TECHNISCHE
UNIVERSITAT
BERLIN
S
SRg
MOBIUS
ISOPARAMETRIC
HYPERSURFACES
IN
S"*1
WITH
TWO
DISTINCT
PRINCIPAL
CURVATURES
Haizhong
LI
Huili
LIU
Changping
WANG
Guosong
ZHAO
Preprint
No.
652/1999
PREPRINT
REIHE
MATHEMATIK
FACHBEREICH
3
MOBIUS
ISOPARAMETRIC
HYPERSURFACES
IN
§"+!
WITH
TWO
DISTINCT
PRINCIPAL
CURVATURES
HaizHone
Lr),
Hur
Liv)2)5),
Cuanepinc
Wane)3)4),
Guosona
ZHAO!)3)
ABSTRACT.
A
hypersurface
z
:
M
>
S”++
without
umbilic
point
is
called
a
Mébius
isoparametric
hypersurface
if
its
Mébius
form
®
=
—p~?
7,
(e,(H)
+
225
(Aig
—
H4i;)ej
(log
p))w;
vanishes
and
its
Mébius
shape
operator
S
=
p~+(S
—
Hid)
has
constant
eigenvalues.
Here
{e;}
is
a
local
orthonormal
basis
for
I
=
da
-
dz
with
dual
basis
{w;},
IJ
=
ag
hijw;
@
wz
is
the
second
fundamental
form,
H
=
=
Yhap
=
85
(I|L7||?
-
nH)
and
S
is
the
shape
operator
of
z.
It
is
clear
that
any
(Euclidean)
isoparametric
hypersurface
is
also
a
Mobius
isoparametric
hypersurface,
but
the
converse
is
not
true.
In
this
paper
we
classify
all
Mébius
isoparametric
hypersurfaces
in
S"+!
with
two
distinct
principal
curvatures
up
to
Mobius
transformations.
By
using
a
theorem
of
Thorbergsson
([T])
we
show
also
that
the
number
of
distinct
principal
curvatures
of
a
compact
Mo6bius
isoparametric
hypersurface
embedded
in
S"*++
can
only take
the
values
2, 3,
4,
6.
§1.
Introduction.
An
important
class
of
hypersurfaces
for
Mobius
differential
geometry
is
the
so-called
Mobius
isoparametric
hypersurfaces
in
S"*?,
It
is
a
hypersurface
x:
M
—
S”+!
such
that
the
Mdbius
invariant
1-form
(1.1)
D
=
—p-?
dei)
+
do
(his
—
Hd;;)e;
(log
p))w;
Jj
vanishes
and
all
eigenvalues
of
the
Mobius
shape
operator
(1.2)
S
:=
p~'(S
—
Hid)
are
constant.
Standard
examples
of
Mobius
isoparametric
hypersurfaces
are
the
images
of
(Euclidean)
isoparametric
hypersurfaces
in
S"+!
under
Mébius
transformations.
But
there
are
some
examples
of
(noncompact)
Mobius
isoparametric
hypersurfaces
which
cann’t
obtained
by
this
way.
It
is
easy
to
show
that
(see
Proposition
3.2)
any
Mobius
isoparametric
hypersurface
1991
Mathematics
Subject
Classification.
53A30, 53C21,
53C40.
Key
words
and
phrases.
Mobius
geometry,
isoparametric
hypersurface,
principal
curvature.
1)
Partially
supported
by
DFG466-CHV-II3/127/0.
2)
Partially
supported
by
the
Technische
Universitat
Berlin.
3)
Partially
supported
by
NSFC.
*)
Partially
supported
by
Qiushi
Award.
5)
Partially
supported
by
SRF
for
ROCS,
SEM;
the
SRF
of
Liaoning
and
the
Northeastern
University.
Typeset
by
A,4S-Tpx
variation
formula
for
minimal
surfaces,
which
is
a
special
class
of
Willmore
surfaces
(cf.
[9]).
But
it
is
too
complicated
to
give
the
second
variation
formula.
for
general
Willmore
surfaces
by
using
euclidean
invariants.
Since
the
Willmore
functional
defined
by
(0.1)
is
invariant
under
the
Moebius
group
(cf.
[3],
[8]),
one
can
use
the
framework
of
Moebius
geometry
and
Moebius
invariants
to
calculate
the
second
variation
formula.
It
is
the
key
point
of
this
paper.
For
any
submanifold
M
in
S” 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
[8].
Submanifolds
in
S”
satisfying
these
equations
are
called
Willmore
submanifolds
or
Moebius
minimal
submanifolds.
In
the
paper
we
give
the
second
variation
formula
of
Willmore
functional
for
submanifolds
in
S”
by
using
Moebius
invariants.
Although
this
formula
looks
very
complicated,
in
case
of
surfaces
in
S?
(which
is
the
most
important
case)
the
formula
is
rather
simple
(cf.
§2,
(2.44)).
Using
the
Euler-Lagrange
equations
we
find
the
standard
examples
of
Willmore
hypersurfaces
{W?
:=
S*(./(n
—
k)/n)
x
S"-*(/k]/n),
1
<
k
<n—1}
in
S"*1,
which
is
(euclidean)
minimal
if
and
only
if
2k
=
n
—
1.
It
is
somehow
the
dual
hypersurface
to
the
standard
minimal
hypersurface
S*(\/k/n)
x
S"—*(,/(n
—k)/n)
in
S"+1,
We
show
that
W?’
are
stable
Willmore
hypersurfaces.
We
organize
this
paper
as
follows.
In
§1
we
give
Moebius
invariants
and
local
formulas
in
Moebius
geometry
for
submanifolds
in
S".
In
§2
we
calculate
the
second
variation
formula
for
Willmore
submanifolds
in
S".
As
an
application
we
prove
in
§3
that
{(W7}
are
stable
Willmore
hypersurfaces.
§1.
Moebius
invariants
and
local
formulas
for
submanifolds
in
S$”
Let
zo
:
M
—
S$”
be
an
m-dimensional
compact
submanifold
with
boundary
0M,
{€1,++;€m}
be
a
local
orthonormal
basis
of
TM
with
respective
to
the
induced
met-
ric
dx
-
dxo
and {6},...,0m}
be
its
dual
basis.
Let
{€m+1,+--;€n}
be
the
local
normal
orthonormal
vector
field.
We
make
using
of
the
following
convention
on
the
ranges
of
indices:
1<i4,9,k,---<m;
m+1<a,B,7,...<n
and
we
shall
agree
that
repeated
indices
are
summed
over
the
respective
ranges.
Then
the
structure
equation
of
x9
can
be
written
as
dxo
=
6;e;
de;
=O6,;e;
+
hz,O;€a
—
9x9
dég
=
—
he,
05
e;
+
Gapep
The
quantities
I
=
dzo
-
dzy,
II
=
he
9i
®@
6;e,
and
H
=
+
h&eg
are
the
first,
the
second
fundamental
form
and
the
mean
curvature
vector
of
Xo
in
S”,
respectively.
We
define
function
p:
M
+
R
by
m
=
,/——_||II
-
1.1
p=
("oI
-
wn
(1.1)
The
metric
g
=
p*dzq
-
dzo
is
called
a
Moebius
metric
which
is
invariant
under
Moebius
transformations
in
S”
(cf.[8])
and
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)
:=
[
p™dM
=
Vol,(M),
(1.2)
where
dM
is
the
volume
element
for
the
metric
dro
-dxg.
Our
purpose
is
to
calculate
the
second
variation
in
the
framework
of
Moebius
geometry.
We
need
the
following
notation
and
local
formulas.
For
more
detail
we
refer
to
[8].
Let
Ret
be
the
Lorentz
space
with
the
inner
product
<,>
given
by
<X,Y
>=
-a®y?
+
alytt...
gp
grtlyntl
where X
=
(2°,c1,---,2"11),
y
=
(y°,y1,---,y"*)
©
R"+?,
The
half
cone
in
Rr?
is
defined
as
CUt*
=
{X
€
RPM)
<
X,X
>=0,2°
>
0}.
For
the
immersion
x9
:
M
—
S”
we
define
Y
=
p*(1,29):
M
>
Cet
(1.3)
If
Zo
:
M
—
S™
is
Moebius
equivalent
to
x9,
then
we
have
Y
=
YT
for
some
Lorentz
matrix
7’.
Thus
g
=<
dY,dY
>=
p?dzg
-
dzo
(1.4)
is
a
Moebius
invariant.
In
the
following
we
assume
that
zo
is
an
immersion
without
umbilical
point,
which
implies
that
g
is
positive
definite
on
M.
Let
FE;
=
p~'e;,
then
{£;}
is
an
orthonormal
basis
with
respect
to
metric
g,
with
dual
basis
{w;
=
p6;}.
Set
1
Y,:=
E,(Y),
N
:=
-—AY
-
to
<
AY,
AY
>
Y,
m
2m?
Eq
:=
(H%,
eq
+
H°
x9),
(1.5)
where
A
is
the
Laplacian
operator
for
metric
g
and
H
=
He,
is
the
mean
curvature
vector.
Lemma
1.1((8])
{Y,N,Y;,
Ea}
satisfy
conditions
<Y,Y
>=<
N,N
>=0,<
Y,N
>=1,
<
Yi,
Yj
>=
O53;
<Y,Yi
>=<
N,Y;
>=<
Y,
Eg
>=<
N,
Ey
>=
0;
<
fa,
Yi
>=
0,
<
Eg,
Eg
>=
bag.
Lemma
1.1
shows
that
{Y,N,Y;,£}
forms
a
Moebius
moving
frame
in
Rit?
along
M.
The
structure
equations
can
be
written
as
.
dY
=
wY,
aN
=
WY;
+
Paka
(1
6)
d¥, =
—YiY
—
wiN
+
wig¥j
+
wie
|
dEa
=
—PaY
—
Wig
Y;
+
Wop
Eg
By
differenting
these
equations
and
using
Cartan
lemma,
we
obtain
Wi
=>
Ajj,
Aij
=
Aji;
Wig
=
By
Wj, Bi;
=
Bos
da
=>
Cr
w;.
We
have
the
following
equations:
3
m—1
a a
>
BE
=0,
>
(Bg)
=
——,
3)
B34
=
—(m-
CF.
(1.7)
i
aij
j
The
relations
between
these
Moebius
invariants
and
euclidean
invariants
are
given
by
Aij
=
—p~?(Hess;;(logp)
—
e;(logp)e;
(loge)
—
H*h¥)
1
_
—5P
*(|Vlogp|?
—
1+
S°(H%)*)
655;
(1.8)
Be
=
p-*(h&
—
H%6,;)
(1.9)
a
1
a
==
a a
|
§2.
The
second
variation
formula
of
the
Moebius
volume
functional
In
this
section
we
caculate
the
second
variation
of
the
Willmore
functional
defined
by
(1.2)
or
(0.1).
Since
the
volume
variation
depends
only
on
the
normal
component
of
the
variation
vector
field
(cf.
[8]),
we
will
consider
the
normal
variation.
Let
¢:
Mx
R
-
S”
be
asmooth
variation
of
zp
such
that
«(-,t)
=
zo
and
dz.(TM)
=
dxo(T'M)
on
OM
for
each
(small)
¢.
These
two
boundary
conditions
disppear
if
OM
=
@.
For
each
¢
we
denote
by{e;}
a
local
orthonormal
basis
for
TM
with
respect
to
dz,
-
dz;
with
dual
basis
{6;}
and
by
{e.}
a
local
orthonormal
basis
for
the
normal
bundle
of
2.
Let
Y
=
p(1,z)
:
M
x
R
Ct"
be
the
canonical
lift
of
x;
and
g,
=<
dY,dY
>
be
the
Moebius
metric
of
2;.
Let
{E;
:=
p~te;}
be
a
local
orthonormal
basis
for
gt
with
dual
basis
{w;
=
p0;}.
Then
the
volume
for
g:
can
be
write
as
W(t)
:=
Voly,(M)
=
[
wt
Ao
Nam,
(2.1)
From
Lemma
1.1
in
section
1,
we
can
choose
a
moving
frame
{Y,N,Vi,---,
Ym,
Emsi,°°-
,
En}
in
Rv
along
M
x
R,
which
satisfy
the
conditions
in
Lamma
1.1
for
each
t.
Let
d
denote
the
differential
operator
on
M
x
R,
then
we
can
find
1-forms
{V,
Va;
Wi,
Ou,
Q;,
5;,
Qin,
Qap}
on
M
x
R
with
Q|;
=
—O,;
and
Qa
=
—OQgq
such
that
dY
=VY+Qi:Y;+
VaEo,
(2.2
)
dN
=-VN+WiY;+
©.Eo,
(2.3)
dY,
=
—UiY
—O,N
+
04;Y;
+
QiaEa,
(2.4)
dEg,
=
—Bq
—
VaN
—
Qia¥i
+
Qap
Ep.
(2.5)
‘Taking
differential
of
these
equations
we
get
dV
=VU;AQ+
2A
Vo,
(2.6)
dQ;
=
04;
A
Q;+V
AQ:
-
Va
A
Qia,
(2.7)
AV
=
Qap
NVg
+2:
A
Qig
+V
A
Va,
(2.8)
4
dv;
=
5;
A
VU;
—
By
AQiog
+
UA
V,
(2.9)
d®,
=
Qog
NOg
+
Ui
A
Dig
+
On
A
V,
(2.10)
dQ;
=
Dig
A
Qn;
+
Qia
A
Qa;
—WV;
AQ;
-QAY;,
(2.11)
dOs«,
=
03;
A
Qa
i
Qis
A
Qba
—WUiAVg
-Q;A
Bq,
(2.12)
AQoagB
=
Qay
A
048
+
Qa;
Nip
—Pa
A
Ve
—-ViA
Op.
(2.13)
Since
Y
=
p(1,2),
if
we
write
the
normal
variation
vector
field
of
x
in
T'S”
by
Ox
_
Ot
=?p
‘Vala;
(2.14)
then
by
(1.5)
we
can
find
a
function
v:
M
x
R
-
R
such
that
OY
OL
=
oY
+
Vola:
(2.15)
From
(2.2),
(2.15)
and
the
fact
that
d
=
w,;E;
+
dtZ
on
C®(M
x
R)
we
get
V
=vdt,
Va
=
vadt,
0;
=
aj.
(2.16)
Since
T*(M
x
R)
=
T*M
@T*R
we
can
write
VU;
=;
+
ajdt,
By
=
do
+
bedt,
(2.17)
Qi
=
Wig
+
Piydt,
Qie
=
wig
+
Liadt,
Qag
=
Wag
+
Qapadt,
(2.18)
where
{a;,
ba,
Pij,Qog}
are
local
functions
with
Pi;
=
—Pji
and
Qag
=
—Qga.
Let
{
Bf,
Aij,
C2}
be
Moebius
invariants
for
«x;
defined
in
§1.
If
we
denote
by
dy,
the
exterior
differential
operator
on
T*M,
then
we
have
d
=
dy
+
dt
A
=
on
T*(M
x
R).
It
follows
from
(2.6),
(2.16)
and
(2.17)
(comparing
the
terms
in
T*(M)
A
dé
)
that
A,
=
—Vi+
UQCF,
where
v,;
:=
Ej(v).
Similarly
we
get
from
(2.8),
(2.16),
(2.17),
(2.18)
that
Lia
=
Va,
and
get
from
(2.7),
(2.16),
(2.18)
that
Ow;
Ot
By
directly
calculation
in
a
similar
way,
we
obtain
from
(2.9)~(2.13)
that
=
(Py
+
di;
-
Va
Bi
)w;.
Ow;;
a
=
(Pij,e
+
Bivaj
—
Brvai
—
adn;
+
aj5ix)wp,
Oi
a
a
0
fe
=
(be,
+
QagCh
=
AijVoa,j
+
a;
By;
_
vC)wy,
(2.19)
(2.20)
(2.21)
(2.22)
(2.23)
(2.24)
OWie,
a
(va,ij
+
PirBej
-
BY
Qpa
+
Ajjva
+
babij)w;
(2.25)
OW
B
a
8
AE
=
(Qapi
+
06,5
BY
—
Vag
BF; +
UgCP
—
Val?
ui,
(2.26)
where
{Uq,i}
are
covariant
derivatives
of
{vg}.
Since
dg
=
Crw;
and
p;
=
Ajj;w;,
from
(2.21),
(2.23)
and
(2.24)
we
have
a0%
_
ay
=
bai
+
QapCh
+
a;
BE
—
Aijtaj
+
PO?
+
BECPug
—
WvC?,
(2.27)
OAij
_
ia
a
_
Ba
_
a
Aig
+
Pip-Agj
“en
Pri
Agi
+
Va,iCy
a
By
ba
+
Ain
BE
Va
aes
2vAj;.
(2.28)
Since
Wig
=
Bfw,;,
from
(2.21)
and
(2.25)
we
have
aBe
ms
=
Va,ij
—
UBS
+
Pir
Be;
—
Pri
BR,
-
BE
Qa
+
up
Bi
i
Bry
+
Aijta
+
badij-
(2.29)
Multiplying
Bf
to
(2.29)
and
using
(1.7)
we
get
m—l1
US
Bio,
gt
BG,
Be,
BS
iUg
+
Ay
Byv
(2.30)
Making
use
of
(2.21),
(2.30),
(1.7),
Green’s
formula
and
the
conditions
on
0M
,
we
get
(cf.[8])
—
|
{BS.j
+
Bi.
BeBe
+
Ai;
BS}
vadMy,
(2.31)
where
dM;
is
volume
element
of
g;.
Formula
(2.31)
shows
that
29
:
M
—
S™
is
Moebius
minimal
(its
Moebius
volume
is
stationary)
if
and
only
if
BS
i;
+
BY.
BE,
BY,
+
Aig
BS
=
0.
(2.32)
Since
2
=
0du
—dyo
g=
0
(or
equivalently
d?
=
(djyy
+
dt
A
2)
=
0)
and
we
get
OCT;
ag
OWE
Oc
e
Ow;
Ow
tg
8
E;)
+
(+),
+
cok
p
Po
(B,),
2.33
Ot
ck
>
|
3)
+
(
ot
i
k
Ot
(E
i)
+
+
C;
Ot
—,
(
5)
(
)
By
a
direct
calculation
and
using
(2.21),
(2.22),
(2.26)
and
(2.33)
we
get
oc.
ace
5
Vg
=
—(CP
Pri)
kVa
—
vcs
Va
+
vpBy,Ce
Va
+
(
ry
Ey
Vo
+CEBE,0p
Va
+
apCR
Va
—
mMayCE
Va
—
CP
Qap,iva
—
BECP
ug
pve
(2.34)
+
BECP
ve
be
—
CPCS
ugva
4
S-
(of)"
.
ve
a6
a
By
a
straightforward
calculation
and
using
(2.27),
(2.19)
we
get
from
(2.34)
that
a
OCR,
oc?
a
ve
=
—(CP
Prive)
i
+
(
:
va)
+
(VBE
Va,i),k
Ot
+(m
—
1)(uCP
va)
b
-
(C?
Qapva)
i
—
(bava,i),i
—
MUCP
Ve
+CPQapva
+
Ch,
Be
vars
+
bave,ii
—
VBE
Ve,ik
+
Aaah
tng
2
~2C2
Be
ugvai
+
20CP
ve,
—_m
S>
(=
Civ.
k
a:
+CP
BE
vanUa
+
S-(cP)?
S-
ve.
1,8
a
(2.35)
From
the
conditions
that
x(.,t)
=
x
and
dz;(TM)
=
dzo(TM)
on
OM
for
each
t,
we
see
that
valayz
=
0
and
0
=
—(dz,)
=
d(—)
=
p-'dugeg
(2.36)
on
boundary
0M.
It
follows
from
(2.35)
and
Green’s
fomula
that
Pps
r=
LG
{az
clt=
o(
Ce
ees
2.37
WBE
Vex
ak
+"
a
Z
—
2C7¢B
P
uBUa,
t+
2uC3*
Va,i
( )
—m
S-
(=
Cie.
+
Ci
P
BP
va
kUa
+
S-(c?)?
S-
v2
}dM.
k
a
2,8
a
Using
(2.28), (2.29),
(2.19)
and
the
facts
that
Py
=
—Py
and
3);
BS;
=
—(m
—
1)C?,
we
get
0
Hy
(Bie
BE
Bh
+
AijzBS)
va
=
(0Bij2a5):
~
(m
~
1)
0G),
“Bi
—2(m
—
Cis
Va
+
2BF
7
PiRYB,
ikVa
+
pens.
+
Ai;
)
Va,
ijVa
—
VB;
Va,ij
+(m
—
1)C?
iQapla
+
2(m
—
cn,
+
CP
BV,
jVa
—
CY
BE
va,j0y
+
(3Bf.
BS,
BY
BE
+
4Be
eBay
Agi
+
(m
—
1)CFC?)
vary
(2.38)
+L
LBiiba(X
Biva)
+
(D1
bava)
(Se
A
+a)
Bij
‘La
da
+
BE
A,B
2
22)
Using
the
boundary
conditions
and
Green’s
theorem
we
have
To:=
[Ls
{Slix0(Bi,
BEB)
+
AijBS)}vadM
=
=[
{-2(m
—
1)008ivq
+
2BE
B%.ug
anda
+
(BEBE,
+
Aig
Ua,ijPa
—-UBiWva,ij
+(m—
1)CPQapra
+
2(m
—
1)uC
iva,
+
C7
By
v4,5Va
;
(2.39)
—C}
BSva,j0y
+
(3BEBS,BUBy
+
4BG.By,
Aj
+
(m
—
1)C2C7)
vav,
©
m—1
+
37D
(Bijbs)
(Do
Bia)
+
(Daa)
——
+
tr(A))
ij
B
o B
+
QAy
2%
+
BE
Ang
Bi(S_
v2)
}dM,
a
where
1
1
ba
=
on
=
(Ave
+
Bo.
Be
iUB
+
mt
In
2)U
x)
(2.40)
1
=
—
A
tr(A)
=
=
fo
zh
(2.41)
where
«
is
the
normalized
scalar
curvature
of
the
metric
g
=
p*dxq
-
dap.
The
function
v
is
given
by
(2.15)
and
(2.30).
Thus
for
a
Moebius
minimal
submanifold
zo
we
get
from
(2.31)
and
(1.7)
that
w"(0)
=-—m
af
*
ris
0’odM
-
wf
(lee
0(Bi, BE,
Be,
+
AyB9))
vadM.
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”
be
a
compact
Moebius
minimal
submanifold
without
boundary.
Then
the
second
variation
formula
of
the
Moebius
volume
functional
is
given
win
2)
a
i
vB;
5
Vojig
4,j-~
jt
Ww"(0)
=
|
{6
—
2)uCZ;0q
—
mCP.
Be
vaug
+
M
2
2(2m
—1
=m?
Aijva,iVa,j
+
re
CP
BB
Up
ive
+m2(m
+1)
>
(=
cv.
—m?
CP
Bi
va,jVa
—m?
a
(of)"
2%
+
27"
BE
Biv,
ikVa
m*
(pb
pb
m*
ob
pe
+
(Bi
Bry
+
Aij)va,ij0a
—
movi
BiiVa,j¥B
2
—
1
(337,
Br
Bh
+
4Bi,
¢
Bey
Aji)
VaUB
ToT
ba
d%
+
m?
mim
—
2
+——
Bi,
Anj
By,
(=
2)
+
m
>
(Avg)?
+
mn
—)
pope
Avovg
a
a
2
+
(Me
in)
a
1)
Va
Ave
—
—7
i
X
(x
BE
Bev
|
18
-(
—_ir(
4)
(Dag
2)
~
tr(
(A)
(
4))
Soak
an
(2.42)
In
case
of
the
surface
in
S?,
the
formula
is
reduced
to
a
simple
form.
We
omit
all
a
and
8
because
the
codimension
now
is
one.
Corollary
2.2
For
a
surface
in
S®
the
second
variation
formula
is
given
by
w"(0)
=
[
2an
+
2fAF
+
12CiBij
ff
+
4Aaj
fig
f
(2.43)
AAG
RSs
+
(AD
Ay
H+
5
+
K
—2K*)}dM,
where
K
is
Gaussian
curvature
of
the
Moebius
metric
g
=
pdzo
-
daxo.
Remark
2.3:
The
second
variation
formula
for
Willmore
surfaces
in
S?
meight
be
im-
portant
towards
the
Willmore
conjecture.
As
we
know
sofar
the
only
stable
example
of
Willmore
torus
is
the
Clifford
torus.
Combining
the
existence
result
of
L.
Simon
in
[7]
we
know
that
the
Willmore
conjecture
is
true
if
one
can
show
that
the
only
stable
Willmore
torus
embedded
in
S°
is
the
Clifford
torus.
§3.
Moebius
tori
in
S™*!
and
their
stability
In
this
section
we
present
a
class
of
important
examples
of
Moebius
minimal
hypersurfaces
called
Moebius
tori.
As
an
application
of
Theorem
2.1
we
show
that
they
are
stable
Willmore
hypersurfaces.
Let
R™*?
be
(m+2)-dimensional
Euclidean
space
with
inner
product
<,
>.
We
write
Ret?
=
RET
x
R™*L
<<
m-—1.
For
any
vector
€
€
R™+?
there
is
unique
decomposition
€
=
&
+
fo
with
€;
€
R**!
and
&
€
R™-*+1_
For
another
vector
7
=
+12
the
inner
product
of
them
can
be
written
as
<
€,n
>=<
&1,m1
>
+
<
0,1
>.
Let
€;
:
S*
=
R¥+1
and
2
:
S™—-*
=
R™-*+1
be
standard
embedding
of
unit
spheres.
Let
x:
S*(a
i)
x
S™-*
(a2)
>
S™t+!
C
R™+2
be
the
embedded
hypersurface
x
=
a
£1
+
ago
with
a?
+
a3
=
1.
It
is
easy
to
check
that
(i)
the
unit
normal
vector
of
M
:=
S*(a,)
x
S™-*
(ag)
in
S™+!
is
given
by
CEm+1
=
—
4281
+
0182;
(ii)
the
second
fundamental
form
of
M
is
given
by
II
=
—
<
daz,
dem41
>=
ayae(<
df,
dé,
> —
<
d&g,
df
>);
(iii)
the
induced
metric
of
M
is
given
by
I
=
alldéy|?
+
a3
|déo).
If
we
take
{e;}
and
{w;}such
that
d(ay€1)
=
Yee
a
(a2&2)
=
3
Wij,
j=k+1
then
we
have
9
a2
Ww?
—
“a4
2
J
ex
Sw,
If
=
ye
S-
ad
=
hijwiw;,
(3.1)
i=1
jak+1
“2
where
oj,
If
l<ig<k,
hi
=
(3.2)
—
bis,
if
k+1<ig<n
Thesrens
3.
1.
Let
W
=
S*(a,)
x
S™-*(ag)
be
the
hypersurface
imbedded
into
gmt
where
a?
+
az
=1.
Then
W
is
Moebius
minimal
ifand
if
m—k
k
a,
=
\|
——,
a2
=
\/—.
m m
Proof.
From
(3.1)
we
see
that
$=
oN
=k
(2)
+
tm—¥)
(1),
(3.3)
Him
=
Sha
=
>
(62
—(m—
2),
(3.4)
p°
=
——(S
—
mH”).
(3.5)
Substituting
(3.2)
and
(3.5)
into
(1.8),
(1.9)
we
know
that
Moebius
minimal
condition
Bis
ij
+
Bip
By
Byi
+
AijByi
=
0
is
equivalent
to
that
(m
—
k)
(22)"
+
(2m
—
38
(22)
+
(msn)
(2)
-n=0.
(3.6)
Q\
Qa)
From
the
equations
(3.6)
and
a?
+
a3
=
1
we
get
a,
=
\/
mae
and
az
=
Ve.
Q.E.D.
We
call
west
ay
x
gmk
(VE)
.actem-
™m
mM
Remark
3.1
It
is
remarkable
that
Moebius
torus
Wi”
is
(euclidean)
minimal
if
and
only
if
2k
=m.
We
note
that
W;”
can
be
obtained
by
exchanging
radii
a,
and
ag
in
the
Clifford
torus
S*
(Vz)
x
gmk
(5),
Remark
3.2
It
is
known
that
any
minimal
surface
in
S™*
is
also
Moebius
minimal.
Theorem
3.1
show
that
for
submanifolds
of
dimension
great
than
2
a
minimal
submanifold
may
not
be
Moebius
minimal.
Moebius
tori.
From
now
on
we
study
the
stability
of
Moebius
minimal
torus
W,”
defined
in
Theorem
3.1.
For
W,"
we
get
from
(3.2),
(3.3),
(3.4)
and
(3.5)
that
k
.
.
—~
7
0:5;
1<i,j<k,
hij
=i
Vink
i
stys
(3.7)
ke
—k)
~
2
g
=
ht
(m
=k)"
He=—-—
ak
p=
(3.8)
k(m
—
k)
Vk(m
—
k)
m—1
From
(1.8)
and
(1.9)
we
have
_
1
_
Ai;
=p
*
hij
Tt
5P
2(1
=
H’)6;;
_
kn?
. .
_
p
STE)
aq)
af
1
Ss
tJ
<
k,
(3.9)
po
mab
Ou,
tf
R+1<i,j
<m,
(m
—
1)(m?
—
3km
+
3k?)
tr(A)
=
3.10
(A)
2km(m
—
k)
ei)
—1,
/m=k
:
.
3
_
p
a
655,
ifl<ij<k
Biz
=
p~"
(hig
—
H6ij)
=
‘re
y
-
(3.11)
—p
\/
muh
oi,
ifk+1<ij7<m,
11
and
>>
B?,
=
™=!.
From
the
last
equation
in
(1.7)
we
obtain
C;
=
0.
We
are
going
to
uz
m
calculate
2
W"(0)
=
—m7T,
+
—
pias
where
ry
=
Su
rut
lo
fdM,
Ty
a
In
{BlecolBuxBrrBa-+
AspBy)}
fm
and
f €
C°(M)
is
the
normal
component
of
the
variation
vector
field.
From
(2.40)
we
have
b=
(ars
m
ey).
(3.12)
2km(m
—
k)
Substituting
(3.9),
(3.11)
and
(3.12)
into
(2.37),
we
get
m
—
1)(m?
2_km
—mP,
=
|
(means
ae
K
Las)
dM
-f
(@
—
1)(3km
—
k?
=
FP
4
(m
—
1)(m?
—
km
—
ives)
dM
2k(m
—
k)
2k(m
—
k)
+m?
|
vBijfijdM.
M
(3.13)
Substituting
(3.9),
(3.11)
and
(3.12)
into
(2.39),
by
a
straightforward
calculation
we
get
2
ma
ke+k
245k?
—
9k
m*
—k*
+km
om*
+
5k*
—
9km
=
ee
fAfdM
+
em)
[
fAifdM
5k
—k
SS
[
fdofaM
-
—"—
|
vBifijdM
(3.14)
+
(2km(m
—
k)(m?
+
7km
—
7K)
m(m?
—
k2
+
km)(R?
+
+
m2?
—
km)
_
m2
—
L2\2
_
2
_
pe
eae
il
ee
2
+
k(3km
—
m*
—
k*)*
+
(m
—k)(m*
—
km
—
k*)
*)
Bank
(m
kp
sft
f°dM.
In
(3.13)
and
(3.14)
we
denote
by
A
the
Laplacian
operator
on
Wj”
with
respect
to
the
Moebius
metric
g
=
p*dz-
dz.
We
write
g
=
gi
®
go
according
to
the
decomposition
Mf"
:=
My
x
My
:=
S*
(=)
x
gmak
(VE).
We
denote
by
{A;,
A}
the
Laplacian
operators
of
{91,92}
and
by
{V1,
V2}
the
gradient
operators
on
M,
and
Mg
respectively.
From
(2.30),
(3.9)
and
(3.11),
we
have
1
m—k
veuts=
x
(y
;
Aif
-
2
K
;oe]
(3.15)
12
From(3.13),
(3.14)
and
(3.15)
we
get
m(k?
+m?
—
km
—
2m)
2k(m
—
k)
As
2
m(m
—
2)
m—-k,
,
k
a1
(,
k
as
cea
m(3km
—
k?
—
m?)
+
6(m
—
k)?
|
2k(m
—
k)
Ais
w"(0)
=
I
(mas?
+
(3.16)
m(m?
—
km
—
k?)
+
6k?
r
2k(m
—
k)
fdgg
+
eo}
)
dM.
™m
We
denote
the
Laplacian
operators
with
respect
to
Euclidean
metric
dz
-
dz
by
Ay,
Am,
and
Ay,
on
Wj”,
M;
and
Mp
respectively.
Since
p
=
constant
and
Moebius
metric
g
=
p’dz
-
dz,
we
have
Ay
=
p’A,
Am,
=
p°Ai,
Am
=
p*A2
and
Ay
=
Am,
+
Am,-
From
(3.8)
and
(3.16),
we
have
m—1
Wr)
=e
[Hn
+m?(k?
+
m?
—
km
—2m)fAmf
+2(m
—
2)
((m—k)Am,
f
—kAm,f)?
(3.17)
+m
(m(3km
—k?
—m?)
+
6(m-—
k)?)
fAm,f
+m
(m(m?
—
km
—
k?)
+
6k”)
fAmaf
+4k(m
—
k)m?
f?)
dM.
Let
;,A;
and
p4;
be
the
eigenvalue
of
Laplacian
operators
A
M,,A4m,
and
Ay
respec-
tively,
then
we
have
n
Mi
=
i(k
+
4-1),
N=
i(n—k+G-NE,
wig
=
+2,
where
2
and
7
are
nonnegative
integers.
Let
f;,
fj
be
eigenfunctions
corresponding
to
A;
and
4;
respectively,
then
gi;(p,q)
=
f;(p)
f;(q)((p,q)
€
My
x
Mg)
is
an
eigenfunc-
tion
corresponding
to
y;;.
For
any
f
€
C™(MiZ")
we
have
the
decomposition
of
f
in
eigenfuctions
f=
>>
aggiz
+0,
(3.18)
i+j
#0
where
c;;
and
co
are
constants.
Thus
we
have
Amf=-
>>
d.ci9i3,
Auf
=
—
S-
NCi5
93,
Amf
=—
D>
pwiyeizgiz.
(3-19)
i+j40
i+j
40
i+jF0
13
Substituting
(3.18)
and
(3.19)
into
formula
(3.17)
we
get
w"(0)
>
SRG
Jue
Di4540
(2k(m
—
k)(m
—
1)
ui,
2
—m?(k?
+
m?
—
km
—
2m)uig
+
2(m
—
2)
((m—
k)Ai
-
kX)
(3.20)
—m
(m(3km
—
k?
—
m?)
+
6(m
—
k)?)
d;
—m
(m(m?
—
km
—
k*)
+
6k*)
X).
+
4k(m
—
k)k?)
.92,dM
=
aEGnak
ym?
Di+j¢0
Sg
{2(m
—
k)(m
—
2m
+
k)d?
—2(2m3
+
km?
—
6km?
+
3k?m)A,
(3.21)
+2k(m?
—
m
—
k)(X)?
—
2m(m3
—
m?
—
km?
+
3k*)X
+4k(m
—
k)AiXi
+
4k(m
—
k)m?}c?.97.dM.
If
we
set
A(i,j)
=
2(m—k)(m?
—
2m
+
k)A?
—
2(2m3
+
km3
—
6em?
+
3k?m)A;j
+2k(m?
—
m
—
k)
(Xj)?
—
2m(m3
—
m?
—
km?
+
3k*)
Xi
+4k(m
—
k)AiXj,
+
4k(m
—
k)m?,
then
one
can
easily
verify
that
A(i,§)
=
(qhtye(m?
—
2m
+
k)i(k
+
4-1)
+
(225(m—-k+5—-D-
2km)
)
(25208
min
—
ej
=1)
+
Bie
1)
26m
Am
_
m?(m?—2m+k)(m?2—m-—k)
.
at
ij(k
+i—1)(m—k4+j-1).
(3.22)
From
(3.22)
it
is
not
diffculty
to
see
that
A(i,j)
>
0
and
A(i,j)
=
0 if
and
only
if
(2,7)
=
(1,0),
(0,1)
or
(1,1).
Thus
we
have
proved
the
main
result
in
this
section:
Theorem
3.2
All
Moebius
tori
S*
(==)
x
gmk
(Vz)
>
S™11<k<m-1,
are
stable.
In
the
end
of
this
paper
we
would
like
to
pose
the
following
generalized
Willmore
conjecture
in
$™+1,
Generalized
Willmore
Conjecture:
Let
M
be
a
m-dimensional
manifold
which
is
diffeomorphic
to
S*
x
S™-*
and
x:
M
—
S™+!
be
an
imbedding,
where
1
<k
<
m—1.
Set
7,
(x)
=
(—",)
>
I.
(5
-
mH”)
?
aM,
(3.23)
m-—-1
14
where
dM
is
the
volum
element
for
induced
metric
dz
-
dz,
S'
the
square
of
the
length
of
the
second
fundamental
form
and
H
the
mean
curvature
of
x.
Then
An"
(m
—k)
tk
3"
m2—?(m
—
1)0
(#41)
r(
eee)
and
equality
holds
if
and
only
if
z(M)
is
Moebius
equivalent
to
W,”.
Here
Tis
gamma
function
and
the
term
on
the
right
of
(3.24)
is
the
Moebius
volume
of
W,.
Tr(a)
>
(3.24)
References
[1]
Blaschke,
W.:Vorlesungen
uber
Differentialgeometrie.
Vol.3,
Springer
Berlin,1929.
[2]
Bryant,R.,
A
duality
theorem
for
Willmore
surfaces,
J.
Differential
Geom.
20(1984),
23-53.
[3]
Chen,B.
Y.:
Some
conformal
invariants
of
submanifolds
and
their
applications.
Bol.
Un.
Math.,
Ital.
(4)10,
380-385.
[4]
Li,P.
and
Yau,S.-T.:
A
new
conformal
invariant
and
its
application
to
Willmore
conjecture
and
the
first
eigenvalue
of
compact
surface.
Invent.Math.
69,
269-291(1982).
[5]
Palmer,B.:
The
conformal
Gauss
map
and
the
stability
of
Willmore
surfaces.
Ann.
Global
Anal.
Geom.
Vol.
9,
No.
3
(1991),
305-317.
[6]
Pinkall,
U.:
Inequalities
of
Willmore
type
for
submanifolds.
Math.
Z.
193,
241-246.
[7]
Simon,
L:
Existence
of
surfaces
minimizing
the
Willmore
energy.
Comm.
Analysis
Geometry
vol.
1,
n.2
(1993),
281-326.
[8]
Wang,
C.P.:
Moebius
geometry
of
submanifolds
in
S$”.
manuscripta
math.
96,
517-534
(1998).
[9]
Weiner,
J.
L.:
On
a
problem
of
Chen,
Willmore,et
al.
Indiana
University
Journal,
Vol.27,
No.1,
19-35(1978).
[10]
Willmore,
T.
J.:
Total
curvature
in
Riemannian
geometry.
Ellis
Horwood
Lim-
itd,1982.
Guo
Zhen
Li
Haizhong
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:[email protected]
e-mail:
Wang
Changping
Department
of
Mathematics
Peking
University
Beijing,100871,
P.
R.of
China
e-mail:[email protected]
15
