The second variational formula for Willmore submanifolds

The second variational formula for Willmore
submanifolds
n ∗ in S
§0. Introduction
Dedicated to Professor S. S. Chern at the occasion of his 90th birthday
Zhen Guo, Haizhong Li, Changping Wang
Abstract: In [17] the third author presented Moebius geometry for submanifolds
in S n and calculated the first variational formula of the Willmore
functional by using Moebius invariants. In this paper we present
the second variational formula for Willmore submanifolds. As an application
of these variational formulas we give the standard examples of
Willmore hypersurfaces {W m k := Sk ( � (m − k)/m)×S m−k ( � k/m), 1 ≤
k ≤ m − 1} in S m+1 (which can be obtained by exchanging radii in the
Clifford tori S k ( � k/m) × S m−k ( � (m − k)/m)) and show that they are
stable Willmore hypersurfaces. In case of surfaces in S 3 , the stability of
the Clifford torus S 1 ( 1
√ 2 ) × S 1 ( 1
√ 2 ) was proved by J. L. Weiner in [18].
We give also some examples of m-dimensional Willmore submanifolds
in an n-dimensional unit sphere S n .
Keywords: Willmore functional, second variational formula, stability,
Willmore conjecture
2000 MR Subject Classification 53A30, 53B25
Let M be an m-dimensional submanifold immersed in Sn . The Willmore functional W is
defined by
W (M) =
� � m
m 2
m − 1
� �
m
S − m�H�
2� 2
dM, (0.1)
M
∗ The second author is partially supported by the Alexander von Humboldt Research Fellowship and the
US-China Cooperative Research NSF Grant INT-9906856 and the third author is partially supported by
973-project, DFG466-CHV-II3/127/0, Qiushi Award and RFDP.
1

which is a conformal invariant under Moebius (or conformal) transformations of S n (see [3],
[17], [19]), where S is the square of the length of the second fundamental form, H is the mean
curvature vector and dM is the volume element of the induced metric on M. In case of a
surface in S 3 , this functional is equivalent to the Willmore functional We(M) := 4 �
M H2 dM
for surfaces in R 3 . The so-called Willmore conjecture states that We(M) ≥ 8π 2 for any
embedded torus in R 3 , well-studied (cf. [2], [9], [19], [20], [21]) since 1965. An equivalent
version of the Willmore conjecture states that W (T 2 ) ≥ 8π 2 for any embedded torus T 2 in
S 3 and the equality holds if and only if T 2 is Moebius equivalent to the Clifford torus. Many
efforts and partial results have been obtained for this conjecture, but up to now it is still
open.
From the analytic point of view, the second variational formula of the Willmore functional
might be very important to solve the Willmore conjecture. In 1978, J. L. Weiner gave the
second variational formula for minimal surfaces, which is a special class of Willmore surfaces
(cf. [18]). But it is rather complicated to give the second variation formula for general
Willmore surfaces by using Euclidean invariants. In [12] M. Peterson derives such a formula.
In [10] and [11] B. Palmer calculated the second variation of a Willmore surface in S 3 by
using its conformal Gauss map and used the formula to study the nonexistence of stable
Willmore surfaces under some conditions.
Since the Willmore functional (0.1) is invariant under the Moebius group (cf. [3], [17]),
one can use the framework of Moebius geometry and Moebius invariants to calculate the
second variational formula. It is the key point of this paper. For any submanifold M in
S n we can introduce a Moebius invariant metric g on M. Then the Willmore functional
is exactly the volume functional of g. The third author computed the first variation and
got the Euler - Lagrange equations in [17]. Submanifolds in S n satisfying these equations
are called Willmore submanifolds or Moebius minimal submanifolds. In this paper we give
the second variational formula of the Willmore functional for submanifolds in S n by using
Moebius invariants. Although this formula looks very complicated, in case of surfaces in
S 3 (which is the most important case) the formula is in good form (cf. §2, (2.43)). Using
the Euler-Lagrange � equations we find the standard examples of Willmore hypersurfaces
(m − k)/m) × Sm−k �
( k/m), 1 ≤ k ≤ m − 1} in Sm+1 , which is (euclidean)
{W m k := Sk (
minimal if and only if 2k = m. In some sense, W m � k can be considered � as the dual hypersurface
of the standard minimal hypersurface S k (
k/m) × S m−k (
that the hypersurface W m k are stable Willmore hypersurfaces.
(m − k)/m) in S m+1 . We show
We organize this paper as follows. In §1 we give Moebius invariants and local formulas in
Moebius geometry for submanifolds in S n . In §2 we calculate the second variation formula
for Willmore submanifolds in S n . As an application we prove in §3 that {W m k } are stable
Willmore hypersurfaces.
§1. Moebius invariants and local formulas for submanifolds in S n
Let x0 : M → S n be an m-dimensional compact submanifold with boundary ∂M, {e1, ..., em}
be a local orthonormal basis of T M with respect to the induced metric dx0 · dx0 and
{θ1, ..., θm} be its dual basis. Let {em+1, ..., en} be the local normal orthonormal vector
field. We make use of the following convention on the ranges of indices:
1 ≤ i, j, k, · · · ≤ m; m + 1 ≤ α, β, γ, . . . ≤ n
2

and we shall agree that repeated indices are summed over the respective ranges. Then the
structure equation of x0 can be written as
dx0 = � θiei
dei = � θijej + � h α ijθjeα − θix0
deα = − � h α ijθjei + � θαβeβ.
� h α iieα are the first, the second
The quantities I = dx0 · dx0, II = � hα ijθi ⊗ θjeα and H = 1
m
fundamental form and the mean curvature vector of x0 in Sn , respectively. We define the
function ρ : M → R by
ρ =
�
m
�II − HI�. (1.1)
m − 1
The metric g = ρ 2 dx0 · dx0 is called a Moebius metric which is invariant under Moebius
transformations in S n (cf.[17]); it is positive definite at any non-umbilical point. Then the
Willmore functional in (0.1) is exactly the Moebius volume functional for g:
�
W (M) :=
M
ρ m dM = V olg(M), (1.2)
where dM is the volume element for the metric dx0 · dx0. In this paper, it is our purpose to
calculate the second variation in the framework of Moebius geometry. We need the following
notation and local formulas. For more details we refer to [17].
be the Lorentz space with the inner product given by
Let R n+2
1
< X, W >= −x 0 w 0 + x 1 w 1 + · · · + x n+1 w n+1 ,
where X = (x 0 , x 1 , · · · , x n+1 ), W = (w 0 , w 1 , · · · , w n+1 ) ∈ R n+2 . The half cone in R n+2
1
defined as
C n+1
+
For the immersion x0 : M → S n we define
= {X ∈ R n+2
1 | < X, X >= 0, x 0 > 0}.
Y = ρ(1, x0) : M → C n+1
+ . (1.3)
If ˜x0 : M → Sn is Moebius equivalent to x0, then we have ˜ Y = Y T for some T ∈ O(n + 1, 1).
Thus
g =< dY, dY >= ρ 2 dx0 · dx0
(1.4)
is a Moebius invariant. In the following we assume that x0 is an immersion without umbilical
point, which implies that g is positive definite on M. Let Ei = ρ −1 ei, then {Ei} is an
orthonormal basis with respect to the metric g, and its dual basis is {ωi = ρθi}. Set
Yi := Ei(Y ), N := − 1 1
∆Y − < ∆Y, ∆Y > Y,
m 2m2 Eα := (H α , eα + H α x0), (1.5)
where ∆ is the Laplacian of the metric g and H = H α eα is the mean curvature vector with
respect to the induced metric dx0 · dx0.
3
is

Lemma 1.1([17]) {Y, N, Yi, Eα} satisfy the following conditions
< Y, Y >=< N, N >= 0, < Y, N >= 1, < Yi, Yj >= δij;
< Y, Yi >=< N, Yi >=< Y, Eα >=< N, Eα >= 0;
< Eα, Yi >= 0, < Eα, Eβ >= δαβ.
Lemma 1.1 shows that {Y, N, Yi, Eα} forms a Moebius moving frame in R n+2
1
The structure equations can be written as
dY = � ωiYi
dN = � ψiYi + � φαEα
dYi = −ψiY − ωiN + � ωijYj + � ωiαEα
dEα = −φαY − � ωiαYi + � ωαβEβ.
By differentiating these equations and using Cartan’s lemma, we obtain
ψi = �
Aijωj, Aij = Aji; ωiα = �
B α ijωj, B α ij = B α ji; φα = �
C α i ωi.
We have the following equations:
�
B α ii = 0, �
i
α,i,j
�
B α �2 ij = m − 1
m
, �
j
along M.
(1.6)
B α ij,j = −(m − 1)C α i . (1.7)
The relations between these Moebius invariants and Euclidean invariants are given by
Aij = −ρ −2 (Hessij(logρ) − ei(logρ)ej(logρ) − H α h α ij)
− 1
2 ρ−2 (|∇logρ| 2 − 1 + �
(H
α
α ) 2 )δij, (1.8)
B α ij = ρ −1 (h α ij − H α δij), (1.9)
C α i = − 1
m − 1 Bα ij,j = −ρ −2 (H α ,i + (h α ij − H α δij)ej(logρ)). (1.10)
§2. The second variational formula of the Willmore functional
In this section we calculate the second variation of the Willmore functional (or the Moebius
volume functional) defined by (1.2) or (0.1). Since the variation of the volume depends only
on the normal component of the variational vector field (cf. [17]), we will consider the normal
variation.
Let x : M × R → S n be a smooth variation of x0 such that x(·, t) = x0 and dxt(T M) =
dx0(T M) on ∂M for each (small) t. These two boundary conditions disappear if ∂M = Ø.
For each t we denote by{ei} a local orthonormal basis for T M with respect to dxt · dxt,
by {θi} its dual basis and by {eα} a local orthonormal basis for the normal bundle of xt.
be the canonical lift of xt and gt =< dY, dY > be the
Let Y = ρ(1, x) : M × R → C n+1
+
4

Moebius metric of xt. Let {Ei := ρ−1ei} be a local orthonormal basis for gt with dual basis
{ωi = ρθi}. Then the volume for gt can be written as
�
W (t) := V olgt(M) =
From Lemma 1.1 in section 1, we can choose a moving frame
in R n+2
1
{Y, N, Y1, · · · , Ym, Em+1, · · · , En}
M
ω1 ∧ · · · ∧ ωm. (2.1)
along M × R, which satisfies the conditions in Lemma 1.1 for each t. Let d denote
the differential operator on M × R, then we can find 1-forms
{V, Vα, Ψi, Φα, Ωi, Ωij, Ωiα, Ωαβ}
on M × R with Ωij = −Ωji and Ωαβ = −Ωβα such that
Taking the differentials of these equations we get
dY = V Y + ΩiYi + VαEα, (2.2)
dN = −V N + ΨiYi + ΦαEα, (2.3)
dYi = −ΨiY − ΩiN + ΩijYj + ΩiαEα, (2.4)
dEα = −Φα − VαN − ΩiαYi + ΩαβEβ. (2.5)
dV = Ψi ∧ Ωi + Φα ∧ Vα, (2.6)
dΩi = Ωij ∧ Ωj + V ∧ Ωi − Vα ∧ Ωiα, (2.7)
dVα = Ωαβ ∧ Vβ + Ωi ∧ Ωiα + V ∧ Vα, (2.8)
dΨi = Ωij ∧ Ψj − Φα ∧ Ωiα + Ψi ∧ V, (2.9)
dΦα = Ωαβ ∧ Φβ + Ψi ∧ Ωiα + Φα ∧ V, (2.10)
dΩij = Ωik ∧ Ωkj + Ωiα ∧ Ωαj − Ψi ∧ Ωj − Ωi ∧ Ψj, (2.11)
dΩiα = Ωij ∧ Ωjα + Ωiβ ∧ Ωβα − Ψi ∧ Vα − Ωi ∧ Φα, (2.12)
dΩαβ = Ωαγ ∧ Ωγβ + Ωαi ∧iβ −Φα ∧ Vβ − Vα ∧ Φβ. (2.13)
Since Y = ρ(1, x), if we write the normal variation vector field of x in T S n by
then by (1.5) we can find a function v : M × R → R such that
∂x
∂t = ρ−1 vαeα, (2.14)
∂Y
∂t = vY + vαEα. (2.15)
From (2.2), (2.15) and the fact that d = ωiEi + dt ∂
∂t on C∞ (M × R), we get
V = vdt, Vα = vαdt, Ωi = ωi. (2.16)
5

Since T ∗ (M × R) = T ∗ M ⊕ T ∗ R we can write
Ψi = ψi + aidt, Φα = φα + bαdt, (2.17)
Ωij = ωij + Pijdt, Ωiα = ωiα + Liαdt, Ωαβ = ωαβ + Qαβdt, (2.18)
where {ai, bα, Pij, Qαβ} are local functions with Pij = −Pji and Qαβ = −Qβα. Let {Bα ij, Aij, Cα i }
be Moebius invariants for xt defined in §1. If we denote by dM the exterior differential operator
on T ∗M, then we have d = dM + dt ∧ ∂
∂t on T ∗ (M × R). It follows from (2.6), (2.16)
and (2.17) (comparing the terms in T ∗ (M) ∧ dt ) that
ai = −v,i + vαC α i , (2.19)
where v,i := Ei(v). Similarly we get from (2.8), (2.16), (2.17), (2.18) that
and get from (2.7), (2.16), (2.18) that
Liα = vα,i
(2.20)
∂ωi
∂t = (Pij + vδij − vαB α ij)ωj. (2.21)
By a direct calculation in a similar way, we obtain from (2.9)∼(2.13) that
∂ωij
∂t = (Pij,k + B α ikvα,j − B α jkvα,i − aiδkj + ajδik)ωk, (2.22)
∂ψi
∂t = (ai,j + PikAkj + vα,iC α j − bαB α ij − vAij)ωj (2.23)
∂φα
∂t = (bα,i + QαβC β
i − Aijvα,j + ajB α ji − vC α i )ωi, (2.24)
∂ωiα
∂t = (vα,ij + PikB α kj − B β
ijQβα + Aijvα + bαδij)ωj
(2.25)
∂ωαβ
∂t = (Qαβ,i + vβ,jB α ji − vα,jB β
ji + vβC α i − vαC β
i )ωi, (2.26)
where {vα,i} are covariant derivatives of {vα}. Since φα = Cα i ωi and ψi = Aijωj, from (2.21),
(2.23) and (2.24) we have
∂C α i
∂t = bα,i + QαβC β
i + ajB α ij − Aijvα,j + PijC α j + B β
ijC α j vβ − 2vC α i , (2.27)
∂Aij
∂t = ai,j + PikAkj − PkjAki + vα,iC α j − B α ijbα + AikB α kjvα − 2vAij. (2.28)
Since ωiα = B α ijωj, from (2.21) and (2.25) we have
∂B α ij
∂t = vα,ij − vB α ij + PikB α kj − PkjB α ki − B β
ijQβα + vβB α ikB β
kj + Aijvα + bαδij. (2.29)
6

Multiplying B α ij to (2.29) and using (1.7) we get
m − 1
m v = Bα ijvα,ij + B α ikB β
kj Bα jivβ + AijB α ijvα. (2.30)
Making use of (2.21), (2.30), (1.7), Green’s formula and the conditions on ∂M, we get by
noting (2.34) of [17]
W ′ (t) = m2
� �
−(m − 1)C
m − 1 M
α i,i + B β
ikBα kjB β
ji + AijB α �
ij vαdMt, (2.31)
where dMt is volume element of gt.
We summarize as follows
Proposition 2.1. ([17]) x0 : M → S n is a Willmore submanifold (i.e., its Willmore
functional is stationary) if and only if
−(m − 1)C α i,i + B β
ik Bα kjB β
ji + AijB α ij = 0. (2.32)
Since ∂
∂t ◦ dM − dM ◦ ∂
∂t = 0 (or equivalently d2 = (dM + dt ∧ ∂
∂t )2 = 0) and
C α i,jωj = dMC α i + C α j ωji + C β
i ωβα,
we get
∂C α i,j
∂t = −Cα ∂ωk
i,k
∂t (Ej) + ( ∂Cα i
∂t ),j + C α ∂ωki
k
∂t (Ej) + C β ∂ωβα
i
∂t (Ej). (2.33)
By a direct calculation and use of (2.21), (2.22), (2.26) and (2.33), we get
∂C α i,i
∂t vα = −(C α i Pki),kvα − vC α i,ivα + vβB β
kiCα �
α ∂Ci i,kvα +
∂t
+C α k B β
ki vβ,ivα + akC α k vα − makC α k vα − C β
i Qαβ,ivα − B α kiC β
i vβ,kvα
+B β
ki Cβ
i vα,kvα − C β
i C α i vβvα + �
i,β
�
C β
i
�2 �
v
α
2 α.
By a straightforward calculation and using (2.27), (2.19), we get from (2.34) that
∂C α i,i
∂t vα = −(C α �
α ∂Ci i Pkivα),k +
∂t vα
�
+ (vB
,i
α kivα,i),k
+(m − 1)(vC α k vα),k − (C β
i Qαβvα),i − (bαvα,i),i − mvC α i,ivα
+C β
i,iQαβvα + C α i,kB β
kivαvβ + bαvα,ii − vB α kivα,ik + Aikvα,kvα,i
−2C α k B β
kivβvα,i + 2vC α i vα,i − m �
�
�
C
k α
α �2 k vα
+C β
i B β
ikvα,kvα + �
(C
i,β
β
�
2
i ) v
α
2 α.
7
�
,i
vα
(2.34)
(2.35)

From the conditions that x(., t) = x0 and dxt(T M) = dx0(T M) on ∂M for each t, we see
that vα|∂M = 0 and
0 = ∂
∂t (dxt) = d( ∂x
∂t ) = ρ−1dvαeα (2.36)
at the boundary ∂M. It follows from (2.35) and Green’s formula that
�
Γ1:= {
M
∂
∂t |t=0(C α i,i)}vαdM
�
= {−mvC
M
α i,ivα + C β
i,iQαβvα + C α i,kB β
kivαvβ + bαvα,ii
−vB α kivα,ik + Aikvα,kvα,i − 2C α k B β
kivβvα,i + 2vC α i vα,i
−m �
�
�
C
k α
α �2 k vα + C β
i B β
ikvα,kvα + �
(C
i,β
β
�
2
i )
α
v 2 α}dM.
(2.37)
Using (2.28), (2.29), (2.19) and the facts that Pij = −Pji and �
j B α ij,j = −(m − 1)C α i , we
get
∂ �
B
∂t
β
ikBα kjB β
ji + AijB α �
ij vα
= (vB α ijvα,j),i − (m − 1)(vC α i vα),i + (aiB α ijvα),j
−2(m − 1)vC α i,ivα + 2B β
ijB α jkvβ,ikvα + (B β
ikBβ kj + Aij)vα,ijvα − vB α ijvα,ij
+(m − 1)C β
i,iQαβvα + 2(m − 1)vC α j vα,j + C γ
i B α ijvγ,jvα − C γ
i B α ijvα,jvγ
+ �
3B β
ijB α jkB γ
klBβ li + 4Bα ikB γ
kjAji + (m − 1)C α i C γ
�
i vαvγ
+ � �
(B
i,j β
β
ijbβ)( �
B
α
α ijvα) + ( �
⎛
bαvα) ⎝
α
�
(B
β,i,j
β
ij) 2 ⎞
+ tr(A) ⎠
+ �
A
i,j
2 �
ij v
α
2 α + B β β
ikAkjBji( �
v
α
2 α).
Using the boundary conditions and Green’s theorem we have
�
Γ2:=
where
�
=
M
M
{ ∂ β
|t=0(Bik ∂t Bα klB β
lj + AijB α ij)}vαdM
{−2(m − 1)vC α i,ivα + 2B β
ijB α jkvβ,ikvα + (B β
ikBβ kj + Aij)vα,ijvα
−vB α ijvα,ij + (m − 1)C β
i,iQαβvα + 2(m − 1)vC α j vα,j + C γ
i B α ijvγ,jvα
−C γ
i B α ijvα,jvγ + �
3B β
ijB α jkB γ
klBβ li + 4Bα ikB γ
kjAji + (m − 1)C α i C γ
�
i vαvγ
+ � �
(B β
ijbβ)( �
B α ijvα) + ( � m − 1
bαvα)( + tr(A))
m
ij
+ �
ij
β
A 2 ij
�
α
α
α
v 2 α + B β β
ikAkjBji( �
α
v 2 α)}dM,
(2.38)
(2.39)
bα = − 1
�
∆vα + B
m
α ikB β
kivβ + 1
2m (1 + m2 �
κ)vα , (2.40)
8

tr(A) = 1
2m
m
+ κ, (2.41)
2
where κ is the normalized scalar curvature of the metric g = ρ 2 dx0 · dx0. The function v
is given by (2.15) and (2.30). Thus for a Willmore submanifold x0, we get from (2.31) and
(1.7) that
W ′′ (0) = −m 2
�
M
∂C α i,i
∂t |t=0vαdM
+ m2
� �
∂ β
|t=0(Bik m − 1 M ∂t Bα kjB β
ji + AijB α �
ij) vαdM.
Thus the formula of the second variation is given by (2.37) and (2.39). We conclude that
Theorem 2.1. Let x : M → S n be a compact Willmore submanifold without boundary.
Then the second variation formula of the Moebius volume functional is given by
W ′′ �
(0) =
M
�
m 2 (m − 2)vC α i,ivα − m 2 C α i,jB β
jivαvβ + m2 (m − 2)
vB α ijvα,ij
m − 1
−m 2 Aijvα,ivα,j + m2 (2m − 1)
m − 1 Cβ i B α ijvβ,jvα + m 2 (m + 1) �
�
�
C
i α
α �2 i vα
−m 2 C β
i B β
�
2
ijvα,jvα − m
i,β
�
C β
i
� 2 �
α
v 2 α + 2m2
m − 1 Bβ
ijB α jkvβ,ikvα
+ m2
m − 1 (Bβ
ik Bβ
kj + Aij)vα,ijvα − m2
m − 1 Cβ
i B α ijvα,jvβ
+ m2 �
3B
m − 1
γ
ijB α jkB β
klBγ li + 4Bα ikB β
kjAji �
vαvβ + m2
m − 1
+ m2
m − 1 Bβ
�
�
β
ikAkjBji v
α
2 �
α
+ m �
(∆vα) 2 +
α
�
A
ij
2 �
ij v
α
2 α
m(m − 2)
m − 1 Bα ijB β
ij∆vαvβ
� �
m(m − 2)
+
tr(A) − 1 vα∆vα −
m − 1 m
⎛
�
⎝
m − 1 α
�
B
ijβ
α ijB β
⎞2
⎠
ijvβ
�
− 1 + 2m
m − 1 tr(A)
� �
� �
B
ij α
α �2 ijvα − tr(A)
�
1 + m
m − 1 tr(A)
�
�
v
α
2 �
α dM.
(2.42)
In case of the surface in S 3 , we omit all α and β because the codimension now is one,
the formula is reduced to
Corollary 2.1. For a Willmore surface x0 : M → S 3 , the second variational formula is
9

given by
W ′′ �
(0) =
M
−4 �
ij
{2(∆f) 2 + 2f∆f + 12 �
Aijfifj + (4 �
ij
ij
A 2 ij + 4 �
CiBijfjf + 4 �
Aijfijf
i
ij
C 2 i + 7
8 + K − 2K2 )}dM,
where K is Gaussian curvature of the Moebius metric g = ρ 2 dx0 · dx0.
(2.43)
Remark 2.1. The second variational formula for Willmore surfaces in S 3 might be important
to solve the Willmore conjecture. As far as we know the only known stable example of a
Willmore torus is the Clifford torus. From the existence result of L. Simon in [15], we know
that the Willmore conjecture is true if one can show that the only stable Willmore torus
embedded in S 3 is the Clifford torus.
§3. Willmore tori in S m+1 and their stabilities
In this section we present a class of important examples of Willmore hypersurfaces called
Willmore tori. As an application of Theorem 2.1 we show that they are stable Willmore
hypersurfaces.
Let R m+2 be the (m+2)-dimensional Euclidean space with inner product . We write
R m+2 = R k+1 × R m−k+1 , 1 ≤ k ≤ m − 1. For any vector ξ ∈ R m+2 there is a unique
decomposition ξ = ξ1 + ξ2 with ξ1 ∈ R k+1 and ξ2 ∈ R m−k+1 . For another vector η = η1 + η2
the inner product of them can be written as < ξ, η >=< ξ1, η1 > + < ξ2, η2 >. Let
ξ1 : S k → R k+1 and ξ2 : S m−k → R m−k+1 be standard embeddings of unit spheres. Let
x : S k (a1) × S m−k (a2) → S m+1 ⊂ R m+2 be the embedded hypersurface x = a1ξ1 + a2ξ2 with
a 2 1 + a 2 2 = 1. It is easy to check that
(i) the unit normal vector of M := S k (a1) × S m−k (a2) in S m+1 is given by
em+1 = −a2ξ1 + a1ξ2;
(ii) the second fundamental form of M is given by
II = − < dx, dem+1 >= a1a2(< dξ1, dξ1 > − < dξ2, dξ2 >);
(iii) the induced metric of M is given by
If we take {ei} and {ωi}such that
I = a 2 1|dξ1| 2 + a 2 2|dξ2| 2 .
k�
d(a1ξ1) = ωiei, d(a2ξ2) =
n�
ωjej,
i=1
j=k+1
10

then we have
where
m�
I = ω
i=1
2 k� a2
i , II = ω
i=1
a1
2 m� a1
i − ω
j=k+1
a2
2 j := hijωiωj, (3.1)
⎧
⎪⎨
hij =
⎪⎩
a2
a1 δij, if 1 ≤ i, j ≤ k,
− a1
a2 δij, if k + 1 ≤ i, j ≤ n.
(3.2)
Theorem 3.1. Let W = S k (a1) × S m−k (a2) be the hypersurface imbedded into S m+1 ,
where a 2 1 + a 2 2 = 1. Then W is a Willmore hypersurface if and if
Proof. From (3.1) we see that
S :=
H := 1
m
a1 =
�
m − k
m , a2
�
k
=
m .
m�
h
i,j
2 � �
a2
2
� �
a1
2
ij = k + (m − k) , (3.3)
a1
a2
m�
i=1
hii = 1
�
k
m
a2
a1
− (m − k) a1
a2
�
, (3.4)
ρ 2 = m
m − 1 (S − mH2 ). (3.5)
Substituting (3.2) and (3.5) into (1.8)-(1.10) we know that the Willmore condition
is equivalent to the equation
(m − k)
� �
a2
6
a1
−(m − 1)Ci,i + BikBkjBji + AijBij = 0
+ (2m − 3k)
� �
a2
4
a1
+ (m − 3k)
� �
a2
2
From the equations (3.6) and a2 1 + a2 �
m−k
2 = 1 we get a1 = m and a2
�
k = m .
We call
W m k = S k
⎛�
⎞
m − k
⎝ ⎠ × S
m
m−k
⎛�
⎞
k
⎝ ⎠ , 1 ≤ k ≤ m − 1,
m
a Willmore torus.
a1
− k = 0. (3.6)
Remark 3.1. It is remarkable that the Willmore torus W m k is (Euclidean) minimal if
and only if 2k = m. We note that W m �
k can be obtained by exchanging the radii k/m and
�
m − k/m in the Clifford minimal torus Sk �� �
k × S m
m−k �� �
m−k . m
Remark 3.2. It is known that any minimal surface in S n is a Willmore surface (see Weiner
[18]) (also see Pinkall’s examples of non-minimal Willmore surfaces [13]). Theorem 3.1
shows that m-dimensional (m ≥ 3) minimal hypersurfaces are not Willmore hypersurfaces
in general.
11

Remark 3.3. In [6], the second author proved an integral inequality of Simons’ type for
m-dimensional compact Willmore hypersurfaces in S m+1 and gave a characterization of Willmore
tori W m k by use of his integral inequality.
From now on we study the stability of Willmore tori W m k defined in Theorem 3.1. For
W m k we get from (3.2), (3.3), (3.4) and (3.5) that
⎧ �
⎨ k
m−k
hij =
⎩
δij, if 1 ≤ i, j ≤ k,
�
m−k − k δij, if k + 1 ≤ i, j ≤ m,
S = k3 + (m − k) 3
k(m − k)
From (1.8) and (1.9) we have
⎧
⎨
=
⎩
Bij = ρ −1 ⎧
⎨
(hij − Hδij) =
⎩
(3.7)
m − 2k
, H = −� k(m − k) , ρ2 = m2
. (3.8)
m − 1
Aij = ρ −2 Hhij + 1
2 ρ−2 (1 − H 2 )δij
ρ −2 3km−k2 −m 2
2k(m−k) δij, if 1 ≤ i, j ≤ k,
ρ −2 m2 −k 2 −km
2k(m−k) δij, if k + 1 ≤ i, j ≤ m,
(3.9)
tr(A) = (m − 1)(m2 − 3km + 3k2 )
, (3.10)
2km(m − k)
ρ −1� m−k
k δij, if 1 ≤ i, j ≤ k
−ρ −1� k
m−k δij, if k + 1 ≤ i, j ≤ m,
(3.11)
and � B 2 ij = m−1
m . From the last equation in (1.7) we obtain Ci = 0. We are going to
calculate
W ′′ (0) = −m 2 Γ1 + m2
m − 1 Γ2,
{ ∂
∂t |t=0(BikBklBli + AijBji)}fdM and f ∈ C ∞ (M) is
where Γ1 = � ∂Ci,i
M |t=0fdM, Γ2 = ∂t �
M
the normal component of the variational vector field. From (2.40) we have
b = − 1
�
∆f +
m
(m − 1)(m2 + k2 �
− km)
f . (3.12)
2km(m − k)
Substituting (3.9), (3.11) and (3.12) into (2.37), we get
−m2 �
Γ1 =
M
�
m(∆f) 2 + (m − 1)(m2 + k2 �
− km)
f∆f
2k(m − k)
� �
2 2 (m − 1)(3km − k − m )
−
|∇1f|
M 2k(m − k)
2 + (m − 1)(m2 − km − k2 )
|∇2f|
2k(m − k)
2
�
dM
+m 2
�
M
vBijfijdM.
12
dM
(3.13)

Substituting (3.9), (3.11) and (3.12) into (2.39), by a straightforward calculation we get
m2 m − 1 Γ2 = − m2 − k2 �
+ km
f∆fdM +
2k(m − k) M
5m2 + 5k2 �
− 9km
f∆1fdM
2k(m − k) M
+ 5k2 + m 2 − km
2k(m − k)
�
M
f∆2fdM − m2
�
vBijfijdM
m − 1 M
+ �
2km(m − k)(m 2 + 7km − 7k 2 ) − m(m 2 − k 2 + km)(k 2 + m 2 − km)
+ k(3km − m 2 − k 2 ) 2 + (m − k)(m 2 − km − k 2 ) 2�
m − 1
4k2m2 (m − k) 2
�
f
M
2 dM.
(3.14)
In (3.13) and (3.14) we denote by ∆ the Laplace operator on W m k with respect to the
Moebius metric g = ρ2dx · dx. � We write g = g1 ⊕ g2 �
according to the decomposition
M m k := M1×M2 := S k �� m−k
m
×S m−k �� k
m
. We denote by {∆1, ∆2} the Laplace operators
of {g1, g2} and by {∇1, ∇2} the gradient operators on M1 and M2, respectively. From (2.30),
(3.9) and (3.11), we have
From(3.13), (3.14) and (3.15) we get
W ′′ �
(0) =
vBijfij = 1
⎛�
m − k
⎝
m k ∆1f
�
k
−
m − k ∆2f
⎞2
⎠ . (3.15)
M
�
m(∆f) 2 + m(k2 + m2 − km − 2m)
f∆f
2k(m − k)
+ m(m − 2)
⎛�
m − k
⎝
m − 1 k ∆1f
�
k
−
m − k ∆2f
⎞2
⎠
+ m(3km − k2 − m2 ) + 6(m − k) 2
f∆1f
2k(m − k)
+ m(m2 − km − k2 ) + 6k2 f∆2f +
2k(m − k)
�
2(m − 1) 2
f dM.
m
(3.16)
We denote the Laplace operators with respect to the Euclidean metric dx · dx by ∆M, ∆M1
and ∆M2 on W m k , M1 and M2, respectively. Since ρ = constant and the Moebius metric
g = ρ 2 dx · dx, we have ∆M = ρ 2 ∆, ∆M1 = ρ 2 ∆1, ∆M2 = ρ 2 ∆2 and ∆M = ∆M1 + ∆M2. From
13

(3.8) and (3.16), we have
W ′′ (0) =
m − 1
2k(m − k)m3 � �
2k(m − k)(m − 1)(∆Mf)
M
2
+m 2 (k 2 + m 2 − km − 2m)f∆Mf
+2(m − 2) ((m − k)∆M1f − k∆M2f) 2
+m �
m(3km − k 2 − m 2 ) + 6(m − k) 2�
f∆M1f
+m �
m(m 2 − km − k 2 ) + 6k 2�
f∆M2f
+4k(m − k)m 2 f 2 ) dM.
(3.17)
Let λi, λ ′ i and µij be the eigenvalues of Laplace operators ∆M1, ∆M2 and ∆M, respectively,
then we have
λi = i(k + i − 1) n
n − k , λ′ j = j(n − k + j − 1) n
k , µij = λi + λ ′ j,
where i and j are nonnegative integers. Let fi, f ′ i be eigenfunctions corresponding to λi and
λ ′ i, respectively, then gij(p, q) = fi(p)f ′ j(q) ((p, q) ∈ M1×M2) is an eigenfunction corresponding
to µij. For any f ∈ C ∞ (M m k ) we have the decomposition of f into eigenfuctions
f = �
i+j�=0
where cij and c0 are constants. Thus we have
∆M1f = − �
λicijgij, ∆M2f = − �
i+j�=0
i+j�=0
Substituting (3.18) and (3.19) into formula (3.17), we get
W ′′ (0) ≥ m−1
2k(m−k)m3 � �
M i+j�=0
cijgij + c0, (3.18)
λ ′ jcijgij, ∆Mf = − �
�
2k(m − k)(m − 1)µ 2 ij
i+j�=0
−m2 (k2 + m2 − km − 2m)µij + 2(m − 2) �
(m − k)λi − kλ ′ �2 j
−m (m(3km − k 2 − m 2 ) + 6(m − k) 2 ) λi
−m (m(m 2 − km − k 2 ) + 6k 2 ) λ ′ j + 4k(m − k)k 2 ) c 2 ijg 2 ijdM
=
m−1
2k(m−k)m3 � �
i+j�=0 M {2(m − k)(m2 − 2m + k)λ2 i
−2(2m 3 + km 3 − 6km 2 + 3k 2 m)λi
+2k(m 2 − m − k)(λ ′ j) 2 − 2m(m 3 − m 2 − km 2 + 3k 2 )λ ′ j
+4k(m − k)λiλ ′ j + 4k(m − k)m 2 }c 2 ijg 2 ijdM.
14
µijcijgij. (3.19)
(3.20)
(3.21)

If we set
A(i, j) = 2(m − k)(m 2 − 2m + k)λ 2 i − 2(2m 3 + km 3 − 6km 2 + 3k 2 m)λi
then one can easily verify that
+2k(m 2 − m − k)(λ ′ j) 2 − 2m(m 3 − m 2 − km 2 + 3k 2 )λ ′ j
+4k(m − k)λiλ ′ j + 4k(m − k)m 2 ,
A(i, j) = � km
(m−k) 2 (m2 − 2m + k)i(k + i − 1) + ( 2km j(m − k + j − 1) − 2km)�
m−k
· � m(m−k)
k 2
(m2 − m − k)j(m − k + j − 1) + ( 2(m−k)m
i(k + i − 1) − 2(m − k)m) k
�
− m2 (m2−2m+k)(m2−m−k) ij(k + i − 1)(m − k + j − 1).
k(m−k)
(3.22)
From (3.22) it is not difficult to see that A(i, j) ≥ 0 and A(i, j) = 0 if and only if (i, j) =
(1, 0), (0, 1) or (1, 1). Thus we have proved the main result in this section:
Theorem 3.2. All Willmore tori Sk �� �
m−k × S m
m−k �� �
k → S m
m+1 , 1 ≤ k ≤ m − 1, are
stable.
Now we would like to propose the following generalized Willmore conjecture in S m+1 (cf.
Pinkall [14] and Kobayashi [5]):
Generalized Willmore Conjecture: Let Gk be a m-dimensional manifold which is diffeomorphic
to S k × S m−k and x : Gk → S m+1 be an imbedding, where 1 ≤ k ≤ m − 1.
Set
τk(x) =
� � m
m 2
m − 1
� �
m
S − mH
2� 2
dM, (3.23)
Gk
where dM is the volume element of Gk with respect to the induced metric dx · dx, S the
square of the length of the second fundamental form and H the mean curvature of x. Then
τk(x) ≥
4π m+2
2 (m − k) k
2 k m−k
2
m m
2 −2 (m − 1)Γ( k+1
2
m−k+1 )Γ( ) 2
(3.24)
and equality holds if and only if x(M) is Moebius equivalent to W m k . Here Γ is the gamma
function and the term on the right hand side of (3.24) is the Moebius volume of W m k .
Remark 3.4. When m = 2, the above generalized Willmore conjecture reduces to the
Willmore conjecture.
§4. Examples of Willmore Submanifolds in S n
In this section, we will give some examples of Willmore submanifolds in a unit sphere S n .
We first prove
15

Theorem 4.1. Let x : M → S n be a m-dimensional submanifold in an n-dimensional unit
sphere. Then x(M) is a m-dimensional Willmore submanifold in S n if and only if, for any
α with m + 1 ≤ α ≤ n,
−ρm−2 [SH α + Hβh β
ijhα ij − hα ijh β
ikhβkj − m| � H| 2H α ]
+(m − 1)∆(ρm−2H α ) − (ρm−2 ),ij(mHαδij − hα ij) = 0,
(4.1)
where ∆ and (.),ij are the Laplacian and covariant derivatives with respect to the induced
metric dx · dx, respectively.
Proof. From [17], we have the following relations between the connections of the Moebius
metric ρ 2 dx · dx and the induced metric dx · dx, respectively,
By use of (1.10), we get
thus
ωij = θij + (lnρ)iθj − (lnρ)jθi, ωαβ = θαβ. (4.2)
ρC α i,jθj = C α i,jωj
= dC α i + C α j ωji + C β
i ωβα
= dC α i + C α j θji + C β
i θβα + [(lnρ)kθi − (lnρ)iθk] · C α k ,
ρC α i,j = 2ρ −3 ρj[H α ,i + (h α ik − H α δik)(lnρ)k] − ρ −2 [H α ,ij + (h α ik,j − H α ,jδik)(lnρ)k]
−ρ −2 (h α ik − H α δik)(lnρ)kj + C α k (lnρ)kδij − (lnρ)iC α j .
Letting i = j and making summation over i in (4.3), we have
�
C
i
α i,i = −ρ−3∆H α − ρ−3 (hα ik − Hαδik)(lnρ)ki −2(m − 2)ρ−3H α ,i (lnρ)i − (m − 3)ρ−3 (lnρ)i(lnρ)j(hα ij − Hαδij). On the other hand, we have from (1.8) and (1.9)
AijB α ij + B β
ikBβ kjBα ij = −ρ−3 (lnρ)ij(hα ij − Hαδij) + ρ−3 (lnρ)i(lnρ)j(hα ij − Hαδij) +ρ−3 [h β
ikhβkj hαij − Hβh β
ijhα ij − SH α + m| � H| 2H α ].
Putting (4.5) into (2.32), we get
m−1
ρ m+1 {− ρm−2
m−1 [SHα + H β h β
ijhα ij − hα ijh β
ikhβkj − m| � H| 2H α ]
+ρm−2∆H α + m−2
m−1ρm−2 (lnρ),ij(H αδij − hα ij) + 2(m − 2)ρm−2 (lnρ)iH α ,i
+ (m−2)2
m−1 ρm−2 (lnρ)i(lnρ)j(hα ij − Hαδij)} = 0.
By a direct computation we can check the following identity
− 1
m−1 (ρm−2 )ij(mHαδij − hα ij) + ∆(ρm−2H α )
= (m−2)2
m−1 ρm−2 (lnρ)i(lnρ)j(hα ij − Hαδij) + m−2
m−1ρm−2 (lnρ)ij(hα ij − Hαδij) +2(m − 2)ρm−2 (lnρ)iH α ,i + ρm−2∆H α .
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
Thus (4.6) is equivalent to (4.1) by use of (4.7). We complete the proof of Theorem 4.1. The
16

Gauss equations of an m-dimensional submanifold x : M → S n give
Rij = (m − 1)δij + mH β h β
ij − h β
ikhβkj , (4.8)
R = m(m − 1) + m 2 | � H| 2 − S, (4.9)
where Rij and R are the Ricci curvature and scalar curvature of M, respectively, with respect
to the induced metric dx · dx. By use of (4.8) and (4.9) we have,
SH α + H β h β
ijh α ij − h α ijh β
ik hβ
kj − n| � H| 2 H α = (Rij − (m − 1)H β h β
ij)(h α ij − H α δij). (4.10)
Combining (4.1) and (4.10), we prove Corollary 4.1. x : M → S n is an m-dimensional
Willmore submanifold in an n-dimensional unit sphere if and only if, for any α with m+1 ≤
α ≤ n,
−ρ m−2 (Rij − (m − 1)H β h β
ij)(h α ij − H α δij)
+(m − 1)∆(ρ m−2 H α ) − (ρ m−2 ),ij(mH α δij − h α ij) = 0.
(4.11)
Remark 4.1. When m = 2, Corollary 4.1 implies that x : M 2 → S n is a Willmore surface
if and only if
∆H α + h α ijh β
ijH β − 2| � H| 2 H α = 0, 3 ≤ α ≤ n,
which was first proved by Weiner in [18]. Thus all minimal surfaces in S n are Willmore
surfaces. In particular, the Veronese surfaces in S n are Willmore surfaces. In [8], Li-Wang-
Wu gave a Moebius characterization of Veronese surfaces in S n .
Remark 4.2. Fix the index α with m + 1 ≤ α ≤ n, define ✷ α : M → R by
✷ α f = (mH α δij − h α ij)fij,
where f is any smooth function on M. We know that ✷ α is a self-adjoint operator (cf.
Cheng-Yau [4] and Li [7]). This operator naturally appears in the Willmore equations (4.1)
or (4.11).
Theorem 4.2. Let x : M → S n be a m-dimensional minimal submanifold in an ndimensional
unit sphere. If M is an Einstein manifold, then x : M → S n is a Willmore
submanifold.
Proof. From our assumption
we have, by use of (4.9),
H α = 0, m + 1 ≤ α ≤ n, Rij = R
m δij, (4.12)
ρ 2 = S − m| � H| 2 = S = m(m − 1) − R = constant. (4.13)
Thus from (4.12) and (4.13) we know that (4.11) holds. We conclude that M is a Willmore
submanifold.
17

Example 4.1. Sm �
2(m+1)
(
S m (
�
m
) → Sm+p with p = 1(m
− 1)(m + 2). Let
2
2(m + 1)
m
) = {(x0, x1, · · · , xm) ∈ R m+1 ; � x 2 i =
2(m + 1)
m
}.
Let E be the space of (m + 1) × (m + 1) symmetric matrices (uij), (i, j = 1, · · · , m), such
that � uii = 0; it is a vector space of dimension 1m(m
+ 3). We define a norm in E by
2
||(uij)|| 2 = � u2 ij. Let Sm+p with p = 1(m
− 1)(m + 2) be the unit hypersphere in E. The
2
mapping of Sm �
2(m+1)
) into Sm+p , defined by
m
uij = 1
�
m
2 m + 1 (xixj − 2
m δij),
is an isometric minimal immersion. (Actually, this gives an imbedding of the real projective
space of m-dimension into Sm+p ). We know that Sm �
2(m+1)
( ) is an Einstein manifold, thus
m
from Theorem 4.2 this immersion is a Willmore submanifold.
Example 4.2. CP m 2m/(m+1) → Sm(m+2)−1 . We can define a minimal immersion of the
with holomorphic sectional curvature
m-dimensional complex projective space CP m 2m/(m+1)
2m/(m+1) into a unit sphere Sm(m+2)−1 such that the usual coordinate functions of Rm(m+2) are all independent hermitian harmonic functions of degree 1 on CP m 2m/(m+1) (see Wallach
[16]). We also know that CP m 2m/(m+1) is an Einstein manifold with Ricci curvature m. There-
fore from Theorem 4.2 we conclude that this immersion is a Willmore submanifold.
Proposition 4.1. Let M = Sm1 (a1) × · · · × Smr (ar) be the submanifold imbedded into
Sm+r−1 , where m1 + · · · + mr = m. Then M is a Willmore submanifold if and only if
Proof. Consider
ai =
�
m − mi
, i = 1, · · · , r. (4.14)
m(r − 1)
R m+r = R m1+1
r�
mr+1
× · · · × R , m =
i=1
S m1 (a1) × · · · × S mr (ar) = {(a1x1, · · · , arxr) ∈ R m+r : |xi| = 1, i = 1, · · · , r},
where Smi (ai) ⊂ Rmi+1 r�
, (i = 1, · · · , r),
a
i=1
2 i = 1, is a m-dimensional submanifold in Sm+r−1 .
Writing x = (a1x1, · · · , arxr) : M = Sm1 (a1) × · · · × Smr (ar) → Sm+r−1 , x1 · x1 = · · · =
xr · xr = 1. Then r − 1 orthogonal normal vector fields of M are
mi,
en+λ = (aλ1x1, · · · , aλrxr), λ = 1, · · · , r − 1
where (aλ1, · · · , aλr) satisfies that (r × r) matrix
⎛
⎜
A = ⎜
⎝
a1
a11
.
· · ·
· · ·
· · ·
ar
a1r
.
ar−11 · · · ar−1r
18
⎞
⎟
⎠

is an orthogonal matrix, thus
r−1
�
λ=1
aλiaλj = δij − aiaj. (4.15)
Therefore the first fundamental form and the second fundamental form of M are given by
I = dx · dx = a 2 1dx1 · dx1 + · · · + a 2 rdxr · dxr,
r−1 �
II = − [a1aλ1dx1 · dx1 + · · · + araλrdxr · dxr]em+λ.
λ=1
We get the components of the second fundamental form
⎛
(h m+λ
⎜
ij ) = ⎜
⎝
− aλ1
a1 E1
. ..
aλr
ar Er
⎞
⎟
⎠ , λ = 1, · · · , r − 1 (4.16)
where Ei is a (mi × mi) unit matrix. It follows that ρ 2 = m(S − mH 2 )/(m − 1) is constant
and ∇ ⊥ eα = 0, ∆H α = 0, α = m + λ.
From (4.1), we know that M is a Willmore submanifold if and only if
SH α + H β h β
ijh α ij − h α ijh β
ik hβ
kj − m| � H| 2 H α = 0, n + 1 ≤ α = m + λ ≤ n + r − 1. (4.17)
Putting
H m+λ = 1
r� aλi
mi,
m i=1
ai
1 ≤ λ ≤ r − 1, S = � 1 − a
mi
i=1
2 i
a2 ,
i
− �
h
µ,i,j,k
m+λ
ij h m+µ
ik hm+µ
� mi
kj =
i a3 aλi(1 − a
i
2 i ), �
h
i,j
m+λ
ij h m+µ
ij = � aλiaµi
mi
i a2 i
into (4.17), by use of (4.15) and (4.16) we can easily check that (4.17) can be reduced to
�
aλi{−
i
mi �
mj(1 −
mai j
1
a2 ) −
j
1 � δij − aiaj
mimj
m j a2 i aj
mi
a3 (1 − a
i
2 i ) + 1
m2 � δjk − ajak
mimjmk } = 0,
j,k
aiajak
where 1 ≤ λ ≤ r − 1. (4.18) is equivalent to
Therefore (4.19) becomes
−2m +
�
mj(1 −
j
1
a2 ) −
j
1 � δij − aiaj
mimj
m j a2 i aj
mi
a3 (1 − a
i
2 i ) + 1
m2 � δjk − ajak
mimjmk = 0.
j,k
aiajak
− mi
mai
m − mi
a 2 i
+ �
j
mj
a 2 j
(4.18)
(4.19)
(1 + mj
) = 0, i = 1, · · · , r. (4.20)
m
19

We solve (4.20) to get
a 2 i =
We complete the proof of Proposition 4.1.
m − mi
, i = 1, · · · , r.
m(r − 1)
Remark 4.3. When r = 2, Proposition 4.1 reduces to Theorem 3.1.
Remark 4.4. We also know that S m1 (a1) × · · · × S mr (ar) in S m+r−1 is a m-dimensional
minimal submanifold if and only if
ai =
� mi
m ,
�
mi = m, i = 1, · · · , r.
i
Thus it follows that S m1 (a1) × · · · × S mr (ar) in S m+r−1 is an m-dimensional minimal and
Willmore submanifold if and only if
m1 = m2 = · · · = mr = m
r , ai
�
=
1
r .
Acknowledgements. The third author would like to thank Professor U. Simon for his
hospitality during his research stay at the TU Berlin. We would like to thank the referee for
his useful suggestions and comments.
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Zhen Guo Haizhong Li
Department of Mathematics Department of Mathematical Sciences
Yunnan Normal University Tsinghua University
Kunming, 650092, P. R. of China Beijing, 100084, P. R. of China
e-mail:guominzhi@ynmail.com e-mail:hli@math.tsinghua.edu.cn
Changping Wang
Department of Mathematics
Peking University
Beijing,100871, P. R.of China
e-mail:wangcp@pku.edu.cn
Eingegangen am 11. April 2001
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