Internat. J. Math. & Math. Sci.
VOL. ii NO. 4 (1988) 675-682
675
AN APPLICATION OF STRESS ENERGY TENSOR TO THE VANISHING
THEOREM OF OIFFERENTIAL FORMS
KAIREN CAI
Department of Mathematics and Statistics
Southern Illinois University at Edwardsville
Edwardsville, Illinois 62026-1653
(Received September 29, 1987)
ABSTRACT.
The author applies the stress energy of differential forms to study the
vanishing theorems
the Liouville type.
of
such as
underlying manifolds
the
complex hyperbolic space form,
stress energy tensor is of
#anishes.
Euclidean
It is shown that f0or a large class of
n-space, the complex n-space, and the
if any vector bundle valued p-form with conservative
finite norm or
slowly divergent norm,
then the p-form
This generalizes the recent results due to Hu and Sealey.
KEYWORDS AND PHRASES.
Conservation, stress energy tensor, complex hyperbolic space
form, vector bundle valued p-form.
1980 AMS SUBJECT CLASSIFICATION CODES. 58AI0.
I.
INTRODUCTION.
The classical Liouville theorem states that any bounded harmonic function defined
on
the entire
theorem.
Yau
plane is a constant
[I]
generalized
it
Later many authors generalized the
function.
to
harmonic
manifolds with non-negative Ricci curvature.
functions
on
complete
Riemannian
It was further extended to harmonic maps
between Riemannian manifolds by Fells and Sampson [2], Cheng [3], Garbar et al. [4],
Hu [5], Xin [6], Sealey [7] and others.
It is well known that if the field equation of a physical system is obtained by
means of calculating the derivative of the action functional I, we can define a so
called stress energy tensor S such that S is conservative at the critical values of
I.
In particular, the stress at a point of an elastic body is described by a tensor
3
in the Euclidean space R
The law of conservation of momentum about some
(Sij)
origin implies
the
stress
equilibrium means that
that
if
Sij
energy
tensor
Sij
be divergence free.
the map between Riemannlan manifolds
tensor is conservative.
be
symmetric
and
the
system
be
in
Recently, Baird and Fells [I] proved
is
harmonic,
then its
stress
energy
This might make the harmonic map a new physical explanation.
K.
676
The purpose of the paper is to apply the stress energy tensor of p-form with
values in a vector bundle over some klnd of Rlemannlan manifolds to the vanishing
theorem of the Liouvllle type.
THEOREM I.I.
U
vector bundle over
slowly
or
finite
U
Let
n
n.
The main result is the following.
be the complex hyperbollc space form and F a Hermltlan
Let
be a p-form with values in F, p < n.
n
U and its stress energy
over
norm
divergent
If
is of
tensor
is
0.
conservatlve, then
It must be pointed out that this is a more general vanishing theorem of vector
valued p-form.
We all know that there are many important maps between
Riemannlan manifolds such as harmonic, relative harmonic [5], holomorphic [8] and
relative affine maps [9] whose differentials possess the free divergent stress energy
tensors.
Sealey [7] proved that if the norm of a harmonic l-form with values In a
bundle
U n is integrable, then the l-form vanishes.
vector bundle over
Therefore, Theorem
means that under a weaker conditlon than Sealey’s,a stronger conclusion still holds
In
section 2 after introducing some notations, we establish an integral formula
of the stress energy tensor and some kind of radial variational formmla for p-forms by
using the Lie derivative.
I, these
In sectlon 3, we prove Theorem I.I. by
Besides being applied to the proof of Theorem
formulas are of interest in its own right.
means of estimating the first variation of the norm of p-form motivated by the method
in
[4].
Finally,
can
manifold
be
expand our result to the general case where the underlying
we
the
complex
n-space
some
or
klnd
of
conformal
flat
Riemannlan
ma ni fo Ids.
2.
NOTATIONS AND LEMMAS.
For simplicity, we only state some notations and fornmlas for the Remannian
vector bundle over real
Riemannlan manifold,
those of Hermtlan vector bundle over
Kaehler manifold can be written correspondently.
,
8
>
Z
<
il<...<iP
where
m, 8 s
^PT*M-F
X
X
APTx *M- Fx
We define an inner product in
over Rlemannian manifold M.
<
Let F be a Riemannian vector bundle
(ell
{e 1,..., e
n
e
i
),
P
8(ell
e
i
>
P
is any orthonormal basis of
this polntwise inner product over M gives an inner product in
the space of sections of
APT*M-F.
For a linear connection
covarlant derivative by
.
by setting
T
x
M.
AP(F),
Integrating
which denotes
V on F we define the
P
(Vy)(X
for Y,
X e A(TM).
d
(X:
Xp) rye(El,..., Xp)
The exterior differential
Xp+l)
(X
i=l
d:AP(F)
,VyX I
A
p+I
,Xp)
(F) is given by
(2.2)
APPLICATION OF STRESS ENERGY TENSOR
p+]
(-I) i+l
V
x
(-)+
+
X
i
:AP(F)
A(TM),,
AP-I(F)
X
,,
i,J
I,
)
p
.
(2.3)
Xp+l
i’’’’’
([x,x], x,
i<j
where
-:(X
i
677
p+I, X i means X i
is
Xp+
The
deleted.
codlfferentlal
is given by
n
6(X
Xp_I)
V
e
i=l
i
(e i, X 1,---,
Xp_ I).
(2.4)
The generalized Laplace-Beltraml operator for p-form with values in F is defined
d6 + 6d.
Any p-form 0 with values in F satisfying the equations dto 0,
by A
6to
0 is called harmonic.
If M is a manifold without boundary and to is of compact
support, then
to is
Vto
harmonic iff
to "to
The symmetric square
of
to
0.
e
AP(F)
is a symmetric F-valued 2-tensor defined
by
"’
to
<
(X,Y)
A(TM),
where X,Y
H
Eto
the norm of
i
x
i to
y
>
to
(2.5)
denotes the contraction with X.
We call
(2.6)
e to *I
over M where
to
i
x
Sto e g
e
Ito[ 2/2"
to
The stress energy tensor of
to
is given by
(2.7)
to "to
in which g is the metric tensor of M.
LEMMA 2.1.
Suppose X be a vector field with compact support over M and
to
p-form
with values in a vector bundle, then
f<
L
M
to
x
to
>
f
*I
div S
M
S0
in which div
to
]
(X) *I +
8M
S
is the divergence of
"m
(x, a/an) *I
8/8n the unit outward normal
and
(2 8)
to
8M.
Since [6
PROOF.
<
Sto(X)
div
i
dto, u
x
>
<
+
6to, i xm
>,
(2.9)
we get
<
L
x
to,
to
>
<
i
x
div
x
Sto(X)
+
<
di
x
m, m
>
< to,
i to
x
>
Integrating and using Green theorem immediately complete the proof.
(2.10)
K. CAI
678
:
M / N be a smooth map
Let us consider a special case of the above formula. Assume
from a Rlemannlan manifold M without boundary to Riemannlan manifold N. According to
Lemma I, we know that
f<
L
slv
S# < z@,
M
so
S@
where
<d#(X), 6d#
(2.12)
n
and
d@
is the torsion field of
The stress energy tensor of a harmonic map from
is a differential submersion and its
Conversely, if
d@
un(p)
Let
un(p).
Suppose
t
is harmonic.
p in the complex n-space
be an open disc with radius
be an F-valued p-form over
and
z
is of conservative stress energy tensor, then
LEMMA 2.2.
C
@
tensor of
COROLLARY (Baird and Eells [i0]).
M to N must be conservative.
differential
(2.1 l)
*l
>
d@
Thus, we easily prove
#.
>
M
the stress energy
is
f
> "I
d#, d#
x
be an l-parameter family of
p-forms such that
m (Z
Z
t
where
un(p),
to z.
n
U
z
,
and
<
(p), 0
t
tP(#t
p
,
(2.13)
tz (ZI’’’" Zp)
[zl<O, {ZI,
Z
are
P
the
constant
vector
fields
over
is induced from parallel transport in F along the straight llne from tz
t
Then we have
d
dt
PROOF.
t
tffil
(2 14)
Lr8/r"
This depends largely on a direct computation.
One can easily see that at
t=
d
d-- (@t
*
(2.15)
Vrs/ r o(ZI,..., Zp)
Zp)
tz (ZI
Therefore at t=l
ddt mt (ZI
Zp)
Zp) Vrs/ rm(z
On the other hand, we may assume z i
It follows that
[r/r,Z i] =-Z
+ pm(Z
/z i or /zi.
....
Thus, [Z
(2.16)
Zp)
i,zj]
0 and
i.
(LrS/Sr )
(Z
,...,
Zp)
.
(i
ra/ar
da +
dirS/ar )
Zp)
(Z
P
=d(ra/Or,Zl,..., Zp)
=Vra/ar
(Z
Z
p
+
(-i)
i+l
+ p (Z I’
Vzlw(ra/ar,Zl,..., Z
Zp).
i
Zp)
(2.17)
679
APPLICATION OF STRESS ENERGY TENSOR
The proof is completed.
From lemmas 2.1, 2.2 we Immedlately obtain
is an F-valued p-form with conservative stress energy tensor,
LEMMA 2.3. If
1o
0 at t
/dt
then
2..
dlltil
Finally, we give the following.
Let F be a Hermltlan vector bundle over
DEFINITION.
values in F.
U
m over
The norm of
n
e
*I
U n and m be a p-form with
U n called slowly divergent if
f(r)
and
dr
0
hold for any positive continuous function f(r) in (0,I) (for example
/ log (l-r) ), we have
f(r)
f(r)e
*I
< -.
THE PROOF OF THEOREM I.I.
After establishing the preceding lemmas,
it is not difficult
We represent the complex hyperbolic space
n
in C with the complete Kaehler metric [II]
(I
lzaz a) (ZdZadZ a)
g=4
r-za a)
(I
It is obvious to see that
+
form
by
the
to prove Theorem
open
(lzadz a) (EZadZ a)
unit
disc
(3 .1)
2
T I’0 (U n) and T 0’I (U n) have respectively the orthonormal
Z
Z
bases
r
2
/l-r
B
’z
2
2
I"-
B
z2
B
r
zn
2
and
r
2
2
8
_-X-’,
Zl
r2
41
2
z2
/I
)
2
z2
n
For convienence, we may asse that
is of degree
(k,h), k+h
p.
Then
m can be
expressed as
a
fll
"IkJl ...JhdZ
A
.Adz^dZjlA...AdZjh.
(3.2)
K.
680
To begin with we calculate the inner product
at the point
(z 1,0,...,0).
At
that time, the Kaehler metric is reduced to
n
4
g
(I
r
do
dz
or dz
3
t
U
n
(3.3)
alt
among which
dz
d
and
consists of such p-forms
those
of
2t
contain
factor
Using (2.13), (3.3), we can compute that
are the remains.
t
Then, since
2 3
ml
not
and
z2 Z dzadz a)
2
+ (I
+ t + t
to
t
t
t
factors
the
contain
We can decompose
that
2)
d
(dz
2
(I
r2) p
(I
t2r2) p+I’
(I
r
2)p+2
(3.4)
is a homogeneous space, the restriction on z can be removed and these
n.
elements of
Denote dV and dS be the volume
z e U
expressions hold for all
n
2
n
Now
We have dV
U and the hypersphere S in U respectively.
2drdS / (l-r).
r
assuming
2
A(r,t)
r
Imtl2ds
2
(3.5)
r
we get
p
2
ilatl
0
-r
2
dr
f la tl2ds 0f
S
r
and
dd_ I[mtll2 -d t2p-2n f l=lp [ll+
where
Gz/t
i
n
2
Ico212
(I
(1
4
det(
(1
r
2)
2
(3.6)
A(r,t)dr
r2/t2) p+I
r2) p+I
(I
2 2
r /t )
+
(
(
r2/t2)P
r2) p
1312 (x(1
aS+
azBlt2])
(3.7)
r2/t2) p+2
r2) p+2 Gz/tdzdz
i
n
2
4n/(l
r2/t2)n+l
(3 .S)
Then at t--I
2
d Iic ll
PA<P, 1) + 2(p
n)11l12p
f
2
(3.9)
APPLICATION OF STRESS ENERGY TENSOR
Here we assne p
<
pA(p,I)
2(n- p)
I=l,p
O
0
such that
0
K
e
AP(F)do
<
0
<
0
--2(n- p)
0
is simpler than the former and can
It follows from Lemma 3.1 that
be treated by the same way.
Suppose that
n-I
n-I because the case of p
681
Ii 2
(n-p- !)r
(I +
(n
p) (l
2
r
2)
zero identically on
not equal
)dV.
U
n.
(3.10)
Thus there exists
and
zI(P0
i,12(
p- 1)r
(
+.
(n
p) (I
2
r
dv
2)
>
0
(3.11)
Therefore
P
K 0 + 2(n- p)
pA(p,I)
(I +
P0
(n
(n
2
l)r
p
r
p) (I
2)
A(r,l) dr.
(3.12)
.According to the method in [7], one can prove that
A(p,I)
for some
K
>
f
f(r)
K
l/
(1-r
0 and all
2)
n-p-I
P0 < p <
I.
(3.13)
Then for any positive continuous function f(r)
with
01-r
dr
we have
f f(r)emdV f
n
U
PO
rf(r)A(r l)dr
K
f
PO
f(r)
r
2dr
This ends the proof.
Finally, instead of considering the complex hyperbolic space form functioning as
the underlying manifolds, we can consider the other kind of underlying manifolds such
n
as the Euclidean n-space
the complex n-space
R
C and some conformal flat
n,
Riemannian manifolds.
Since the corresponding lemmas still exist, one may derive the
corresponding estimations over these manifolds.
For sake of simplicity, we only state
the following.
n
If the stress energy
be a p-form with values in F over C
n
tensor of m is conservative and the norm of m over C is finite or slowly divergent,
THEOREM 3. I.
then
vanishes.
Let
682
K. CAI
A conformal flat Riemannlan metric g on
DEFINITION 3.1.
n
R
is called to be of
type A if
(A)
g2 f2(x)
2
x. and the mean curvature normal H of
En
S
r
never
pointing away from zero. g is called to be of type B if the underlying space is the
n
open unit disc in R and
(B)
f2(r)
g
H of S
dx
curvature normal
r
i
f0 n-2Pdr
where f satisfying
and the mean
never pointing away from zero.
Let M be a conformal flat Riemannian manifold of typed A or B.
THEOREM 3.2.
m be a Riemannian vector bundle valued p-form over M,
stress energy tensor.
If the norm of
p
>
n/2,
Let
with conservative
over M is finite or slowly divergent.
Then
vanishes.
The author would like to thank Professor C.W. Ho and Professor T.
A(XNOWLEDGEMENTS.
Wu
for
their
encouragement
and help.
He
is
grateful
to
the
Southern
lllinois
University at Edwardsville for its hospitality.
RE FERENCE S
I.
YAU, S.T.
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4.
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Harmonic mappings of Riemannian manifolds, Amer.
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Liouville theorem for harmonic maps, Proc. Sm. in Pure Math. 36
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(1980), 147-151.
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