MATEMATIQKI VESNIK
originalni nauqni rad
research paper
65, 4 (2013), 494–498
December 2013
A LIOUVILLE TYPE THEOREM FOR p-HARMONIC FUNCTIONS
ON MINIMAL SUBMANIFOLDS IN Rn+m
Yingbo Han and Shuxiang Feng
Abstract. In this note, we prove that if an n-dimensional complete noncompact minimal
submanifold M in Rn+m has sufficiently small total scalar curvature, and u is a p-harmonic
function on M with |du|2p−2 ∈ L1 (M ), then u is constant.
1. Introduction
In [2], Cao, Shen and Zhu showed that a complete connected stable minimal
hypersurface in Euclidean space must have exactly one end. Its strategy was to
utilize a result of Schoen-Yau asserting that a complete stable minimal hypersurface
in Euclidean space can not admit a non-constant harmonic function with finite
integral [8]. Later Ni [7] proved that if n-dimensional complete minimal submanifold
R
M in Euclidean space has sufficient small total scalar curvature (i.e. M |A|n <
C1 < ∞) then M has only one end. In [9], Seo improved the upper bound C1 .
Due to this connection with harmonic functions, this allows one to estimate the
number of ends of the above submanifold by estimating the dimension of the space
of bounded harmonic function with finite Dirichlet integral [3].
A C 1 -function u : (M, hiM ) → R is said to be p-harmonic function, if it satisfies
the non-linear system div(|du|p−2 du) = −δ(|du|p−2 du) = 0 in [1]. In case p = 2, the
non-linear factor |du|p−2 disappears and the 2-harmonic function is simply called
harmonic function. So the p-harmonic function is the generality of the harmonic
function.
In this paper, following the work due to Ni and Seo, we study the p-harmonic
function on complete noncompact minimal submanifold in Rn+m . We obtain a Kato
type inequality for p-harmonic functions and the following Liouville type result:
2010 AMS Subject Classification: 58E20, 53C42
Keywords and phrases: Minimal submanifolds; p-harmonic function; Liouville type theorem.
Y.H. was supported by NSFC No.10971029, NSFC No.11201400, NSFC-TianYuan Fund
No.11026062, Project of Henan Provincial Department of Education No.2011A110015 and Talent
youth teacher fund of Xinyang Normal University.
494
A Liouville type theorem for p-harmonic functions
495
Theorem 1.1. Let M n be an n-dimensional complete minimal submanifold
in R
, n ≥ 3, and u : M n → R be a p-harmonic function (p ≥ 2). If
sµ
µZ
¶ n1
¶
n
(p − 2)2
n
n
|A| dv
<
−
Cs−1
2
n − 1 (p − 1)
n−1
M
n+m
and |du|2p−2 ∈ L1 (M ), then u is constant. (Here Cs is a Sobolev constant in [3].)
2. Proof of the theorem
We begin with the following useful facts.
Lemma 2.1 (Bochner-Weitzenböck formula) [6] Let u : M n → R be any smooth
function on Riemannian manifold M n . Then
1
4|du|2(p−1) = −h4H (|du|p−2 du), |du|p−2 dui
2
+ |D(|du|p−2 du)|2 + |du|2(p−2) RicM (∇u, ∇u).
where D and 4H = dδ + δd are respectively the covariant derivative and the Hodge
Laplacian of the vector bundle T ∗ M ⊗ u−1 T R.
Lemma 2.2. [3] Let M n be an n-dimensional complete immersed minimal
submanifold in Rn+m , n ≥ 3. Then for any φ ∈ W01,2 (M ) we have
¶ n−2
n
µZ
|φ|
2n
n−2
dv
Z
|∇φ|2 dv,
≤ Cs
M
(1)
M
where Cs depends only on n.
Lemma 2.3. [3] Let M be an n-dimensional complete immersed minimal submanifold in space form F n+m (c). Then the Ricci curvature of M satisfies
RicM ≥ (n − 1)c −
n−1 2
|A| .
n
(2)
We give a Kato type inequality for p-harmonic function.
Lemma 2.4. Let u : M n → R be a p-harmonic function on Riemannian
manifold M , then we have
n
|D(|du|p−2 du)|2 ≥
|∇|du|p−1 |2 .
(3)
n−1
where p ≥ 2.
Proof. When p = 2, u is a 2-harmonic function i.e. harmonic function, (3) is
true. So we only need to prove the case for p ≥ 3. Since u is p-harmonic function,
we have
δ(|du|p−2 du) = 0,
(4)
496
Y. Han, Sh. Feng
We choose a local frame field {e1 , · · · , en } of M n near a fixed point q ∈ M such
that ∇ei ej (q) = 0, u1 (p) = du(e1 )(q) = |∇u|(q) = |du|(q) 6= 0 and du(ei )(q) = 0
for i ≥ 2. Then we have
P
qP
ui uij
∇ej |∇u| = ∇j |∇u| = ∇j (
u2i ) =
= u1j .
|∇u|
From (4), we have
0 = δ(|du|p−2 du)(q) = −
n
P
i=1
n
P
= −|du|p−2
Dei (|du|p−2 du)(ei )(q)
[(p − 2)∇i (ln(|∇u|))ui + uii ](q),
i=1
so we have
n
P
[(p − 2)∇i (ln(|∇u|))ui + uii ] = 0.
(5)
i=1
Now we compute,
|D(|du|p−2 du)|2 − |∇|du|p−1 |2 =
−
=
P
ij
i
P
|du|2(p−2) [(p − 2)∇i (ln |∇u|)|∇u| + ∇i |∇u|]2
|du|2(p−2) [(p − 2)∇i (ln |∇u|)u1 + u1i ]2
|du|2(p−2) [(p − 2)∇i (ln |∇u|)u1 + ui1 ]2
+
P
P
i
i6=1
≥
ij
|du|2(p−2) [(p − 2)∇i (ln |∇u|)uj + uij ]2
|du|2(p−2) [(p − 2)∇i (ln |∇u|)uj + uij ]2
−
≥
P
P
P
|du|2(p−2) [(p − 2)∇i (ln |∇u|)ui + uii ]2
i6=1
|du|2(p−2) [(p − 2)∇i (ln |∇u|)u1 + ui1 ]2
i6=1
+ |du|2(p−2)
1 P
[ [(p − 2)∇i (ln |∇u|)ui + uii ]]2
n − 1 i6=1
1 P
|du|2(p−2) [(p − 2)∇i (ln |∇u|)u1 + ui1 ]2
n−1 i
1
=
|∇|du|p−1 |2 .
n−1
≥
(6)
where we use the equation (5) in the last inequality. From (6), we have
n
|D(|du|p−2 du)|2 ≥
|∇|du|p−1 |2 .
n−1
Proof of Theorem 1.1. Since u is a p-harmonic function and (1), we have
|du|p−1 4|du|p−1 = −hδd(|du|p−2 du), |du|p−2 dui
+ |D(|du|p−2 du)|2 − |∇|du|p−1 |2 + |du|2(p−2) RicM (∇u, ∇u)
497
A Liouville type theorem for p-harmonic functions
From (2) and (3), we have
|du|p−1 4|du|p−1 +
n−1 2
|A| |du|2(p−1)
n
≥ −hδd(|du|p−2 du), |du|p−2 dui +
1
|∇|du|p−1 |2
n−1
Let f = |du|p−1 . Then we have
1
n−1 2 2
|A| f ≥ −hδd(|du|p−2 du), |du|p−2 dui +
|∇f |2
n
n−1
f 4f +
(7)
Fix a point x0 ∈ M and take a cut-off function η satisfying 0 ≤ η ≤ 1, η = 1 on
Bx0 (r), η = 0 on M − Bx0 (2r) and |dη| ≤ 1r . Multiplying (7) by η 2 and integrating
over M , we get
Z
Z
n−1 2 2
2
η f 4f +
η2
|A| f
n
M
M
Z
Z
1
≥−
η 2 hδd(|du|p−2 du), |du|p−2 dui +
η 2 |∇f |2 (8)
n
−
1
M
M
Since for a function f on M , |d(f dφ)| ≤ |df ||dφ|. Hence we have
¯Z
¯
¯
¯
2
p−2
p−2
¯
hη |du| du, δd(|du| du)i¯¯
¯
M
¯Z
¯
¯
¯
2
p−2
p−2
¯
=¯
hd(η |du| du), d(|du| du)i¯¯
¯ ZM
¯
¯
¯
≤ ¯¯
|d(η 2 |du|p−2 )||du||d|du|p−2 ||du|¯¯
ZM
Z
≤2
η|dη||du|p−2 |d|du|p−2 ||du|2 +
η 2 |d|du|p−2 |2 |du|2
M
M
Z
Z
p−2 2
p−2
p−1
p−1
η|dη||du| |d|du| | + (
)
η 2 |d|du|p−1 |2
=2
p−1 M
p−1
M
Z
Z
p−2 2
p−2
η|dη|f |df | + (
)
η 2 |df |2 .
=2
p−1 M
p−1
M
Using integration by parts for (8) and using (9), we have
Z
Z
Z
n−1 2 2
−
η 2 |df |2 − 2
ηf hdf, dηi +
η2
|A| f
n
M
M
M
Z
Z
Z
p−2
p−2 2
1
≥ −2
η|dη|f |df | − (
)
η 2 |df |2 +
η 2 |df |2
p−1 M
p−1
n−1 M
M
On the other hand, it follows from (2) and Hölder inequality that
n−1
n
Z
n−1
η |A| f ≤
n
M
2
µZ
2 2
|A|
M
n
¶ n2 µ Z
(ηf )
M
2n
n−2
¶ n−2
n
(9)
(10)
498
Y. Han, Sh. Feng
n−1
≤
n
n−1
≤
n
µZ
|A|
µZ
n
M
|A|
M
n
¶ n2
Z
|d(ηf )|2
Cs
¶ n2
Z
M
(f 2 |dη|2 + η 2 |df |2 )
Cs
M
µZ
¶ n2
Z
n−1
|A|n
Cs
2ηf hdη, df i.
(11)
n
M
M
From (10), (11) and Schwarz inequality, we have
µ
·
¶2
µZ
¶ n2
p−2
n−1
n
−
−
|A|n
Cs
n−1
p−1
n
M
¶ n2
µZ
¶¸ Z
µ
p−2
n−1
|A|n
Cs +
+1
η 2 |df |2
−ε
n
p−1
M
M
"
µZ
µ
µZ
¶# Z
¶ n2
¶ n2
n−1
1 n−1
p−2
n
n
≤
|A|
Cs +
|A|
Cs +
+1
f 2 |dη|2
n
ε
n
p−1
M
M
M
"
µZ
¶ n2
µ
µZ
¶ n2
¶#
Z
n−1
1
n
−
1
p
−
2
1
n
n
≤
|A|
|A|
+1
f2
Cs +
Cs +
n
ε
n
p−1
r2 M
M
M
q
R
2
1
−1
n
n
Since ( M |A|n dv) n < ( n−1
− (p−2)
and |du|2p−2 ∈ L1 (M ), i.e. |f |2 ∈
(p−1)2 ) n−1 Cs
+
L1 (M ), choosing ε > 0 small and letting r → ∞, we have |df | = 0 on M , i.e. f is
constant. Since |f |2 ∈ L1 (M ) and the volume of M is infinite, we have f = 0, then
u is constant. This completes the proof of Theorem 1.1.
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Math. J. 31 (2008), 113–119.
(received 06.12.2011; in revised form 21.06.2012; available online 10.09.2012)
College of Mathematics and Information Science, Xinyang Normal University, Xinyang, 464000,
Henan, P. R. China
E-mail: yingbhan@yahoo.com.cn, fsxhyb@yahoo.com.cn