Electronic Journal of Differential Equations, Vol. 2006(2006), No. 15, pp. 1–9.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
A LIOUVILLE THEOREM FOR F -HARMONIC MAPS WITH
FINITE F -ENERGY
M’HAMED KASSI
Abstract. Let (M, g) be a m-dimensional complete Riemannian manifold
with a pole, and (N, h) a Riemannian manifold. Let F : R+ → R+ be a strictly
increasing C 2 function such that F (0) = 0 and dF := sup(tF 0 (t)(F (t))−1 ) <
∞. We show that if dF < m/2, then every F -harmonic
map u : M → N with
R
finite F -energy (i.e a local extremal of EF (u) := M F (|du|2 /2)dVg and EF (u)
is finite) is a constant map provided that the radial curvature of M satisfies a
pinching condition depending to dF .
1. Introduction and statement of result
Let (M, g) and (N, h) be two Riemannian manifolds and F be a given C 2 function
F : R+ → R+ . Then, a map u : M → N of class C 2 is said to be F -harmonic if for
every compact K of M , the map u is extremal of F -energy:
Z
|du|2
EF (u) :=
F(
)dVg .
2
K
In a normal coordinate system, the tension field associated with EF (u) by the
Euler-Lagrange equations is
m
n
X
|du|2
|du|2 o
|du|2
τF (u) :=
(∇ei (F 0 (
)du))ei = F 0 (
)τ (u) + du. grad F 0 (
)
2
2
2
i=1
where τ (u) is the usual tension field of u defined by
τ (u)k = ∆M uk +
n;m
X
N
Γkαγ (u)g ij
β,γ;i,j
∂uβ ∂uγ
,
∂xi ∂xj
k = 1, . . . , n .
Then, the map u is F -harmonic if τF (u) = 0. For further properties of F -harmonic
maps, we refer the reader to [1, 2]. For the particular case of F (t) = t, the Liouville
problem for harmonic maps with finite energy have been studied in [4, 6, 7, 8, 9].
While for F (t) = p2 tp/2 , with p ≥ 2, this is the problem of p-harmonic maps
√
with finite p-energy (corollary 1.2. If F (t) = 1 + 2t − 1 corresponding to the
minimal graph (corollary 1.3). In this paper, we study the same problem for F harmonic maps with finite F -energy without condition on the curvature for the
2000 Mathematics Subject Classification. 58E20, 53C21, 58J05.
Key words and phrases. F -harmonic maps; Liouville propriety; Stokes formula;
comparison theorem.
c
2006
Texas State University - San Marcos.
Submitted March 24, 2005. Published January 31, 2006.
1
2
M. KASSI
EJDE-2006/15
target manifold. We assume that F is strictly increasing, F (0) = 0, and dF =
0
(t)
sup tF
F (t) < ∞, “the degree of F ”. For x in M , we set r(x) = dg (x, x0 ).
Theorem 1.1. Let (M, g) be a m-dimensional complete Riemannian manifold,
m > 2, with a pole x0 , and let (N, h) be a Riemannian manifold. If dF < m/2,
then every F -harmonic map of M into N with finite F -energy is constant provided
that the radial curvature Kr of M satisfies one of the following two conditions:
(i) −α2 ≤ Kr ≤ −β 2 with α > 0, β > 0 and 1 + (m − 1)β − 2dF α > 0
β
1
α
≤ Kr ≤ 1+r
(ii) − 1+r
2 with α ≥ 0 and β ∈ [0, 4 ] such that 2 + (m − 1)(1 +
√
√ 2
1 − 4β) − 2dF (1 + 1 + 4α) > 0.
Furthermore, we have the following corollaries.
Corollary 1.2. Let (M, g) and (N, h) be as in the theorem. Then, every C 2 pharmonic map of M into N with finite p-energy, for p < m, is constant.
Corollary 1.3. Let (M, g) and (N, h) be as in the theorem. Then, for m > 2,
every C 2 map u of M into N , with finite energy, solution of
n
o
τ (u)
1
p
+ du. grad p
=0
1 + |du|2
1 + |du|2
is constant.
For m = 2, the statement of the theorem is false in general. In fact, for the
case (i), there exist holomorphic maps of the hyperbolic disc with finite energy [9].
While for the case (ii) there exist holomorphic maps of C into P1 with finite energy
[8].
2. Proof of Theorem 1.1
Let X and Y be two vector fields on M . It is well-known [3, 6], that the stressenergy for harmonic maps is
Su :=
|du|2
hX, Y ig − hdu(X), du(Y )ih
2
and satisfies
(div Su )(X) = −hτ (u), du(X)ih .
Following [2], we define the stress-energy of F -harmonic maps by
|du|2
|du|2
)hX, Y ig − F 0 (
)hdu(X), du(Y )ih .
2
2
When F (t) := t we have SF,u := Su . Also (div SF,u )(X) = −hτF (u), du(X)ih
thanks to the following lemma.
SF,u (X, Y ) := F (
Lemma 2.1. For every vector field X on M , we have
(div SF,u )(X) = −hτF (u), du(X)ih ,
(2.1)
2
|du|
)X)
2
|du|2
= div(F 0 (
)hdu(X), du(ei )ih ei ) − hτF (u), du(X)ih + [SF,u , X],
2
div(F (
(2.2)
EJDE-2006/15
A LIOUVILLE THEOREM FOR F -HARMONIC MAPS
3
where
[SF,u , X](x) =
m X
F(
i,j=1
|du|2
|du|2
)δij − F 0 (
)hdu(ei ), du(ej )ih h∇ei X, ej ig .
2
2
In particular, if u is F -harmonic and D ⊂⊂ M is a C 1 boundary domain, then we
have
Z
Z
SF,u (X, ν)dσg =
[SF,u , X]dVg
∂D
D
where ν is the normal to ∂D.
Proof. Let x ∈ M . Chose a normal coordinate system such that at x. gij (x) = δij
dg(x) = 0, where (e1 , . . . , em ) being a normal basis, we have ∇ej ek = 0 for all j, k
and
(div SF,u )(X)
m n
o
X
=
∇ei SF,u (ei , X) − SF,u (ei , ∇ei X) − SF,u (∇ei ei , X)
i=1
m n
X
|du|2
|du|2
|du|2
)hei , Xi − hF 0 (
)du(ei ), du(X)i − F (
)hei , ∇ei Xi
=
∇ ei F (
2
2
2
i=1
o
|du|2
+ F 0(
)hdu(ei ), du(∇ei X)i − SF,u (∇ei ei , X)
2
m n
X
|du|2
=
∇ ei F (
)hei , Xi
2
i=1
|du|2
|du|2
)du(ei ), du(X)i − F (
)hei , ∇ei Xi
2
2
o
2
|du|
+ F 0(
)hdu(ei ), du(∇ei X)i − SF,u (∇ei ei , X)
2
m n X
m
X
|du|2
=
F 0(
)h∇ei (du(ej )), du(ej )i hei , Xi
2
i=1
j=1
− ∇ei hF 0 (
|du|2
|du|2
)∇ei hei , Xi − h∇ei (F 0 (
)du(ei )), du(X)i
2
2
|du|2
− F 0(
)hdu(ei ), ∇ei (du(X))i
2
2
|du|
|du|2
− F(
)hei , ∇ei Xi + F 0 (
)hdu(ei ), du(∇ei X)i
2
2
o
+ F(
− SF,u (∇ei ei , X) .
Thus
(div SF,u )(X) =
m n
o
X
|du|2
F 0(
)h∇ei (du(ej )), du(ej )iXi
2
i,j=1
+
m n
X
|du|2
|du|2
F(
)h∇ei ei , Xi + F (
)hei , ∇ei Xi
2
2
i=1
− h∇ei (F 0 (
|du|2
)du(ei )), du(X)i
2
4
M. KASSI
EJDE-2006/15
|du|2
|du|2
)hdu(ei ), ∇ei (du(X))i − F (
)hei , ∇ei Xi
2
2
o
|du|2
)hdu(ei ), du(∇ei X)i − SF,u (∇ei ei , X)
+ F 0(
2
m n
o
X
|du|2
=
F 0(
)hXi ∇ei (du(ej )), du(ej )i
2
i,j=1
− F 0(
−
m n
X
|du|2
)hdu(ei ), ∇ei (du(X))i
F 0(
2
i=1
|du|2
|du|2
)hdu(ei ), du(∇ei X)i + F (
)h∇ei ei , Xi
2
2
|du|2
|du|2
+ F(
)hei , ∇ei Xi − F (
)hei , ∇ei Xi
2
2
o
|du|2
)du(ei )), du(X)i − SF,u (∇ei ei , X) .
− h∇ei (F 0 (
2
+ F 0(
Since ∇ei ei = 0, with (∇ei du)(X) = ∇ei (du(X)) − du(∇ei X) and by symmetry
(∇ei du)(X) = (∇X du)(ei ), we have
div(SF,u )(X) =
m n
o
X
|du|2
)h∇X (du(ej ), du(ej )i
F 0(
2
j=1
m n
X
|du|2
−
F 0(
)hdu(ei ), ∇ei (du(X)) − du(∇ei X)i
2
i=1
o
|du|2
− h∇ei (F 0 (
)du(ei )), du(X)i .
2
Finally,
div(SF,u )(X) = −hτF (u), du(X)i.
Also
m
div(F (
X
|du|2
|du|2
)X) =
h∇ei (F (
)X), ei i
2
2
i=1
=
m n
o
X
|du|2
|du|2
h∇ei (F (
))X, ei i + F (
)h∇ei X, ei i
2
2
i=1
= ∇X F (
m
X
|du|2
|du|2
)+
F(
)h∇ei X, ei i.
2
2
i=1
Then, by straightforward computation, we obtain
m
∇X F (
X1
|du|2
|du|2
)=
F 0(
)∇X hdu(ei ), du(ei )i
2
2
2
i=1
=
m
X
F 0(
|du|2
)h∇X (du(ei )), du(ei )i
2
F 0(
|du|2
)h(∇X du)(ei ) + du(∇X ei ), du(ei )i
2
i=1
=
m
X
i=1
EJDE-2006/15
A LIOUVILLE THEOREM FOR F -HARMONIC MAPS
m
X
=
F 0(
|du|2
)h(∇X du)(ei ), du(ei )i
2
F 0(
|du|2
)h(∇ei du)(X), du(ei )i (by symmetry)
2
i=1
m
X
=
i=1
5
m n
X
|du|2
)du(ei )i
=
h∇ei (du(X)), F 0 (
2
i=1
o
|du|2
− F 0(
)hdu(∇ei X), du(ei )i
2
Thus
m
∇X F (
Xn
|du|2
|du|2
)=
∇ei hdu(X), F 0 (
)du(ei )i
2
2
i=1
|du|2
)du(ei ))i
2
o
|du|2
− F 0(
)idu(∇ei X), du(ei )i
2
m n
X
|du|2
)idu(X), du(ei )iei
=
div F 0 (
2
i=1
− hdu(X), ∇ei (F 0 (
|du|2
)du(ei ))i
2
o
2
|du|
− F 0(
)hdu(∇ei X), du(ei )i
2
m n
X
o
|du|2
)hdu(X), du(ei )iei
=
div F 0 (
2
i=1
+ hdu(X), −∇ei (F 0 (
− hdu(X), τF (u)i −
m
X
F 0(
i=1
|du|2
)hdu(∇ei X), du(ei )i
2
Thus
m
div(F (
Xn
o
|du|2
|du|2
)X) =
)hdu(X), du(ei )iei
div F 0 (
2
2
i=1
− hdu(X), τF (u)i + [SF,u , X]
with
[SF,u , X] =
m X
i,j=1
F(
|du|2
|du|2
)δij − F 0 (
)hdu(ei ), du(ej )ih h∇ei X, ej ig
2
2
because ∇ei X = h∇ei X, ej iej . If D ⊂⊂ M is a C 1 boundary domain, we get by
the use of Stokes formula
Z
Z
(div SF,u )(X) +
[SF,u , X]
D
D
|du|2
=
div(F (
)X) −
2
D
Z
Z X
m
D i=1
div F 0 (
|du|2
) < du(X), du(ei ) > ei
2
6
M. KASSI
Z
=
F(
∂D
|du|2
)hX, νi −
2
Z
F 0(
∂D
EJDE-2006/15
|du|2
)hdu(X), du(ν)i .
2
Thus, if u is F -harmonic:
Z Z
|du|2
|du|2
)hX, νi − F 0 (
)hdu(X), du(ν)i =
F(
[SF,u , X].
2
2
∂D
D
This completes the proof.
Lemma 2.2. Let u : M → N be a F -harmonic with finite F -energy and X a vector
field on M such that |X| ≤ φ(r) for φ : R+ → R+ satisfying
Z +∞
dt
= +∞.
φ(t)
1
Then there exists an increasing strictly sequence (Rn ) such that
Z
lim
[SF,u , X]dVg = 0.
n→∞
B(x0 ,Rn )
0
Proof. Since tF (t) ≤ dF F (t) we have
Z
[SF,u , X]
B(x0 ,R)
Z
|du|2
|du|2
F(
)hX, νi + F 0(
)idu(X), du(ν)i
2
2
∂B(x0 ,R)
∂B(x0 ,R)
Z
Z
|du|2
|du|2
≤
F(
)|hX, νi| +
F 0(
)|idu(X), du(ν)i|
2
2
∂B(x0 ,R)
∂B(x0 ,R)
Z
|du|2
)|X| .
≤ (1 + 2dF )
F(
2
∂B(x0 ,R)
Z
≤
By the Co-area formula and |X| ≤ φ(r(x)),
Z ∞
Z
Z
|du|2
1 |X||∇r| |du|2
F(
)|X| dt =
F(
)
φ(t) ∂B(x0 ,t)
2
φ(r)
2
0
M
Z
|du|2
≤
F(
)<∞
2
M
R ∞ dt
Since 1 φ(t)
= ∞, there exists a increasing strictly sequence (Rn ) such that
R
2
limn→∞ ∂B(x0 ,Rn ) F ( |du|
2 )|X| = 0. Hence
Z
lim
[SF,u , X]dVg = 0.
n→∞
B(x0 ,Rn )
This completes the proof of Lemma 2.2.
For the theorem, it suffices to choose X satisfying Lemma 2.2 and the condition
[SF,u , X] ≥ cF (|du|2 /2) where c > 0 is a constant. For that we take X = r∇r and
using the comparison theorem of the Hessian [5].
Theorem 2.3 (Comparison theorem). Let (M, g) be a complete Riemannian manifold with a pole x0 and k1 , k2 be two continuous functions on R+ such that
EJDE-2006/15
A LIOUVILLE THEOREM FOR F -HARMONIC MAPS
7
k2 (r) ≤ Kr ≤ k1 (r), where Kr is the radial curvature of M , i.e., the sectional curvature of the tangent planes containing the radial vector ∇r. Also, let Ji (i = 1, 2)
be the solution of classical Jacobi equation
Ji00 + ki Ji = 0;
Ji (0) = 0
and
Ji0 (0) = 1.
Then, if J1 > 0 on R+ , we have on M \ {x0 }
J 0 (r)
J10 (r)
(g − dr ⊗ dr) ≤ Hess(r) ≤ 2 (g − dr ⊗ dr) .
J1 (r)
J2 (r)
Case (i) of Theorem 2.3: With k1 (r) = −β 2 and k2 (r) = −α2 , we have
β coth(βr)(g − dr ⊗ dr) ≤ Hess(r) ≤ α coth(αr)(g − dr ⊗ dr) .
Case (ii) of Theorem 2.3: With k1 (r) = rβ2 and k2 (r) = − rα2 , and the fact that on
M \ {x0 },
α
α
β
β
− 2 ≤−
≤ Kr ≤
≤ 2
2
2
r
1+r
1+r
r
we have
1 + √1 + 4α 1 + √1 − 4β (g − dr ⊗ dr) ≤ Hess(r) ≤
(g − dr ⊗ dr) .
2r
2r
Lemma 2.4. Under hypothesis of Theorem 2.3, in case (1), we have
[SF,u , X] ≥ (1 + (m − 1)β − 2dF α)F (
|du|2
)
2
and in case (ii),
p
√
1
|du|2
(2 + (m − 1)(1 + 1 − 4β) − 2dF (1 + 1 + 4α)F (
).
2
2
Proof. First note that
m X
|du|2
|du|2
)δij − F 0 (
)hdu(ei ), du(ej )ih < ∇ei X, ej ig ,
[SF,u , X] =
F(
2
2
i,j=1
[SF,u , X] ≥
∂
∂
where (e1 , . . . , em−1 , ∂r
) with em = ∂r
, being a normal basis on B(x0 , R). Then,
∂
∂
since X = r ∂r , it follows that ∇ ∂ X = ∂r
and so we get
∂r
∂
ig = 1 ,
∂r
h∇ei X, ei ig = r Hess(r)(ei , ei ), for i = 1, . . . , m − 1,
h∇ ∂ X,
∂r
∇ ei X =
m−1
X
r Hess(r)(ei , ej )ej ,
for i = 1, . . . , m − 1.
j=1
Therefore,
[SF,u , X] = F (
−
m−1
X
|du|2
)(1 +
r Hess(r)(ei , ei ))
2
i=1
m−1
X
i,j=1
− F 0(
F 0(
|du|2
)hdu(ei ), du(ej )ih h∇ei X, ej ig
2
|du|2
∂
∂
∂
)hdu( ), du( )ih h∇ ∂ X, ig
∂r
2
∂r
∂r
∂r
8
M. KASSI
−
m−1
X
F 0(
∂
|du|2
)hdu( ), du(ej )ih h∇ ∂ X, ej ig
∂r
2
∂r
F 0(
∂
∂
|du|2
)hdu(ei ), du( )ih h∇ei X, ig
2
∂r
∂r
j=1
−
m−1
X
i=1
= F(
−
EJDE-2006/15
m−1
X
|du|2
)(1 +
r Hess(r)(ei , ei ))
2
i=1
m−1
X
i,j=1
− F 0(
F 0(
|du|2
)hdu(ei ), du(ej )ir Hess(r)(ei , ej )
2
|du|2
∂
∂
)hdu( ), du( )i
2
∂r
∂r
For the case (i), we have
|du|2
|du|2
) + (m − 1)(βr) coth(βr)F (
)
2
2
2
|du|
)|du|2 (αr) coth(αr)
− F 0(
2
|du|2
∂
∂
+ F 0(
)((αr) coth(αr) − 1)hdu( ), du( )i
2
∂r
∂r
|du|2
|du|2
≥ F(
) + F(
)((m − 1)(βr) coth(βr) − 2dF (αr) coth(αr))
2
2
|du|2
coth(αr)
|du|2
) + F(
)r coth(βr)((m − 1)β − 2dF α
).
≥ F(
2
2
coth(βr)
[SF,u , X] ≥ F (
Since the function coth(x) is decreasing and, x coth(x) is bounded below by a
positive constant in R+ , we have
[SF,u , X] ≥ (1 + (m − 1)β − 2dF α)F (
|du|2
)
2
For the case (ii), we have
|du|2
|du|2
|du|2
) + (m − 1)aF (
) − bF 0 (
)|du|2
2
2
2
|du|2
∂
∂
+ (b − 1)F 0 (
)hdu( ), du( )i
2
∂r
∂r
|du|2
≥ (1 + (m − 1)a − 2dF b)F (
),
2
[SF,u , X] ≥ F (
where we have set
a=
1+
√
1 − 4β
2
and b =
1+
√
1 + 4α
≥ 1.
2
Acknowledgements. I would like to express my gratitude to Professor S. Asserda
for his valuable suggestions and warm encouragement. Also I thank the anonymous
referee for many valuable comments.
EJDE-2006/15
A LIOUVILLE THEOREM FOR F -HARMONIC MAPS
9
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M’hamed Kassi
Equipe d’Analyse Complexe, Laboratoire d’Analyse Fonctionnelle, Harmonique et Complexe, Département de Mathématiques, Faculté des Sciences, Université Ibn Tofail,
Kénitra, Maroc
E-mail address: mhamedkassi@yahoo.fr