rights reserved.
Trang 2When 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
ρ = |A|2+ 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, , ω n } be a positively oriented orthonormal coframe on T∗
x (M ), for ℓ ≥ 1,
the Hodge star operator is given by
∗(ω i1∧ · · · ∧ ω i ℓ ) = ω j1∧ · · · ∧ ω j n−ℓ ,
where j1, , j n−ℓ is selected such that {ω i1, , ω i ℓ , ω j1, , ω j n−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 L2harmonic 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 L q 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])
Trang 3In this paper, we will prove the following theorem.
Theorem 1.1 Suppose that M is a Riemannian manifold satisfying the weighted Poincar´ e inequal-ity (1.1) with the positive weighted function ρ If the Weitzenb¨ ock curvature operator K ℓ > −aρ, and
a < 4(p−1) p2 then every p-harmonic ℓ-form ( 2 ≤ ℓ ≤ n − 2) with finite L p 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 R n+k with flat normal bundle Denote by H the mean curvature vector of M If one of the following conditions
1 |A|2≤ max{l,n−ℓ} n2|H|2 ,
2 the total curvature ∥A∥ n is bounded by
∥A∥2n < min
max {ℓ, n − ℓ} p2C S
2
2 max {ℓ, n − ℓ} C S
,
3 supM |A| is bounded and the fundamental tone satisfies
λ1(M ) > max {ℓ, n − ℓ}
sup
M
|A|2p2
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
ϕ q
p q
≤
for some constant λ and for every ϕ ∈ C0∞(M ) Here p, q are real numbers satisfying 1 < p ≤ q < ∞ Defining the p-Laplacian of a function u ∈ W loc 1,p (M ) by
∆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 ∈ C1(M ) The variational principle tells us that
λ ≤ inf
M |∇ϕ| p
M ϕ p : ϕ ∈ C0∞(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.
Trang 4Definition 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
E
|∇f | p = 0,
where the infimum is taken among all f ∈ C c∞(E) such that f ≥ 1 on K Otherwise, the end E is called
p-nonparabolic.
The paper has three sections In Section 2, we will recall some auxiliary lemmas then give proofs of
Theorems 1.1 and1.2 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, , e n } be a local orthonormal frame on M on M with dual coframe {ω1, , ω n} Given a
ℓ-form ω on M , the Weitzenb¨ ock curvature operator K ℓ acting on ω is defined by
K ℓ (ω, ω) =
n
j.k=1
ω k ∧ i e j R(e k , e j )ω.
Using the Weitzenb¨ock curvature operator, we have the following Bochner type formula for ℓ-forms.
Lemma 2.1 ([ 16 , 23 ]) Let ω =
I a I ω I be a ℓ-form on M Then
∆|ω|2= 2 ⟨∆ω, ω⟩ + 2|∇ω|2+ 2 ⟨E ℓ (ω), ω⟩
= 2 ⟨∆ω, ω⟩ + 2|∇ω|2+ 2K ℓ (ω, ω)
where E ℓ (ω) =n
j,k=1 ω k ∧ i e j R(e k , e j )ω.
Apply the above Bochner formula to the form |ω| p−2 ω we obtain
1
2∆|ω|
2(p−1) = |∇(|ω| p−2 ω)|2−(δd + dδ)(|ω| p−2 ω), |ω| p−2 ω + K ℓ (|ω| p−2 ω, |ω| p−2 ω)
= |∇(|ω| p−2 ω)|2−δd(|ω| p−2 ω), |ω| p−2 ω + |ω| 2(p−2) K ℓ (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) K ℓ (ω, ω).
By Kato type inequality |∇(|ω| p−2 ω)|2≥ |∇|ω| p−1|2 (see [3]) and K ℓ ≥ −aρ, this implies
|ω|∆|ω| p−1≥ −δd(|ω| p−2
ω), ω − aρ|ω| p
Now we will give a proof ofTheorem 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), ϕ2ω − a
ρϕ2|ω| p
Trang 5Integration by parts implies
M
∇(ϕ2|ω|), ∇|ω| p−1 ≤
M
d(|ω| p−2
ω), d(ϕ2ω) + a
M
ρϕ2|ω| p
=
M
d(|ω| p−2
) ∧ ω, d(ϕ2) ∧ ω + a
M
ρϕ2|ω| p
≤
M
d(|ω| p−2 ) ∧ ω.d(ϕ2) ∧ ω+ a
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
|α ∧ β| ≤ c|α| · |β|, where c =
ℓ + m ℓ
.
Therefore the first term in the right hand side of(2.4)is estimated by
M
d(|ω| p−2 ) ∧ ω.d(ϕ2) ∧ ω≤ c2
M
∇(|ω| p−2)|ω| · |d(ϕ2)||ω|
= 2c2|p − 2|
M
ϕ|ω| p−1 |∇ϕ| · |∇(|ω|)|
≤ c2|p − 2|ε
M
|ω| p−2 |∇(|ω|)|2ϕ2+c
2|p − 2|
ε
M
|ω| p |∇ϕ|2 (2.5)
for any ε > 0 Here we used the elementary inequality 2AB ≤ εA2+ ε−1B2 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
M
ρϕ2|ω|2=
M
ρ
ϕ|ω| p2
2
≤
M
∇ϕ|ω| p2
2
≤ (1 + ε)
M
ϕ2∇|ω| p/22
+
1 +1
ε
M
|ω| p |∇ϕ|2
= (1 + ε)p
2
4
M
ϕ2|ω| p−2
∇|ω|2
+
1 +1
ε
M
for any ε > 0.
On the other hand, we compute the left hand side of(2.4)as follows
M
∇(ϕ2
|ω|), ∇|ω| p−1 ≥ (p − 1)
M
|ω| p−2 |∇|ω||2ϕ2− 2(p − 1)
M
ϕ|ω| p−1 |∇ϕ||∇|ω||
≥ (p − 1)
M
|ω| p−2 |∇|ω||2ϕ2− (p − 1)ε
M
|ω| p−2 |∇|ω||2ϕ2
− p − 1
ε
M
By(2.5)–(2.7), it turns out that there exist A, B ∈ R such that
A
|ω| p−2 |∇|ω||2ϕ2≤ B
|ω| p |∇ϕ|2
Trang 6A = (1 − ε)(p − 1) − a(1 + ε)p
2
4 − c2|p − 2|ε
B = a
1 +1
ε
+c
2|p − 2|
p − 1
ε .
Since a < 4(p−1) p2 , choosing ε small enough, this implies there exists a positive constant C = C(p, ε) > 0 such
that
M
|ω| p−2 |∇|ω||2ϕ2≤ C
M
|ω| p |∇ϕ|2.
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
4
p2
B(o,R)
∇|ω| p/2
2
=
B(o,R)
|ω| p−2 |∇|ω||2≤4C
R2
M
|ω| p
Letting R → ∞, we conclude that |∇|ω| p/2 | = 0 Hence |ω| is constant, but since |ω| ∈ L p (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
|∇(|ω| p−2 ω)|2≥
(p − 1)2
|∇|ω| p−1|2,
where
κ = min
1, (p − 1)
2
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 L pnorm 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 R n+k
M is said to be satisfied a δ-super stability condition for 0 < δ ≤ 1 if
δ
M
|A|2ϕ2≤
M
|∇ϕ|2,
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 R n+k If M is super-stable then there is no non-trivial p-harmonic 1-form on M with finite L p norm provided that
2(n −√
n)
n − 1 < p <
2(n +√
n)
n − 1 . Consequently, for p = 2, every harmonic 1-form on M is trivial.
Trang 7Proof By Leung’s estimate for Ric M (see [14]), we have that
Ric M ≥ −n − 1
n |A|2.
Since M satisfies the stable condition, this implies that a weighted Poincar´ e inequality holds true on M , with the weighted function ρ = |A|2 Therefore, byTheorem 1.1, M does not admit any non-trivial p-harmonic
ℓ-form if
n − 1
n <
4(p − 1)
p2
or equivalently,
2(n −√
n)
n − 1 < p <
2(n +√
n)
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
M
|f | n−2 2n dv
n−2 n
≤ C S
M
where C S is the Sobolev constant which depends only on n.
Let us recallTheorem 1.2
Theorem 2.4 Let M n
, n ≥ 3 be a complete non-compact immersed submanifold of R n+k with flat normal bundle If one of the following conditions
1 |A|2≤ max{ℓ,n−ℓ} n2|H|2 ,
2 the total curvature ∥A∥ n is bounded by
∥A∥2n < min
max {ℓ, n − ℓ} p2C S
2
2 max {ℓ, n − ℓ} C S
,
3 supM |A| is bounded and the fundamental tone satisfies
λ1> max {ℓ, n − ℓ}
sup
M
|A|2p2
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 L p norm By Bochner formula and Kato inequality, we have
|ω|∆|ω| p−1≥ −δd(|ω| p−2 ω), ω + K ℓ |ω| p
It is proved in [20] that
K ℓ≥ 1
2n2|H|2− max {ℓ, n − ℓ} |A|2
Trang 8|ω|∆|ω| p−1≥ −δd(|ω| p−2 ω), ω +1
2n2|H|2− max {ℓ, n − ℓ} |A|2 |ω| p (2.9)
1 If |A|2≤max{ℓ,n−ℓ} n2|H|2 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
M
ϕ2|ω|∆|ω| p−1≥ −
M
δd(|ω| p−2 ω), ϕ2ω
Consequently,
M
∇(ϕ2|ω|), ∇|ω| p−1 ≤
M
d(|ω| p−2 ω), d(ϕ2ω)
Using(2.5)and(2.7), we infer for any ε > 0
((p − 1)(1 − ε) − c(p − 2)ε)
M
ϕ2|ω| p−2 |∇|ω||2≤ c(p − 2)
p − 1 ε
M
|ω| p |∇ϕ|2.
Choosing ε > 0 small enough, this implies there exist a constant C > 0 such that
4
p2
B(o,R)
∇|ω| p/2
2
R2
M
|ω| p
Let R → ∞, we conclude that |ω| is constant Since |ω| ∈ L p (M ), it turns out that ω is zero.
2 By the previous part, we may assume |A|2> max{ℓ,n−ℓ} n2|H|2 then by(2.9), we have
M
∇(ϕ2|ω|), ∇|ω| p−1 +n
2
2
M
ϕ2|H|2|ω| p
≤
M
δd(|ω| p−2 ω), ϕ2ω + max {ℓ, n − ℓ}
M
Now, we will estimate the last term of the right hand side of(2.10) ByLemma 2.3and H¨older inequality,
we have
M
|A|2ϕ2|ω| p ≤ ∥A∥2n
M
|ω| p2ϕ
2n
n−2 n−2 n
≤ C S ∥A∥2
n
∇|ω| p2ϕ
2
+
|H|2|ω| p ϕ2
,
Trang 9where C S is the Sobolev constant depending only on n Since
M
∇|ω| p2ϕ
2
=
M
ϕ∇|ω| p2 + |ω| p2∇ϕ
2
≤ (1 + ε)
B x (R)
ϕ2
∇|ω| p2
2
+
1 + 1
ε
M
|ω| p |∇ϕ|2
≤(1 + ε)p
2
4
M
ϕ2|ω| p−2 |∇|ω||2+
1 +1
ε
M
|ω| p |∇ϕ|2,
we get
M
|A|2|ω| p ϕ2≤ C S ∥A∥2
n
(1 + ε)p2
4
M
ϕ2|ω| p−2 |∇|ω||2
+ C S ∥A∥2
n
1 +1
ε
M
|ω| p |∇ϕ|2+ C S ∥A∥2
n
M
|H|2|ω| p ϕ2.
Combining the above inequality,(2.5),(2.7)and(2.10), we have
A1
M
|ω| p−2 |∇|ω||2+ n2
2 − max {ℓ, n − ℓ} C S ∥A∥2
n
M
|H|2|ω| p ϕ2≤ B1
M
|ω| p |∇ϕ|2,
where
A1:= (p − 1)(1 − ε) − c|p − 2|ε − max {ℓ, n − ℓ} C S ∥A∥2
n
(1 + ε)p2
B1:= max {ℓ, n − ℓ} C S ∥A∥2
n
1 +1
ε
+p − 1
c(p − 2)
Since ∥A∥2
n < minmax{ℓ,n−ℓ}p 4(p−1) 2C
S , n2
2 max{ℓ,n−ℓ}C S
, choosing ε > 0 small enough, we conclude that there exist two positive constants C1, C2> 0 such that
B(o,R)
|ω| p−2 |∇|ω||2+ C1
B(o,R)
|H|2|ω| p≤ C2
R2
M
|ω| p
Let R → ∞, we infer |ω| is constant and |H||ω| = 0 Since |ω| ∈ L p (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.
M
|A|2ϕ2|ω| p≤
sup
M
|A|2
λ1
M
|∇(|ω| p/2 ϕ)|2
≤
sup
M
|A|2
λ1
(1 + ε)p2
4
M
ϕ2|ω| p−2 |∇|ω||2+
1 + 1
ε
M
|ω| p |∇ϕ|2
.
Combining this inequality,(2.5),(2.7)and (2.10), we infer
A2
M
|ω| p−2 |∇|ω||2+n
2
2
M
|H|2|ω| p ϕ2≤ B2
M
|ω| p |∇ϕ|2,
where
A2:= (p − 1)(1 − ε) − c|p − 2|ε − max {ℓ, n − ℓ}
sup
M
|A|2
λ1
(1 + ε)p2
B2:= max {ℓ, n − ℓ}
sup
M
|A|2
λ
1 +1
ε
+p − 1
c(p − 2)
Trang 10Since λ1> max {ℓ, n − ℓ}supM |A|2p2
4(p−1) , choose ε > 0 small enough, we conclude that there exist two positive constants D1, D2> 0 such that
B(o,R)
|ω| p−2 |∇|ω||2+ D1
B(o,R)
|H|2|ω| p≤ D2
R2
M
|ω| p
Let R → ∞, we infer |ω| is constant and |H||ω| = 0 Since |ω| ∈ L p (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 inTheorem 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, , e n } of the tangent space T p N such that R ijkl := ⟨R(e i , e j )e k , e l ⟩ = 0 whenever the set {i, j, k, l} contains more than two elements Here R ijkl 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
K ℓ≥1
2n2|H|2− max {ℓ, n − ℓ} |A|2.
Therefore, the results inTheorem 1.2are 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 H2
O Since M is noncompact, its universal
covering is H2
O The following theorem is proved in [12]
Theorem 3.1 Let M be a locally symmetric space with universal covering H2
O 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)) ≤ Ce 22R , where B(R) stands for the geodesic ball centered at the point o ∈ M with radius R.
Recall that for any function u ∈ W loc 1,p (M ) and p > 1, the p-Laplacian operator is defined by
∆p := div(|∇u| p−2 ∇u).
If u is a positive function u ∈ W loc 1,p (M ) such that for any ϕ ∈ W01,p (M ) we have
|∇u| p−2 ∇u, ∇ϕ = λ 1,p
ϕu p−1