The Complex-Analyticity of Harmonic Maps and the - bogomolov-lab

Annals of Mathematics
The Complex-Analyticity of Harmonic Maps and the Strong Rigidity of Compact Kahler
Manifolds
Author(s): Yum-Tong Siu
Reviewed work(s):
Source: The Annals of Mathematics, Second Series, Vol. 112, No. 1 (Jul., 1980), pp. 73-111
Published by: Annals of Mathematics
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The complex-analyticity
of harmonic
maps and the strongrigidity
of
compactKahlermanifolds'
By YUM-TONGSIU
In 1960 Calabi and Vesentini [3] proved that compact quotients of
boundedsymmetricdomains are rigid in the sense that they do not admit
any nontrivial infinitesimalholomorphicdeformation. In 1970 Mostow
discovered
the phenomenonofstrongrigidity[7]. He provedthat the fundamentalgroup of a compact locally symmetricRiemannian manifold of
nonpositivecurvature determinesthe manifoldup to an isometryand a
choiceof normalizingconstantsif the manifoldadmitsno closed one or two
dimensionalgeodesic submanifoldswhich are locally direct factors. In
two compactquotientsof the ball of complexdimension?2 with
particular,
isomorphicfundamental groups are either biholomorphicor conjugate
Yau conjecturedthat this phenomenonof strong rigidity
biholomorphic.
shouldhold also for compact Kihler manifoldsof complexdimension>2
withnegativesectionalcurvature. That is, two compactKihler manifolds
ofcomplexdimension?2 with negative sectionalcurvatureare biholomorphicor conjugatebiholomorphic
if they are of the same homotopytype. In
thispaperwe provethat Yau's conjectureis true when the curvaturetensor
ofoneofthe two compactKihler manifoldsis stronglynegativein the sense
in Section 2 with no curvature assumptionon the othermanifold.
defined
Thestrongnegativityof the curvaturetensoris a conditionstrongerthan
thenegativityof the sectionalcurvature. This strongnegative curvature
is satisfiedby quotientsof the ball and also by the compactKghler
condition
surfacerecentlyconstructedby Mostow and Siu [8] which has negative
curvatureand whose universal covering is not biholomorphicto
sectional
theball. Until now there is no knownexample of a compactKaihlermanifoldofnegativesectionalcurvaturewhichdoes not admit a Kghler metric
withstronglynegative curvaturetensor.
Ourresultis provedby showingthat a harmonicmap of compactKiihler
is eitherholomorphicor conjugate holomorphicif the rank over
manifolds
$ 01.95/1
0003-486X/80/0112-1/0073/039
C 1980by PrincetonUniversity(MathematicsDepartment)
Forcopyinginformation,
see inside back cover.
1 Researchpartiallysupportedby an NSF grant.
74
YUM-TONG SIU
R of the differentialof the map is > 4 at some pointand if the curvature
tensorofthe image manifoldis stronglynegative. This resulton thecomplexanalyticityof harmonicmaps is obtained by a Bochnertype argument. The
usual techniqueof proving propertiesof a harmonicmap f is to obtain a
Bochnertype formulaby consideringthe Laplacian of the pointwisesquare
norm of df. For the complex manifoldcase the pointwisesquare normof
df is replaced by the pointwisesquare normof Afin this technique(see [91).
The metric tensor of the image manifoldand the inverse matrix of the
metrictensorof the domain manifoldappear in the pointwise square norm
of af. Hence in this Bochner type formula an expression involvingthe
differenceof the curvaturetensors of the domain manifoldand the image
manifoldappears. This prevents one fromdrawing any conclusionwhen
both manifoldshave negative curvature. In our proof we overcomethis
difficulty
by replacingthe pointwisesquare normof af by the contractionof
af A af with the metrictensorof the image manifoldand by replacing the
Laplacian by ha. This methodenables us to get rid of the curvaturetensor
of the domain manifoldin the Bochner type formula. Our result on the
complex-analyticityof harmonicmaps can be applied to the problem of
representinghomologyclasses by complex-analyticsubvarieties. Unfortunately it is still far frombeing able to prove the Hodge conjectureeven
forthe case of the compactquotientsof the ball.
Our methodof provingthe complex-analyticityof harmonicmaps and
strong rigidity can be applied to a much wider class of compact Kihler
manifoldsthan the class of those with stronglynegative curvature tensor.
As examples, we apply our methodto compactquotientsof the fourtypes
of classical bounded symmetricdomains and obtain the following result.
Any harmonicmap froma compactKihler manifoldto a compactquotient
of an irreducibleclassical bounded symmetricdomain of dimension>2 is
either holomorphicor conjugate holomorphicif the map is a submersion
at some point. In particular,any compactKihler manifoldwhich is of the
same homotopytypeas a compactquotientof an irreducibleclassical bounded
symmetricdomain of dimension ?2 is either biholomorphicor conjugate
to it. This rigidityresult is strongerthan the corresponding
biholomorphic
strongrigiditytheoremof Mostow [71, because here only one manifoldis
assumed to be locally symmetricwhereas in Mostow's theoremboth manifoldshave to be assumed locally symmetric.
These results were announcedin [101.
I would like to thank S.-T. Yau for introducingme to harmonicmaps
and his conjecture and for many conversationsin connectionwith his con-
75
COMPLEX-ANALYTICITY OF HARMONIC MAPS
jecture. E. Bedforddrew my attentionto the inequality that the second
elementary symmetricfunction of a finite number of real numbers is
dominatedby the square of their sum. He showed me how to use such
an inequalityand the complexBernsteinformula[2, p. 378] to obtain, in the
pseudoconvexcase, a new proofof Bochner'stheoremon extendingholomorphicfunctionsfromboundaries. This simpleinequalityis used in our proof
of the complex-analyticityof harmonicmaps and its use is inspired by
Bedford's proof of the special case of Bochner's theorem. I would like to
express my indebtednessto him.
Table of Contents
Section
1. Harmonicmaps.......................................
2. Curvatureconditionsand statementof results ........
3. A Bochnertype identity
..............................
of harmonicmaps................
4. Complex-analyticity
...............
5. Boundedsymmetricdomains...........
6. Adequate negativityof the curvatureof DI.. .........
.
7. Adequate negativity of the curvature of D'. ..........
75
76
79
81
90
93
98
8. Adequate negativityof the curvatureof Dii" ......... 102
9. Adequate negativityof the curvatureof D.V ......... . 107
110
.................
10. Strongrigidityand other applications
1. Harmonic
maps
Let f: N -R M be a map of Riemannian manifoldswhose metrics are
respectively
EpgapdxadxP
ds82 -
hjdid
ds~y=L
The energyE(f) of f is definedto be
2
2
N
(f *d
traced%2
N
)
that is,
E(f) = -
2
agaph'i afi afP
Li~j,"IP
ayi 6yj
in terms of local coordinatecharts. The Euler-Lagrange equation for the
energyfunctionalE is
c
f
fo l ,where
f f af h j = O
ayi ay t
is the Laplace-Beltramioperator of N and
ANf?a + AfMip
AX is the
AN
Mra
MJ3a,
is the
76
YUM-TONG SIU
Christoffel
symbolof M. The map f is said to be harmonicif f satisfiesthe
Euler-Lagrangeequation for the energyfunctional.
Eells-Sampson[4] proved that when M and N are both compact and M
has nonpositivesectional curvature,every continuousmap fromN to M is
homotopicto a harmonicmap. Hartman [6] proved that the harmonicmap
is uniquein each homotopyclass if M has strictlynegative curvature. Since
the Euler-Lagrange equation for the energy functionalis a second-order
quasilinear elliptic system of partial differentialequations,it followsthat
any harmonicmap between smoothRiemannianmanifoldsis smooth. The
reader is referredto [4] for a surveyof the theoryof harmonicmaps.
2. Curvature conditionsand statementof results
Suppose M is a complexmanifoldwith Kihler metric
ds' = 2Rea
pg, dzadzP.
The curvaturetensoris given by
g
=
gARa-ir- Art-
agr- awid
The sectionalcurvatureat the 2-planespannedby the two tangent vectors
p = 2Re E
da
a
aza
q = 2ReEa "az a
is given by
- II~~~
A j12E,,p
~
A qj
-j[P
where
II
A q1
r~d5Rasr-~d_7a)er
-
ay
g
)ggr4(bank
=
+ (Wyr -
3-
a
_ imr)
_
-
r2TWS))JT-
.
Nonpositivityof the sectionalcurvatureis equivalent to
Ea
vi se
Rapr-s(t-a7)7-7aVf)(V8)7r
-_vr
)7F ?
Roarar-
0
forall complexnumbers a, p The sectional curvature is negative if, in
additionin the above inequality,equality holds if and onlyif
Bag
-
t-ar =
0
for all a, fl,
- rasp is equivalent to the vanishing of
because the vanishing of all fatsP
Ip A q
as one can easily verify by diagonalizing the matrix (ga) and
makinguse of the identity
2
COMPLEX-ANALYTICITY OF HARMONIC MAPS
a-P
2
=
I
bans _ rags
12
_
(
-aya
_
-ae)(ePp
77
_ LiP)
We now introducesomenotionsof negativecurvaturestrongerthan the
negativityof sectionalcurvature.
Definition. The curvatureRpa is said to be stronglynegative(respectively stronglyseminegative)if
Lul jsRak7 (AaB7
-
C"D%)(A6Br - CaDr)
is positive (respectively nonnegative) for arbitrary complex numbers
Aa, Ba, C5, Da when AaBP - CDP #0 for at least one pair of indices (a, fi).
Definition. The curvature tensor Rri, is said to be very strongly
negative(respectivelyverystronglyseminegative)if
is positive (respectivelynonnegative) for arbitrary complex numbers a
when A # 0 for at least one pair of indices (a, /3).
Clearly very strongnegativityof the curvature tensor implies strong
negativity of the curvature tensor. Strong negativity of the curvature
tensorimpliesnegativityof the sectionalcurvature.
THEOREM1. Suppose M and N are compactKdhler manifolds and the
curvaturetensorof Mis stronglynegative. Supposef: N-AMis a harmonic
map and therank over R of the differentialdf of f is at least 4 at some
point of N. Thenf is eitherholomorphicor conjugate holomorphic.
In Theorem 1 the curvature conditionon M can be weakened to the
following. The curvaturetensorof M is stronglyseminegativeeverywhere
and is stronglynegative at f(P) for some P C N with rankRdf> 4 at P.
The followingstrongrigiditytheoremis a consequenceof Theorem1.
THEOREM2. Let M bea compactKdhler manifoldofcomplexdimension
at least two whosecurvature tensor is stronglynegative. Then a compact
Kdhler manifoldof thesame homotopytypeas M must be eitherbiholomorphic or conjugate biholomorphicto M.
Though most of the boundedsymmetricdomains do not have strongly
negative curvaturetensor,our methodcan also yield the strongrigidityof
compactquotientsof classical boundedsymmetricdomainsand the complexanalyticityof harmonicmaps into them. The precise statementsare contained in the followingtwo theorems.
THEOREM3. Suppose N is a compact Kdhler manifold and M is a
compactquotientof a boundedsymmetricdomain of typeImn(mn > 2), IIn,
78
YUM-TONG
SIU
(n > 2), or IV, (n > 3) (whoseprecise descriptionsare given in
(n.> 3), JJJn
? 5). Suppose f: N -> M is a harmonicmap whichis a submersionat some
point of N. Then f is eitherholomorphicor conjugate holomorphic.
THEOREM
4. Let Mbe a compactquotientof a boundedsymmetricdomain
of type 'mn (mn > 2), IIJ (n > 3), IIIJ (n > 2), or IV, (n > 3). Then any
compactKdhler manifold of the same homotopytype as M must be either
biholomorphicor conjugate biholomorphicto M.
The mainpart of this paper will devotedto the proofsof these theorems.
As examples of compactKihler manifoldswith very stronglynegative
curvaturetensor,we show that any compactquotientof the ball has very
stronglynegative curvaturetensor.
1. The curvature tensor of the invariant metric of the
PROPOSITION
ball is verystronglynegative.
Proof. The invariantmetric2 Re
of the ball B of C' is
gffdzadzp
given by
gaf= aa _(-log (1 - I z 12))
Since
IZ 12) =
-log(1_
+
+
112
3 6?
it followsthat at the originall the componentsof the curvaturetensor
RaAr-= aaa-ara-(-log
(1
-
I Z 12))
are zero except the following
Raaaa- = 2,
Hence at the origin
~Ap,rR
-a$aPr
Ra-pp= Rapa-= 1
=
=
E
~a(ErRaa-r~RaaraT)
La~e
=
4iardar
?E
=2~~
E
for a #zf.
a~p(~r^ciRar
+
~erer2?
Ia2
+?EL
I aa 12+ I
a daaI12+ Ea
which is >0 and is zero if and only if all dap- 0.
R
? LaI
eerr
E
)
ap 12
I
ap
12
aa 12
Q.E.D
Compact quotients of the ball are not the only examples in complex
dimension >2 of compact Kihler manifoldswith very stronglynegative
curvaturetensor. Recently Mostow-Siu [8] constructeda compact Kihler
OF HARMONIC MAPS
COMPLEX-ANALYTICITY
79
surface with very strongly negative curvature tensor whose universal
coveringis not biholomorphicto the ball of complexdimensiontwo.
3. A Bochner type identity
Suppose M is a Kahler manifoldwith metrictensor
gap dzydzf
g - 2ReL,,
and suppose N is a complexmanifoldandf: N-- M is a smooth(i.e., C-) map.
Let TMdenote the real tangent bundle of M when M is regarded as a real
manifold. The complexstructureof M gives a decompositionof TM0 C into
tangent vectorsof type (1, 0) and type (0, 1),
TM?&C=
TM&TMVI.
The differentialdf: TA-XTM of f given rise to a map
df?(C:
TN
TM&C.
C->
Composingthis with the projectionmap
TM(S C
141,O:
>TM
we obtain
[II
oo(df (gC):
TN?
C-
TTM.
This is equivalent to a bundle map
TN?(C
f*TlM
of C-vectorbundlesover N. Composingthis bundle map with the inclusion
map
TN
> TN?C,
l
we obtain a bundle map fromTO
f *T1Owhichwe denote by af. Hence
af is a smoothsectionof the C-vectorbundle
f * TM
O) = (TNPO)*
?& f * T1O
Homc (TNl',
over N. In otherwords 5f is an f* T9V-valued(0, 1)-formon N.
Let (wi) be a local holomorphiccoordinatechart of N. Then in terms
of the local coordinatecharts (za) and (wi) of M and N, af is simply represented by (a-fa),where a- = 3/3wi.In Sections 1-5 we will use the notation
and
a-= a/law
=
In
also the notations ai a/awla, = alaza, and Aa =iaza.
these notationswe may substitute another lower case italic (respectively
Greek) for i (respectivelya).
Likewise we defineaf, af, and 5f. af is an f* T"0-valued(1, 0)-formon
N representedby (aifa). if is an f *T? l-valued(1, 0)-formon N represented
by (aifa). af-is an f *T, '-valued (0, 1)-formon N representedby (3tfa). It
YUM-TONG SIU
80
is clear that af is the complex conjugate of af and aV is the complex conjugate of af.
Let V be the Riemannianconnectionof M definedby its Kahler metric.
It inducesa connectionf*V on the vector bundle f* T1'0. This connection
togetherwiththe a operatorof N enable us to define,for any f* T"0?-valued
(0, 1)-formwvon N, the a exteriorderivative of ,),which we denoteby D).
It is an f* T10-valued(1, 1)-formon N. In terms of local coordinates,if
@ = (a)o), then
D) = (E;,,,da.dw' A dwi)
with
ji
G2il As)+ Eparm
)
where M'F is the Christoffelsymbol of M (evaluated at f(Q) when the
equation is consideredat the point Q of N).
Likewise we define,for any f* T?l-valued (1, 0)-formco on N, the a
exteriorderivative D(o' of A! which is an f* T0l"-valued(1, 1)-formon N.
Let As be the bundle over M of (complex-valued)tensors of contravariant orderr and covariantorder s. Let a be a section of A on M and
let r be an f*As-valued p-formon N. We denote by <a, r> the p-formon
N obtainedby the contractionwhichcontractselementsof A withelements
of As to formscalars.
Let R = (Raotr)denote the curvature tensor of M. We are now ready
to state our Bochnertype identity.
PROPOSITION 2.
fA af> = KR,afA
A3f A afA (f> - <g, Df A Daf>.
a35<g,p
In local coordinates
9a
A3L
ga-fa A afffi=EpraRa.rjfa
-a
where
D~a~f= (afa +
Daffi = (afP+
P
gasDafa
A
affiA afTA
Afa
A D aff,
MI"fiaf A &fT,
P,
LarM m afa AA f
Proof. Fix a point Q of N where we want to verifythe equation. Let
P = f(Q). We choose a holomorphiclocal coordinatesystem (za) of M at P
symbolsMFi vanish.
such that dga = 0 at P. Then at P all the Christoffel
It followsthat at P,
= ar(aga,
Rafir-
and at Q
Daf
81
OF HARMONIC MAPS
COMPLEX-ANALYTICITY
a
af
D af a=awfa
Using arg,
=
we have at Q
a
0 at P and recalling that in the equation gaA stands for ga
a pgajaf/Af
-
LaB,
+
Ma
-,aiarg
-
Since a^ag,
2 Re
A-af/
afaA
af
A afr A af A af
af A afT A afa A
+4-- ar8M-g"A
+
rA
/
Af
,5aaargas af
A
Ealpl,5aaargap
of,
f
f
f
+ ,gai/aafa Aaaf.f
is symmetricin a, -r (due to the fact that the metric
gcArdzadzflis Kahler) and since af 'A 5fr A 5f" A af P is skew-symmetric
it followsthat the firsttermon the right-handside of the preceding
in a,
equation in zero. Likewise, it followsfromthe symmetryof adarga"in a, -'
and the skew-symmetryof af' A afr A afa A af P in a, -' that the second
term on the right-handside is zero. And it followsfromthe symmetryof
of af A af A 5fa A af Pthat the last
a-a-gaBin ,3,3 and the skew-symmetry
termon the right-handside is zero. Hence
a a~p g.Aafa A afp
Ea'P'r'a-EA3aaf5A
-
aPga9Dafa
afr A afa A afj'
aA DafP.
Q.E.D.
4. Complex-analyticity of harmonic maps
PROPOSITION3. Let M, N be compact Kdhler manifolds and f: N-+ M
be a harmonic map. Let m = dimM, n -dimN, and f" (1 < a < m) be
the componentsof f with respect to a local coordinate systemof M. Let
w&(1 < i < n) be a local coordinatesystemof N and let
dafi=
(aif)(aif) - (aff)(aifP)
If thecurvaturetensorRafiraof M is stronglyseminegative,thenfor 1 < i,
j < n,
=0
and h - 2 Re
zgap
,hij-dwidwi be respectively the Kahler metrics of M and N. Denote the Kihler form
V/-1Ei hij-dw'dwiof N by w. We use the notationsof Section 3. First
we show that
Proof. Let g - 2Re
82
YUM-TONG SIU
KglDaf A DVf> A af)-2 = Xwfn
for some nonnegativefunctionX on N.
(4.1)
Fix a point Q of N and let P = f(Q). To prove (4.1) at Q, we choose local
holomorphiccoordinate systems (za) at P and (wi) at Q such that with
respectto these coordinatesystems
g, = &,
dga,= O
hij-- aj
dhij-= O
(the Kroneckerdelta) at P,
at P,
at Q,
at Q.
Let ua and va be respectivelythe real and imaginary parts of f.
Xc** ) (respectively ce',
Let
1cec) be the eigenvalues of the complex
*.*,
Hessian of ua (respectivelyva) at Q with respect to the coordinates(wi).
We have
<g,Daf A Daf> A ()n2
=
EAafa A af a A
=
)0n-2
(aua~ A 8au" + A9vaA aav") A
-1
~~~j
n(n - 1) a
)n-2
+n
where the last identityis obtained by diagonalizingthe complexHessians
of ua and va (nonsimultaneously).Now (4.1) followsfromthe identity
=Xj=(Li(X)2
-
Li
2
and the fact that the harmonicityof f implies
E4)
-=
iM-a
=
0.
This proofof (4.1) yields also the statementthat X - 0 at Q if and only if
Dafa = 0 at Q for all a.
We now express
<R, af A af A af A af> A a)f-2
in termsof local coordinates. Fix Q e N. Choosean arbitrarylocal holomorphic coordinatesystem(wi) at Q such that hi--= ij at Q. Then at Q
<R,af A af A af A af> A a)n-2
-
EapraRa-r afa A afP A afr A afVA (n
XE
i< jn A ISkn, k*i,j(dW A dwk)
(n -2)!
+
(V'i-<1)j2
c
firS
I
-
2)!
(1-)n-2
nRafra(3-
( ifa)(a fP)(aifr)(@f5)+ (aj-fa)(aiff)(jfr)Q-fa)
(8f j)Q fP)(afr)[email protected])) A (Al ksn(dwk A dwk))
-
83
OF HARMONIC MAPS
COMPLEX-ANALYTICITY
U
)
la)ptI i.6l~<j~nRabiiv((&ifa)(ajf?)
_ (a-fe)(f
- i)dIt3
n(nn-1
X
((f-fP)(aif
r)- (o3fP)(a3fT))wn
is symmetricin ,8, 3, it followsthat
Since Rarpr
(4.2) KR,
j A aj' A ajfA af> A
-Efl2
n(n A
w
Re',-
I
,6?1<i?n
)
at any point Q of N when hi, = 3ij at Q.
The strongseminegativityof the curvaturetensorRat,-impliesthat
<R
(4.3)
fAafA af Aaf > Aa)n-2- Uan
for some nonpositivefunctiona on N.
By Proposition2,
n-2
aa<gjaf A af> A Ko
=
-
<R, A afAafAaf>A
<g,Df A D af> A
(n
n-2
-2
Since the left-handside is an exact form,it followsthat
N
<RI f A af A af A af> A
n-2
-
<g, Df A Daf> Aa
n-2
0.
From(4.1) and (4.3) we concludethat X and a are identicallyzero. By (4.2),
=O o
(4.4)
r a.
a~aprd-ffd
~
~ /ap rRa
To finishthe proof, we take an arbitrary local holomorphiccoordinate
system(Ci) at Q. Let
af affi
p
7y~
af
a
af,
evaluated at Q. We have to show that
Rar~r
(4.5)
Fix 1 < i, j < n. Since
Q let
-O
=
o
0
0
when i = j, we can assume that i
av-
a
-
j. At
aa
_
Ekak aWkI
Ek
akaWk
a
bkk
There exist uniquelytwo unit orthogonaln-vectors(A() ..., A(")), = 1, 2,
such that
ak
=
cA'l) and bk= c'A ') + c"A , 1 < k < n, for some complex
numbersc, c', c". Constructa unitarymatrix(A(")), 1 < v, k < n whosefirst
two rows are these two unitorthogonaln-vectors.Definea local holomorphic
system(Zk ) at Q so that
coordinate
84
YUM-TONG SIU
arp
at Q. Let
okk
4( aWk
af
f9
afa Of
aF1 aT2 &z d 2
evaluated at Q. By replacingthe coordinatesystem(wk) by the coordinate
system(zk), correspondingto (4.4) we obtain the following:
afa
Rara0R 0rTafl =O.
(4.6)
Now
=)
Ek,
lakblkI
-
-
c
Ek~l
Ak AL
~Ipt ~
~
$k
~~I
+ CC
k
Ak AIl2 I
Lk
where the vanishing of Ekl
A~
l All)
is due to the skew-symmetry
of e
in k, 1. Hence (4.5) followsfrom(4.6).
Q.E.D.
In orderto avoid repeatingpart of the argumentin the case of classical
bounded symmetricdomains, instead of proving directlyTheorem 1, we
prove a moregeneral result. To state this moregeneral result,we have to
introducethe followingdefinition.
Definition. The curvaturetensorRoiA,of a Kiihlermanifoldof complex
dimension m is called negative of order k if it is strongly seminegative and
it enjoys the followingproperty. If A = (Ail),B = (B.) are any two m x k
matrices(1 < a m, 1 < i < k) with
rank
(A
B\
BA
=2k
and if
Eta,,ivr
aRea ijfi sj
forall 1 < i, j < k,where
di.=A'- Bfi- As Bfi
then either A = 0 or B- 0.
The curvaturetensorRadari
is called adequatelynegativeif it is negative
of orderm.
The above definition
is motivatedby the followingequivalent definition
whichis clumsierto state but whichrendersmoretransparentthe motivation behindthe definition.
Definition. The curvature tensorR,,,A,of a Kihler manifoldM is said
to be negative of order k at a point P of M if it is strongly seminegative at
85
COMPLEX-ANALYTICITY OF HARMONIC MAPS
P and if it enjoysthe followingproperty. If f: U M is a smoothmap from
an openneighborhoodU of 0 in Ck to M with f (O) = P and rankRdf 2k at
0 and if at P
-
0
for 1 < i, j < k, where
-A
=
(atfa)(O)(djfP)(O)
(the coordinates of Ck being wi, 1 < i
<
-
(0 fa)(O)(aifP)(0)
k), then either af
=
0 at 0 or af
0
at 0.
is said to be adequatelynegativeat P if the
The curvaturetensorRafir,
above holds with the conditionrankRdf= 2k at 0 replacedby the condition
at 0.
that f is locallydiffeomorphic
We will use only the latter definition,because there the indiceshave
transparentmeaningsand are easier to keep track of.
THEOREM5. Let k > 2. Suppose M and N are compact Kdhler mani-
folds and the curvature tensor of M is negative of order k. Suppose
f: N -o M is a harmonicmap and theran/kover R of the differentialdf of
f is at least 2k at some point of N. Then f is eitherholomorphicor conjugate holomorphic.
Beforewe prove Theorem5, we have to prove firstthe followingvery
simplelemmain linear algebra.
LEMMA1. Suppose V is a vectorspace of dimensionn overC and W is
an R-vectorsubspaceof V and thedimensionof W overR is ?2n - 2k. Let
E be the set of all bases of V overC and let F be thesubsetof E consisting
n W = 0 for all
of all bases (e, ..., en) of V over C such that (k=1Ced)
...
1 ? i1 <
<ik < n. Then F is a dense open subsetof E.
< n let Fi... ik be the subset of E con< ...
Proof. For 1?
<ik
n W o. It sufficesto show that
sistingof all (e1,..., en)with (ok=Ce%)
each Fi, ik is a dense open subset of E. Let H be the set of all C-vector
subspaces L of complex dimensionk in V with L n w = o. It sufficesto
show that H is a dense open subset of the GrassmannianGk(V) of all
C-vectorsubspaces of complexdimensionk in V. ClearlyH is openin Gk(V).
Let K = W n V-1 W be the maximumC-vectorsubspace of V contained
in W. We can choose a basis e1, * ,
basis of K over C and
el,r R, ep, hV/el +
of V over C such that e*,
- eW,ep+l,
d1im
-
ep is a
2,eF
is a basis of W over R. We have I + p = diMRWf< 2n -2k.
First we show
YUM-TONG SIU
86
that H is nonempty. Let q be the largest integer < (
-
p/2). Let Q be the
C-vectorsubspace of V spanned by
eP+1+
/-1epd2,
ep+3
+ V-1
/ep+4,
ep+2q-?
*..,
+ V-lep+2q,
el+,, ...,
e.
= o. We claim that dimcQ > k. Since 2q >1 - p -1
and 2n - 2k > 1 + p, it follows that 2n - 2k + 2q > 21 - 1, whichimplies
n-k + q > 1, because n, k, q, t are integers. Hence dim,Q = q + (n-I) > k.
Choose a C-vectorsubspace L' of complexdimensionk in Q. Then L' e H.
Take L e Gk( V) and let gi, * * *, gk be a basis of L over C. Let
over C. Clearly Q f w
(I < i < k) .
gi = Ej=Jajiej
Let A = (aji)p<j?I,?<<k and B = (aji)1<js? ,l?,k. Then L n W # 0 if and only
if for some nonzerocolumnc of k complexnumbersIm Ac = 0 and Bc 0,
-
i.e.,
Ac--AcBc = O
Be= O
O
which is equivalent to
A -A
rank ( B
O B
< 2k
because, if
=O
Ac-Ad
Bc O
Bd
O
for some c, d not both zero, then by suitably adding the equations to and
subtractingthe equations fromtheir complexconjugates one obtains
Bi-At=O
ford c + d, V1l (c - d), one of whichis nonzero. By replacingL by L'
we obtain g', a'i, A', and B'. For 0 < t < 1, let
-
gi(t) = (1
-
t)gi + tgi,
aji(t) = (I1-t)aji
A(t)
B(t)
(1
(1
-
+ ta'i ,
t)A + tA'
t)B + tB'
COMPLEX-ANALYTICITY OF HARMONIC MAPS
87
The followingtwo inequalities
rank (aji(t))t?fl j<,l?Ik
k
>
A(t) -A(t)
rank B(t) 0
> 2k
0
B(t)
hold at t = 1 and hence hold for all t ej[O,11 - J, where J is somefiniteset.
For t e [0, 11 - J let L(t) be the C-vector subspace of V spanned by g#(t),***,
over C. Then L(t) E H and L(t) approachesL in Gk( V) as t approaches
0. Hence H is dense in Gk( V).
Q.E.D.
gk(t)
Proof of Theorem5. Now rankRdf> 2k at some pointof N and hence
at every pointof some nonemptyconnectedopen subset U of N. We first
prove that
either af _ 0 on U or Af 0 on U.
(4.7)
It sufficesto show that for every point Q of U
either af=O
(4.8)
at Q or af= 0 at Q,
of df on U impliesthat the two closed subbecause the nowhere-vanishing
sets U fa{f =O} and Un {Of=O} are disjoint and since by (4.8) their
unionis the connectedset U, one of themis equal to U.
Since the curvature tensor Rall is stronglyseminegative,it follows
fromProposition3 that for 1 < i, j < n and 1 < a, 73 < m
(4.9)
Ras
8
0 at P
where
J(dfa)(Q)(afP)(Q)- (d3-fa)(Q)(if )(Q)
(wi, 1 < i < m, being local coordinatesof N and fa, 1 ? a < n, being componentsof f with respectto a local coordinatesystemof M).
Let P = f(Q) and let TMP (respectively TNQ) be the tangent space of
M at P (respectivelyN at Q) when M (respectivelyN) is regardedas a real
manifold. Let K be the kernel of df: TNQ -> TMP. The complex structure
of M makes TMP a vectorspace over C. Since rankRdf > 2k, it followsthat
dimR K < 2n - 2k. By Lemma 1, there exists a basis g,, ** , go of TNQ over
C such that for 1 < ij < ... < ik< n the intersectionof K withthe C-vector
subspace of TNQspanned by gi, - * , gik is the zero vector subspace. Choose
a holomorphiccoordinatesystem(wt) of an open neighborhoodW of Q in U
such that gi = 2Re(a/awt). For 1 < ij < ... < ik< n let (i... ik be the
restriction of f to
w n {w- =w(Q)
for 1 <
< n,+il.
*W* ik}-
YUM-TONGSIU
88
2k at Q. Since Ras,, is negative of order k at every
pointof M, it followsfrom(4.9) that for 1 ? i, < ... < ik< n eitherDff 0
Then rankRd$i...ik
m and
j= i, *..,ik or ajfa=O at Q for 1?a
a <mand
j =
Now (4.8) follows fromk > 2, because, if sa # O at Q for
.,
ik.
some 1 < a ? m and some 1 < j < n, and aifi # 0 at Q for some 1 < 73< m
and some I < L<n , then we can select I < ii < *** < ik< n such that both
j and 1 belong to the set {il, ***, ij. The theoremnow follows from(4.8)
Q.E.D.
and the followingproposition.
PROPOSITION4. Suppose M, N are compact Kdhler manifolds and
f: N -o M is a harmonicmap. Let U be a nonemptyopen subsetof N. If f
is holomorphic(respectivelyconjugate holomorphic)on U, then f is holomorphic(respectivelyconjugate holomorphic)on N.
Proof. Since the proofs of the holomorphiccase and the conjugate
holomorphiccase are similar,we prove onlythe holomorphiccase.
Let Q be the largest connectedopen subset of N containing U such
that df vanishes identicallyon Q. It sufficesto show that Q is closed in N.
Suppose the contrary. Q has a boundarypoint Q. Let W be a connected
open neighborhoodof Q in N such that
i) there exists a holomorphiccoordinate system (wi) on some open
neighborhoodof the closure of W and
ii) there exists a holomorphiccoordinate system (za) on some open
neighborhoodof the closure of f (W).
The harmonicityof f is given by the equation
atQforl
ANf
+
Ei
~jo,7M>
fi'7Vaif
')(a-f
0
-)h
where (hi) is the inverse matrix of the matrix (hi--)of the Kdhler metricof
is the Christoffel
N, AN is the Laplace-Beltramioperator of N, and MJF7r
symbolof M. Applyingakto this equation and recallingAN= 2Ei i h e3f3
3
obtain
we
N(dkf
)
?
(2&kh )dfaif
?
-
(
?Ej
(h,,
Hence for some positive numberC
0r-(ift)h
lakad
I
-
0
+ ,
) ? C(.
ai fa I + Ejfr I .f
f
|IANQ8kf
be respectively the real and imaginary parts of a3-fa.
on W. Let uWand vka
Since AN is a real operator(i.e., it maps real functionsto real functions),it
followsthat for some positive numberC'
IANuJ 2 <C Cp
Nk12
< CEP
(Igradufi12
JIgrad
u
+
2 +?
Igradv-I2
+
Iu-I2 + Ivfl2)
+ Iu-12+
gradvfi12
Iv-12)
COMPLEX-ANALYTICITY
OF HARMONIC MAPS
89
on W, where grad is the gradient with respect to the coordinatesystem
whose coordinate functionsare the real and imaginary parts of wi. By
applyingAronszajn's unique continuationtheorem[1, p. 248] to the system
of functionsuW,v' (1 < a < m, 1 < k < n) and to the elliptic operator AN,
we concludefromthe identicalvanishingof uW,vkon wnQ thatua,va vanish
identicallyonW. This contradictsthe fact that Q is a boundarypointof Q.
Q.E.D.
Hence Q = N and 3f 0 on N.
Theorem1 followsfromTheorem5 and the followinglemma.
of a Kdhler manifold M is
LEMMA 2. If the curvature tensor
R,-rstronglynegative,thenit is negativeof order 2.
Proof. Let U be an open neighborhoodof 0 in C2 with coordinatesw1,
w2. Suppose f: U -> M is a smoothmap whichis a local immersionat 0 such
that for 1 i, j < 2
at P = f(O), where
d
Leas7wafiEfr6E
(& fa)(0)G8jfP)(?) -(a
0
O(8f))
We have to show that eitheraf = 0 at 0 or 0f= 0 at 0.
Let m = dimM. Let TM,,pbe the tangentspace of M at P when M is
regarded as a real manifold. The complex structureof M makes TMP a
vectorspace over C. Since f is a local immersionat 0, the image L of df is
an R-vectorsubspace of real dimension4 in TMP. By Lemma 1, we can
finda basis el, ... , enlof TMover C such that for I l a < 13a m the intersection of L with the C-vectorsubspace of TMp spanned by ey,1 : y ? n,
y # a, 18,is the zero vector subspace. Choose a local coordinatesystem (z")
?! m the map
of M at P such that ea = 2 Re (a3/za)at P. For 1 < a <
(fo(Wlt W%) f (W1Y W2))
(%p (w', w') i
at 0.
is locallydiffeomorphic
0 for 1 < a,,? < m
Since Rankis stronglynegative,it followsthat
and 1 < i, j < 2. Assume af#0 at 0. We want to prove that af = O at 0.
Since afo # 0 at 0 for some 1 < a ? m, without loss of generalitywe can
assume that Af' # 0 at 0. Let p be the numberof C-linearlyindependent
(1, 0)-formsamong af'1, - - *, afm at 0. We distinguishbetween two cases.
-
Case 1. p = 2. Withoutloss of generalitywe can assume that af' and
3f2 are C-linearlyindependentat 0. For 1 ? a < m and A = 1, 2, it follows
0 at 0
-0 0 that af
carafA at 0 for some cakE C. Hence, if afa
for some 1 < a < n, then af' = (Ca2/Cal) af2 at 0, contradictingthe C-linear
from
independenceof af' and df2at 0.
90
YUM-TONG SIU
Case 2. p 1. For 1 < a < m there exist complex numbers ra such
that aft rdaf1at 0. For 1 < a
m it follows from
= 0 that cfacadf1
at O for some c, E C. For I < a <j3? < m, df- A df xA df A dfP is a linear
combinationof exterior products of Of' and af1 at 0 and hence vanishes at
0, contradictingthe fact that D,, is locally diffeomorphic
Q.E.D.
at 0.
5. Bounded symmetricdomains
We recall the fourtypes of classical boundedsymmetricdomainswhich
we denote by Di ,, Dl', DI", D'V and computetheircurvaturetensors.
Type Imn The domain D nnis an open subset of Cm"and is the set of
all m x n matrices Z= (zap) with complex entries such that I - ZZ is
positive definite,where I., is the identitymatrix of order n and tZ is the
transposeof the complexconjugate of Z. An invariant Kihler metric has
the potentialfunction
(D
log det(Jn
-
tZZ)1
a
2+ -
zp
+
zge
higherorderterms.
At Z
0 the coordinates(Zap) are normalcoordinatesin the sense that the
firstorderderivativesof the coefficients
of the metric tensor with respect
to these coordinatesvanish at Z = 0. Hence the curvaturetensorat Z
0
is given by
It follows that at Z =
az"a-izjP~az2woaz
0
(5.1)
cra
-a
-
p
tusno
rr~ap<Prp +
?rPs ILaittr PI
12 +
v
E"
tra,
a?
<
P I Er
datrj
12
TypeII.. The domainDn' is an opensubsetof Cn(%-1)'2 and is the set of
all skew-symmetricn x n matrices Z= (zap) with complex entries such
that In - tZZ is positive definite. It is a complexsubmanifoldof D' a. The
invariantKahier metricof Dnn
induces an invariant Khhler metric of DnI,
At Z = 0, the coordinates(zap) fora < 3 are normalcoordinates. Hence the
second fundamentalformof DI, in DI , vanishes at Z- 0 and the curvature
tensorof DI, at Z- 0 is the restrictionof the curvatureof DI n. It follows
that at Z = 0
(5.2)
L
a<r
FPA, r
A<p<0 tf<rRar
ar'ap I
=
r 1=2E ?n
r
COMPLEX-ANALYTICITY
where
is
a
91
OF HARMONIC MAPS
skew-symmetric
in a, Y and skew-symmetric
in /3,p.
Type IIIn. The domain Dn" is an open subset of C'n+"''2and is the set
of all symmetricn x n matrices Z= (zap) with complex entries such that
is positive definite. It is a complex submanifoldof D'n. The
In -ZZ
invariantKahler metricof D'n induces an invariantKahler metricof Dn".
At Z = 0, the coordinates(Zap) for a < ,6 are normalcoordinates. Hence the
second fundamentalformof D'11in Dn vanishesat Z- 0 and the curvature
tensorof Dn" at Z = 0 is the restrictionof the curvaturetensorof Dnn. It
followsthat at Z
0
(5.3)
'<r
Tr
2or,
RaTIFI
-
EnI
12
+
E
En
aXpTr
2
where ar pis symmetricin a, y and symmetricin ,8,p.
Type JVn. The domain D'V is an open subset of Cn and is the set of all
Z = (Za, ** Zn)in Cnsatisfying
341+
2EAz
z2a 12_
12> 0
2<1
An invariantKahler metricis given by the potentialfunction
=2
2
-
q) =-log(1
2aI Za
1ZaI| + | aZ
|12)
+ 2(Ea IZa 12)2 + higher order terms.
IaZ2
_
-
At z = 0 the coordinatesZa (1? a < n) are normal coordinates. Hence the
curvaturetensoris given by
Rass
- -
a4D
It followsthat at z
0
iap
RY
ap
(5.4)
E
_
aZaadZpaZpaZ,
4~aap~a~aP~+
41E
=4
4
1
-4(&ap3p,
41
ag p~a~P-
a
e
f
aap~
44Lapp
a12& 1+
+
dalta
a Bae
aa 2
+ 2 La,'ts
+ 6aaap
p 4
-
6mpo)
41:6,~paP~
4E ,oi(appepa
_ Xpa 2
The curvature tensors of these four types of classical bounded symmetricdomainsare stronglyseminegative,but are not stronglynegative.
In the followingfourSections6-9, we will show that these curvaturetensors
are adequately negative so that Theorem 3 follows from Theorem 5. For
the proofof the adequate negativityof the curvaturetensors,we will need
the followingvery simplelemmain linear algebra.
SIU
YUM-TONG
92
LEMMA3. Suppose 'p is a smooth map from an open neighborhood of 0
, wm
in C"mto an open neighborhood of 0 in Cn with 9(0) = 0. Let wl,
(respectively z1, ..*, zn) be the coordinates of Cm(respectively Cu).
a) If the rank of the n x m matrix (aza/6wi) at 0 is n, then there exists
a nonsingular linear transformation
(1 < i < m)
wi = E,=l bijwi
(5.5)
such that at 0,
for 1
Or=i
awi
a < n, 1 <
< m.
b) If the rank of the n x m matrix (aza/awi) at 0 is p < n, then there
exist a nonsingular linear transformation (5.5) and a unitary transformation
a < n)
(1<
z =>azp
such that at 0,
wi,
6=- , for
1 < a < p, 1 < i < m
=0
p < a < n, 1
and
aZa
wi,
for
i < m.
c) If the rank of the n x m matrix (dzl/wi) at 0 is p < n and azl/awi ? 0
at 0 for some 1 < i < m, then there exist a nonsingular linear transformation (5.5) and a unitary transformation of the form
ZI = Z,
zIt
=
Zp
=2aap
(2
<
a
<
n)
such that at 0
a
-
for
1 < a < p1
-0
for
p < a < n, 2 <
=a
<
m
and
aZa
< m.
Proof. Let $D:Cm-> C" be theC-linearmap definedby the n x m matrix
(aza/awi) at 0. Let el, *..., en(respectivelyu,, * , ur) be the standard unit
basis vectorsof C" (respectivelyCt).
a) Take a basis v"+1, **, v. of Ker (.
ID(v,) = ej (1 < ii< n). Let
_i =
in
bju
Let vi be a vector in Cmwith
1 :!< i <
-)
COMPLEX-ANALYTICITY
93
OF HARMONIC MAPS
Then (biq)satisfiesthe requirement.
b) Take a unitary basis d, ** *, dn of Cn such that da e Im (D for 1 < a < p.
Take a basis vp+?, **, V. of Ker (D. Let vi be a vector in Cmwith (D(vi) = e
(1 <
< p). Let
7i=
and
(1 <
inubj
?<
m)
da = En>a,,pep
(1<a<n).
Then (bij) and (actp)satisfythe requirement.
c) LetIT: Cn -? Ce, denote the orthogonalprojection. Since az1/awi? 0
o(D). Let
at 0 for some 1 < i _ m, 7To(D is surjective. Let K = Ker (wT
': K -Ker wT= e)<f=,Ce<be induced by (D. Then rank If = p - 1. Take a
unitary basis d2, ***, d? in eL=2Ce, such that da,e Im P for 2 < a < p. Let
di = el. Let w' denote the orthogonalprojection fromC" onto the linear
o (D: Cm-- V is surjective. Choose
subspace V spanned by d, ** *, dp. Then wT'
V of Ker (wT'
o (P). Let vi be a vector in Cmwith (wT'o (D)(vi) =di
vm
a basis vp+l, *
for 1 < i < p. Let
and
Vi = EIn1 bsiuj
(1 ? vi< m)
da
(2<a<p).
=
En>aa,
ep
Then (brj)and (acp) satisfythe requirements,because (D(vi)= di for2 < i < p
whichfollowsfromthe fact that (P(vi) - di belongs both to Im (P and to the
Q.E.D.
linear subspace spanned by dp+l, .*., dn.
6. Adequate negativity of the curvature of
PROPOSITION 5. For mn > 2 the curvature tensor Rr,
me
<m, 1 <_y,p, a, r < n) of D'. is adequately negative.
mnn
- (1 < a, ,3,?,
Proof. Let U be an open neighborhood of 0 in Cmn. Let (Wij),ism,?ni,<n
be the coordinatesof Cmff.Denote d/6wij,6/6wijby Fiji, 6- respectively. Let
f: U -> Dnn be a smoothmap which maps the originto the zero matrixand
at 0. Let f (1
whichis locally diffeomorphic
J< a < m, 1 < ,i < n) be the
componentsand let
Assume that
(6.1)
Aid =
(a- faP)(0)(klf"r)(O) ERar'pploypr-I
(a fa')(0)(i3ifT)(0)
kLitLLk
= 0
at Z = 0 for all (i, j), (k, 1). We have to prove that
= 0 at 0 for all (i, j) and (a, mu)
1'eitheraf
(6.2)
0 at 0 for all (i, j) and (a, p3).
tor af f
-
94
YUM-TONG SIU
The following three conventions will be used in this proof and also in the
proofs of Propositions 6 and 7. Moreover, in the proofs of Propositions 6
will carry analogous meanings.
and 7 the notations aij, a-, fa, and t
i) For notational simplicity we will denote (ajjfOP)(O), (azfaP)(O),
([email protected])(O), (01jf )(O) simplyby ajjfcP, &ffi i
&-f respectively.
ii) After we apply linear transformations to the coordinates (wij) and
(ztap),we will use the same symbols for the new coordinates.
iii) We use the lexicographical ordering for the double indices (i, j).
By (i, j) > (k,1) we mean either i = k and j > I or i > k. By (i, j) > (k,1)
we mean (i, j) > (k, 1) or (i, j) = (k, 1).
From (6.1) and (5.1) it follows that
(6.3)
~~aitjrka
=
0
for all (,y,p),
0
for all
(a, A).
We will prove (6.2) from the equations (6.3). The equations (6.3) are invariant
under the following transformations of Dm'n
Zv->
(6.4)
Z.
zl
)tz
ZF
> AZB,
where A and B are fixed unitary matrices respectively of orders m and n.
To prove the proposition from (6.3) we can assume without loss of
tZ if necessary) that m > 2.
generality (after the transformation Z
Assume that aijffi ? 0 for some (i, j) and some (a, p3). We want to
prove that ai-f" - 0 for all (i, j) and (a, j). By applying a unitary transformation to the rows of Z (i.e., a transformation of the form (6.4) with
B = In), we can assume withoutloss of generality that aijf1P# 0 for some
3 and some (i, j). Let the rank of the n x (mn) matrix
be p. By applying Lemma 3b) to the smooth map
(f
f 1 ): U
11 .
) Cn
we conclude that, after we apply a linear transformation to the coordinates
(wij) and apply a unitary transformation to the columns of Z (i.e., a transformation of the form (6.4) with A = In), we can assume without loss of
<
generality that for 1 < A ?n,
(6.5)
{8ajflP
=
0
for (i,
i)> (1, p)
95
COMPLEX-ANALYTICITY OF HARMONIC MAPS
For 1 < i < p and (k, 1) > (1, p), consider
EAi=
that is,
EV,
kl=?;
1li
(&3f1J dklf'A
-
0 and 6 af1P (i3, it followsthat
Since okif
-
(6.6)
for 1 < i < p and (k, 1) > (1,p)
ak-cfli =O
We claim that the rank of the (n
P:
is n
-
0
a6f1Aaiif1A)
-
p) x (mn
p) matrix
-
(6i01f'T)pp<T<n,(k,l)>(1,p)
p. Suppose the contrary. Then at 0,
af 1P+l
A ...
A af1n = 0
<
By (6.5), af'A = o for p <, 3n.
df"1P+' A ...
(6.7)
A
mod (dw-N, * * *, di-10)
Hence at 0,
-
dfln
*.
mod(dwh11,
0
,
d iv) P
By (6.5) and (6.6), df'A, 1 < /3< p, is a linear combinationof dwII,
d
, ...,
dwIlp at 0.
*..,
dwjP,
It follows that at 0,
AfP=,(df1'A df')
-
c AsP=,(dw1,
A dwiA)
forsome constantc. This togetherwith (6.7) impliesthat at 0,
A/ =1(df1A A dfl)
=
0
at 0. Hence the
contradictingthe assumptionthat f is locally diffeomorphic
fin)
rank of P is n - p. By applying Lemma 3 a) to the map (fl Pl,
whose domain variables are (Wkl), (k, 1) > (1, p) (the other variables being
fixed),we conclude that after a linear coordinatechange in the variables
?< n,
(wkl), (k, 1) > (1, p), we have for p <
...,
(6.8)
k-flr
<j <
for p
(&f1T =jr
n
for 2<k<m,
0
1<1<n.
Since this last coordinatechange involves only Wkl with (k, 1) > (1, p), the
validityof (6.5) is not affected.
Consider
Take any 1 < a<m. Let 1< k<m, 1<I <n, and p<i<n.
that is,
=(@~liflA
aklf
A-
a
I
fJ)
=
0
Since ak-jf'A= 0 and a&f'A= 6jAby (6.6) and (6.8), it followsthat
aklfoi = 0 for 1 < a < m,1 < k < m, 1 < l < n, and p < i < n .
(6.9)
Considernow
YUM-TONG SIU
96
54
D
Ap
(8-f
kllj
< m, and 1 < j < p. That is,
where 1 < a < m, 1 < k < m, 1 <?
f'l
-
aklf p)
aj
=
0
Since 6klf'fi = 0 and a6jflfi= 6j, by (6.5), it followsthat
(6.10)
aT-fedj= O for 1 < a
l ?<nI< n, and 1 < j <. p .
m, 1
< m, 1 < k
We claim that, after a linear coordinate change in (Wkl) for k > 1 the
followinghold for 1 < i ? m:
{
,
(6.11)
(aif
fis= alp
for 1 <I<p
aklfip= 0
for (k,I)>(i,p)
a-fifi = 0
for k>i,
ifi= alp
and 1<,l<n.
for p < 1, 8 < n .
1<1,
and 1?<,<n.
3<n.
We prove this by inductionon i. When i = 1, (6.11)i followsfrom(6.5) and
(6.8). Suppose (6.11)i holds for 1 < i < j and j $ m and we want to show
that (6.11)j holds. From (6.11)i, 1 < i < j, it followsthat
(dwkl,
dfaP,1 < a < j, 1 <o < n, is a linear combinationof
< k j, 1 < 1 < n, at 0.
diikl, 1
From (6.10) it followsthat
1(63
of dWkl, 1 < k < m,
{dfi'P 1 _ /3? p, is a linearcombination
1 _ n, and diw,_1< r < n at 0.
The p x (m - j + 1)n matrix
('3k
fi)1
p ~p,3?lfm,1lfln
must have rank p, otherwisefrom(6.12) and (6.13) it followsthat
(A a< j,i pn (dfau A dfep)) A (A :? p (dfjfiA dfrp))
vanishes at 0, contradictingthe assumptionthat f is locally diffeomorphic
at 0. By applying Lemma 3 a) to the map (fj31..., fiP) whose domain
k < m, 1 < 1 < n (the other variables being fixed),
variables are (Wkl), j
we conclude that, after a linear coordinate change in (Wkl), j ?k < m
1 1 < n, we have the firsttwo equations of (6.11)j. This impliesthat
<
(dffip
(6.14)
1 ? / ? p, is a linear combinationof dwkl,
dikl,
p) ~~L
(Ic,1),at 0.
~~(j6,
From (6.9) it followsthat
(df
(1
of diIkl, 1 < k < m,
K < n, is a linearcombination
j, p <,
_ I <n, and dw1, 1 < r < n, at 0.
OF HARMONIC MAPS
COMPLEX-ANALYTICITY
The rank of the (n
p) x ((m
-
-
j + 1)n - p)
-
matrix
f)jp-<fi<n, (k,l) >(j,p)
00kf
must be n
97
p, otherwise it follows from (6.12), (6.14), and (6.15) that
n (dfafi
Aia -j,!fi<
A dafg)
vanishes at 0, contradicting the assumption that f is locally diffeomorphic
at 0. By applying Lemma 3 a) to the map (fjiP+l, *. ., fin)whose domain
variables are (Wkl) with (k, 1) > (j, p) (the other variables being fixed), we
conclude that, after a linear coordinate change in (Wkl) with (k, 1) > (j, p),
we have the last two equations of (6.11)j.
We distinguish between two cases.
Case 1. p = n. By the first two equations of (6.11)i we have for all
(a, 3)
=1
fa
& fa-O
for (k,1) > (a, 3).
Hence the (mn) x (mn) matrix
(a ijfP)
iaiam ,i?jin
is nonsingular. By Lemma 3 a) we can apply a linear change of coordinates
to (wij), 1 < i <m , 1 ? j < n so that after this linear coordinate change we
have
(6.16)
-
ai jf
3iajfi;
but statements derived above concerning _f&afi may no longer remain valid,
i.e., (6.6), (6.8), (6.10), and the last two equations of (6.11)i may no longer
hold.
We are now ready to show that
f afi = 0 for all a, h3,i, j. Fix arbitrarily (a, h3)and (i, j). Consider firstthe case i # a. We have
n
i.e.,
n>(13ii
(a~f
-afr
iif
arafar)
r_ a-~f
fipaaFr
1:1.=1
By (6.16),
-
0 and aafi
remaining case i - a.
We have
= 0
ir. Hence
=
=0
afi = 0.
n
arkr=
0
that is,
ET=
Consider now the
Since m > 2 there exists 1 < k < m with k + a.
(aajf
akf
By (6.16), aajfkr = 0 and akfifkr=-ir.
fr
-
araf
kr)=
Hence Aa-fafi 0.
98
YUM-TONG SIU
Case 2. p < n. From the last two equations of (6.11)i, we have
(a.-fin=1
?
taint 0
for i > j.
It followsthat the rank of the m x (mn) matrix
is m. Since p < n, we must have n > 2. Afterwe apply the transformations
Z IAtZ
where A is the matrix obtained fromLm by interchangingthe firstand last
rows, we reduce this case to Case 1 and we thereforeconclude, by the
argumentsof Case 1, thataijfa = O for all 1 < i, a < m and 1 < j, d< n,
which is a contradiction. Hence Case 2 cannot occur and a af = 0 forall
1 < i, a < m and 1 < j, i ? n.
Q.E.D.
7. Adequate negativity of the curvature of D~n
PROPOSITION6. For n ? 3 thecurvature tensorRumorqS (1?a<y<n,
1?<3 < p < n, 1 < X a ?< n, 1 _ , < ?_ n) of D"' is adequately negative.
Proof. Let U be an openneighborhoodof 0 in C'1/2)'nn-1'.
Let (wjj)1?i<1j?
be the coordinatesof C'1/2)nfn-1).
Let f: U-> D"' be a smoothmap whichmaps
the originto the zero matrix and which is locally diffeomorphic
at 0. The
componentsfaPof the map f satisfyf " = - f Pafor 1 < a, i < n. Assume
(7.1)
A T<Tp
<
at< R
=
0
at Z = 0. We have to prove that
(7.2)
Ieither afaP = o
at 0 for all 1 < a < f < n
orafaP=O0 at 0 forall 1<a<3
?n.
It followsfrom(7.1) and (5.2) that
(7.3)
=0
for 1 < a, i < n .
ET=
We will prove (7.2) from the equations (7.3). The equations (7.3) are
invariantunder the followingtransformationsof D':
Z )-. tAZA
(7.4)
where A is a fixedunitarymatrixof ordern. Sincef is locallydiffeomorphic
at 0, we have either
aijfli
0 for some i < j and some (3
OF HARMONIC MAPS
COMPLEX-ANALYTICITY
or
99
t 0 for some i < j and some if
jflP
(otherwisedf iP- 0 at 0 for all i3,contradictingA,< (dfhPA dfEP)f 0 at 0).
Since the equations (7.3) are invariantunder the transformationZ -* Z, we
can assume without loss of generality that deifP 0 for some i < j and
some hi. Let the rank of the (n - 1) x (1/2)n(n - 1) matrix
(aiifP)1<P
n.l-<i<j1.
be p
-
1. By applyingLemma 3 b) to the smoothmap
(f2
fin): Ui
> Cn
we concludethat, after we apply a linear transformationto the coordinates
(wij) and apply a transformationto Z of the form(7.4) with
/1 0
NOB
where B is a unitarymatrix of order n - 1, we can assume withoutloss of
?< n
generalitythat for 1 <
=ip
al3flP
(7.5)
p
for 1<j
for (i, j) > (1, p) .
=0
a~Aijf1
For 1 < i < p and (k, 1) > (1, p), consider
n
rai~l
Since aklf1
=
ii, kl
0 and aif 1P= 6ip,it followsthat
aklf 1" =
(7.6)
2,2~, = 0
The rank of the (n
-
0
for 1 < i _ p and (k, 1) > (1, p) .
p) x ((1/2)n(n
-
1)
-
p + 1) matrix
7 n,(k,1)> (1,p)
(a 0 f ") p<r
must be n
-
p, otherwiseit followsfrom(7.5) and (7.6) that
A dfi')
A>&=1(df1P
vanishes at 0, contradictingthe assumptionthat f is locally diffeomorphic
at 0. By applying Lemma 3 a) to the map (flP+l,
fls) whose domain
.,
variables are (wkl) with (k, 1) > (1, p) (the other variables being fixed),we
concludethat, after a linear change in the coordinates (Wk,), (k, 1) > (1, p),
we have for p <K < n,
(7.7)
Tso
ftaft
=
{a-flr
a1fro
h2e
vi
=
jr
for p<j<n
o
?k
ln
5for
This does not affectthe validityof (7.5).
k<
< n.
YUM-TONGSIU
100
For 1 < a < n, 1 < k < 1 < n, and p < i < n, consider
~n
pL
Since aciflp
(7.8)
=
i iki
0 by (7.7) and aj-f1P =-i
=
aklfai
=
0
by (7.6) and (7.7), it followsthat
and p<i<n.
for 1<a<n,1<k<I<n,
0
For 1 < a < n, 1 < k < 1 < n, and 1 < j < p, consider
Since aklf1
(7.9)
[P
0 and aif3
akcfai =
0
by (7.5), it followsthat
-p
for 1 < a < n, 1 < k < l < n, and 1 < j < p .
We claimthat, aftera linearcoordinatechange in (Wkt) for 1 < k < I < n
the followingholds for 1 < i < n:
(7.10)
(aTf'P
p and i<13
Tilfif dip
for i<l
aklfifi= 0
for (k, 1) > (i, p)
a1iAip= 0
for i<k<l<n
< n
= dip for max(i, p) < I < n and i < S
<
n
and i<,?<n.
We prove it by inductionon i. When i = 1, (7.10)i follows from (7.5) and
(7.7). Suppose (7.10)i holds for 1 < i < j and j ? n and we want to show
that (7.10)j holds. From (7.10)i, 1 < i < j, it followsthat
dfa, 1 < a < j, a < 3 < n, is a linear combinationof
1 < k < j, k < I < n, at 0.
dwkldikl
(7.1)
From (7.9) it followsthat
(7.12)
{df iP,j
<,i _ p, is a linear combinationof dWkl, 1
_
k < 1 < n,
and dwv,,1 < r < n.
When p > j, the (p
-
j) x (1/2)(n- j + 1)(n - j) matrix
j5k< <-n
(aklfj) j<Pi5pp
must have rank p
-
j, otherwisefrom(7.11) and (7.12) it followsthat
(A/<a<ja<pn(dfaP
A df a))
A (A j<Ais(dfjP A df P))
vanishes at 0, contradictingthe assumptionthat f is locally diffeomorphic
at 0. By applying Lemma 3 a) to the map (fi?1, **., fil) whose domain
variables are (Wkl), j < k < 1 < n (the other variables being fixed),we conclude that, after a linear coordinatechange in (Wkl), j ? k < 1 < n, we have
the firsttwo equations of (7.10)j. This impliesthat
(7.13)
df'j, j < , ? p, is a linear combinationof dwkl,dwkl,
(j, p) > (k, 1) at 0.
101
OF HARMONIC MAPS
COMPLEX-ANALYTICITY
From (7.8) it followsthat
df'i, max (j, p) <,8 < n, is a linear combination of
dikl, 1 < k < I < n, and dw, 1 < r < n, at 0.
(7.14)
The rank of the (n
-
max(j, p)) x (1/2(n - j + 1)(n
-
j)
-
max(0, p
-
j))
matrix
(k,l)
00-kf
>(j,p)
<P<-n,
j)mas(j,v)
must be n
that
-
max(j, p), otherwise it follows from (7.11), (7.13), and (7.14)
Ala<ja<,<n(dfaPd fA
dfa)
vanishes at 0, contradictingthe assumption that f is locally diffeomorphic
at 0. By applyingLemma 3 a) to the map (f jmax(j)+l, I * fjn) whose domain
variables are (Wkl) with (k, 1) > (j, p) (the other variables being fixed),we
conclude that, after a linear coordinatechange in (Wkl) with (k, 1) > (j, p),
we have the last two equations of (7.10)j.
We distinguishbetween the followingtwo cases.
,
Case 1. p = n. By the firsttwo equations of (7.10)i the ((1/2)n(n - 1)) x
((1/2)n(n - 1)) matrix
(ai f
)~i<j <-n, Ia<
P
n
is nonsingular. By Lemma 3 a) we can apply a linear coordinatechange to
< j < n, so that after this coordinatechange we have
(wij), 1
= 6iajp for 1 < i < j < n and 1 < a < 3 <?n,
(7.15)
ajjfc1P
at the expense of possibly sacrificingthe validity of (7.6), (7.7), (7.9), and
the last two equations of (7.10)i.
We want to show that a-fi = 0 for 1 < i < j ? n and 1 ? a <( 3? n.
and
n, a.=
For notational convenience, we define, for ?i <j
-ai
=- =-&. First we show that
(7.16)
a-yfaP=0
for 1 < a, 8, j < n with j
a and
+ a .
Since n > 3, we can choose 1 ? k < n with k + a and k + (3. Consider
n
Lar:
Er
kr _0
a--
-k
that is,
(7.17)
=>l(a
faTakPfkT
-
aifara
jfkr)
= 0
0 if y + a. On the other
Since a # k, it follows from (7.15) that ajfkT
0. Hence in any case a-far ajfkr = 0. Since
then fa
hand, if y=a,
by (7.15), it followsfrom(7.17) that Aa-j = 0.
akPf r = 6py
We want to prove
Now fix 1 < a < S _ n and 19 i,pj < n with i +j.
-
SIU
YUM-TONG
102
0. Because of (7.16), we can assume withoutlossofgenerality
a-f" =
that a, ,, i, j are distinct. Consider
that
that is,
(7.18)
_
(f
f
0
aifar
-
0) that P-fa'
It follows from (7.16) (and the fact that fa
Aasfar = 6r by (7.15), we have &afaP= 0 from(7.18).
-
0. Since
Case 2. p < n. From the last two equations of (7.10)i, we have
jafi
Since fni
matrix
for 1 < i < j < n .
1
- -fi-, it follows that the rank of the (n
(fe~jf'")<i
is n
-
?<j?,1?a<
-
1) x (1/2)n(n
-
1)
n
1. Afterwe apply the transformation
Z
tAZA
-'
where A is the matrixobtained fromI,, by interchangingthe firstand last
columns,we reduce this case to Case 1 and we thereforeconclude,by the
argumentsof Case 1, that aijf"P= 0 forall 1 < i < j < n and 1 < a < i3< n,
which is a contradiction. Hence Case 2 cannotoccur and a3f0 = 0 for all
Q.E.D.
1< i< j ? n and 1< a <B_ ? n.
8. Adequate negativity of the curvature of DE"'
7. For n > 2 thecurvaturetensorRay,V, AS - (1 < a _ ? n,
< a < n, 1 ?< < z < n) of DnI" is adequately negative.
lf3?p <n n< 1
PROPOSITION
Proof. Let U be an open neighborhoodof 0 in C'1/2'ff'n+1'.Let (Wij),,<<-,
be the coordinatesof C 12'"'+. Let f: U-->Dl" be a smooth map which
at 0.
maps the origin to the zero matrix and whichis locally diffeomorphic
f P off satisfyfap = f Pofor1 < a, 3< n. Assume
The components
(8.1)
/a-yi?p,
R,.
A
at Z = 0. We have to prove that
(8.2)
Lp7
L.j~el =
0
iP
either 0f'i=0 at 0 for 1 ? a < A < n
or
afaP= Oat 0 for 1 < a <A? < n .
From (8.1) and (5.3) it followsthat
(8.3)
InskL
=0
for 1 <a,
3< n.
We will prove(8.2) fromthe equations(8.3). The equations(8.3) are invariant
COMPLEX-ANALYTICITY
103
OF HARMONIC MAPS
under the following transformations of D,"':
ZF -
(8.4)
tAZA,
where A is a fixed unitary matrix of order n.
Since f is locally diffeomorphicat 0, we have either
atJf' #0
for some
i _ j
j f1L
W0
for some
i
or
< j
(otherwise df1 = 0 at 0, contradicting A a<?(dfaP A dfr) # 0 at 0). Since
the equations (8.3) are invariant under the transformation Z ---i Z, we can
0 for some i < j. Let the
assume without loss of generality that aijf
rank of the n x (1/2)n(n + 1) matrix
(a~ijf 1,
nlijn
be p. By applying Lemma 3 c) to the smooth map
(f
f17):
*...
U-
Cla
we conclude that, after we apply a linear transformation to the coordinates
(wij) and apply a transformation to Z of the form (8.4) with
/1 0
0 B
where B is a unitary matrix of order n - 1, we can assume without loss of
generality that
p
jflp= ajar
for 1 _ j < p, 1
(8.5)
for 1 < j
a l1jf1 =0
aij
for
=
p,
p<
(i, j) > (1, p), 1
n
< 3?
n.
For 1 < i < p and (k, 1) > (1, p), consider
n
that is,
f
EPliflp>klflp
- alflifp)
= 0
= 6ip and aklf'p - 0 by (8.5), it follows that
Since a1if1P
(8.6)
and (kl)>(1,p).
alfli = O. 1 < i < p
Let q be the rank of the (n - p) x ((1/2)n(n + 1) - p) matrix
P
Select p < il <
...
=
([email protected])p<i<<.,(k,l)>(l,p)
-
< iq < n such that the rank of the q x ((1/2)n(n + 1) - p)
104
YUM-TONG SIU
matrix
1i&')J<_v<q, (ksl) > (1, p)
(akjf
whose domain
is q. By applying Lemma 3 a) to the map (f l,
fliq)
variables are (Wkl), (k, 1) > (1, p) (the other variables being fixed),we conclude that,aftera linear change in the variables (Wkl), (k, 1) > (1, p), we can
assume withoutloss of generalitythat
(8.7)
1
for 1 : v, j < q
for (k,l)>(1, p+q),1<
,
3lj~pF- -=
akf1i=0
q.
We must have
(8.8)
fl
for (k, 1)> (1,p + q), p < i< n,
_0
For (k, 1) > (1, p + q), consider
otherwisethe rank of the matrix P is >q.
n0
this is,
Er
Since aklflr
that
(aflraifir
-
)
a11f'talf
o
?
0 and allft'1- ( for 1 < Y < p by (8.5), it follows from (8.8)
for (k, 1)> (1, p + q) .
When p + q < n, this impliestogetherwith (8.5), (8.6), (8.7), and (8.8) that
at 0,
(8.9)
0f1Wz=0
df' A ... A df in A dfF' A ... A dfIn
is a linear combinationof exteriorproductsof dwl,,dwv,for 1 < I ? p + q
and hence must be zero, which contradictsthe fact that f is locally diffeomorphicto 0. Thereforewe must have q = n - p. Thus i, = p + v and
(8.7), (8.8), (8.9) read
for p<1,f3<n
lf = alp
for 1 < kk<?I < n, p < 3 < n
(8.10)
-f'i= 0
I, f11= 0
for 1 < k < I n .
Takeany l <caxn.
that is,
nand p<i<n.
Letl<k<I<
En (0-f1Paklfi
-
a fif
0
Since akf' = 0 by (8.6) and (8.10) and since a-f'j
and (8.10), it followsthat
afif
llaklfal
+
aklf
i
Consider
0
?
-
6ipfor 1 <,3 < n by (8.6)
105
COMPLEX-ANALYTICITY OF HARMONIC MAPS
Because
f 1 = flo,by (8.5) we have
(8.11)
=O
aklfai
for
1 <
aklfal =
a
where 1<a<n,
Since
=i
0
Thatis,
1<k<1<n,and1<jp.
0 and aljf 'P = bjpby (8.5), it follows that
aklf1f
(8.12)
c 0B
t
ak,t
n.
n, p < i < n, 1 < kk<I?
<
Consider now
n
Epn=
0. Hence
-=O for 1<a<n,1<k
a(3f
l<n,
and 1<j<p.
We are going to prove by induction on i for 1 < i < n that after a linear
coordinate change in (Wkl) for 1 < k I < n, the following hold:
{ ft
aklfi
(8.13)i
=
-o
p for i
for i
AiP= 31g for
a
ak-ifi- = 0
<,i < n
and i < I < p
< n and (k, l) >
<
max(i, p + 1) <,SI
<fl
for i <
k? I < n
(i, p)
n
and i < 8 <
n.
To avoid a repetitious initial step of the induction process, we agree to mean
by (8.13), the vacuous statement and prove (8.13)i by induction on i for
1 ? i < n. Now we prove (8.13)j-1
(8.13)i for 1 < i < n. From (8.5), (8.6),
(8.10), it follows that
(8.14)
df",
df'" is a linear combination of dw11,..,
.,
*
dw,
di1, ... , div. at 0.
From (8.13), (1
< v<
(8.15)*
df
i) it follows that for 1 < v <i
dfPn is a linear combination of dwkl,
1<k<,
k<I
diwkl,
<n.
From (8.11) and (8.12) it follows that
(8.16)
df
dftP is a linear combination of dWkl, 1 < k ? I ? n,
,
and di&j, 1
<
j < n, at 0,
and
(8.17)
df 'P,max(i, p + 1) < ,8 < n, can be expressed in terms of
diwkl, 1 < k < I < n, and dwlj, 1 < j < n at 0.
When p > i, the (p
-
i + 1) x (1/2)(n - i + 1)(n - i + 2) matrix
Q:
must have rank p
it follows that
-
-=8l
f<<
g~g
i + 1; otherwise from (8.14), (8.15)v, 1 < v < i, and (8.16)
YUM-TONG SIU
106
A df _))A (AP=X(dfipA dfi))
vanishes at 0, contradictingthe assumptionthat f is locally diffeomorphic
..,
* fiP) whose domain
at 0. By applying Lemma 3a) to the map (f i
variables are Wkl, i < k < I ? n (the othervariables being fixed),we conclude
that, after a linear coordinatechange in Wkl, i < k < 1 < n, the firsttwo
equations of (8.13)i are satisfied. This impliesthat
(Ai<a<i
(8.18)
df
ar!4<n(df-\
**,
dftP is a linear combination of dWkl,
diikl,
(li,p) > (k, 1).
The (n
-
max(i, p + 1) + 1) x ((1/2)(n - i + 1)(n
matrix
L:
(
-
i + 2)- max(O, p
-
i + 1))
f )max(ifp?l, (k.l,(i.max(i.p?1))
musthave rankn - max (i, p + 1) + 1; otherwisefrom(8.14), (8.15)",1 < v <i,
(8.18), and (8.17) it followsthat
Ai<a<i "pP<n(dfaP A dfo)
vanishes at 0, contradictingthe assumptionthat f is locally diffeomorphic
at 0. By applying Lemma 3a) to the map (fi maax(i
'P1) *I
fin) whose domain
variables are Wkl, (k, I) > (i, max(i, p + 1)) (the othervariables being fixed),
,
we conclude that, after a linear coordinatechange in
p + 1)), the last two equations of (8.13)i are satisfied.
We distinguishbetween two cases.
Wkl,
(k, 1) > (max(i,
Case 1. p = n. By (8.5) and (8.13)i, 1 < i < n, the (1/2)n(n+ 1) x
(1/2)n(n+ 1) matrix
is nonsingular. By Lemma 3a) we can apply a linear change of coordinates
to (wij), 1 < i < j < n so that after this linear coordinatechange we have
(8.19)
aijf-fi= 3j
for 1< i<j<n
and 1<a<,<
n,
at the expense of possiblysacrificingthe validity of (8.6), (8.10), (8.12), and
the last two equations of (8.13)i.
We want to show that da-jf" = 0 for 1 < i < j < n and 1 < a < ,3 < n.
.
For notationalconvenience,we define,for1 ? i < j < n, aji aij and 82-i
<
Take 1 < a, s, j, k, I n with +k, 1. Consider
brlkl,,pj=
i.e.,
fl_
(6Rkfaf
aafi-
far
It followsfromH3
UB k, I and (8.19) that aklf
Pr
0f)
=.
0. From (8.19) we also have
107
COMPLEX-ANALYTICITY OF HARMONIC MAPS
daijf':2=
b.
Hence
for 1 < a, j, ky1 <n ,
if there exists 1 <?!3?<n with 83/ k, 1. When n > 3, for any given 1 < k,
I < n, we can always find1 < 38< n with 38# k, l. It remainsto prove the
(8.20)
=0
akfa
case n = 2. Consider
for (a, 3, i, j, k, t) = (1, 1, 1, 2, 1, 1, (1, 2, 1, 2, 2, 2), (2, 2, 1, 2, 2, 2).
That
is,
(8.21)
ii
a-f
11
1
isj l
(12f
12f21 (2f2-
0
+f"h2f"? a&-f 12allf12 -aiif'a2f12-
-
a312f32f2'
? (f2(2f
_
ff21' 12f2'
+
&f22
al2f'2 0
-
a22f22
a-f22
a12f22
-
0
By (8.20),
afi
for 1 < a, ? < 2 .
=0
=-fi
By (8.19)
f
alfla
228-
ajf12 =
=
a
J22=
121- 0 .
It followsfrom(8.21) that
aij1
12
= al2f
=
a2f 22
=
0
Case 2. p < n. It followsfrom(8.10) and (8.13)i, 1 < i <n , that
for 1 < i < j < n
.fin =a
Since fi"
ft",
it followsthat the rank of the n x (1/2)n(n+ 1) matrix
(a-i-f)
Z3
1e;a
<n'
19i:9j:!-~n
is n. Afterwe apply the transformation
Z ,
tAZA,
where A is the matrix obtained fromI, by interchangingthe firstand last
columns,we reduce this case to Case 1 and we thereforeconclude,by the
argumentsof Case 1, that ajfaiiS = 0 forall 1 < i ? j ? n and 1 < a < ,8 < n,
which is a contradiction. Hence Case 2 cannot occur and a(-3Pfai - 0 for all
1 ? i ? j ? n and 1 <
Q.E.D.
<? 8 < n.
9. Adequate negativity of the curvature of D"V
PROPOSITION8. For n > 3 thecurvaturetensorR,,r,-(1< a, p, by,a < n)
of DnVis adequately negative.
Proof. Let U be an open neighborhoodof 0 in Cn. Let wi(1 ? i < n) be
Let f: U -> DV be a
the coordinates of Cn. Denote l/awi, l/aw'by ai, (.
YUM-TONG SIU
108
smooth map which sends the origin to the origin and which is locally diffeomorphic at 0. Let f(1 < a ? n) be the components of f and let
=d(8f-f)(0)(a8f)(0)-(adf)(0)(8sf)
Assume
0
L
(9.1)
at 0 for all 1 < i, j _ n. We have to show that
n.
n or faf 0 at 0 for 1<a<
0 at 0 for 1<a<
eitheraf
(9.2)
For notational simplicity we will denote (3ifa)(0), (aff)(0), (aifa)(0), (j f)(0)
simply by aif, a3-fa, afa,a-f respectively. From (9.1) and (5.4) it follows
that
{
=0
t:=la$
(9.3)..-i~-J=0
for all
ij
for all
i, j, a, h.
of (9.3) is invariant
equation
under the automorphism
The second
of (9.3).
We will prove (9.2) only from the second equation
z a -z of
DAV.
For 1?< a, < a2< a,< n let Vala2a3 be the vector subspace of the tangent
space TL-Oof U at 0 consisting all tangent vectors whose images under df
are annihilated by dz,,p,d,,,, 1 <
i-
< 3. Since f is locally diffeomorphicat
is of real codimension 6 in Tu,0. By Lemma 1, we can choose a basis
...
, en of Tf,,0over C such that
el,
0,
VaIa2cf3
nf(el-31Cei) = 0
Vana2a3
n. We denote again
n and all I < il < ... < in3?<
by wi, 1 < i < n, the linear coordinate system of U such that ej = 2 Re (a/awj)
for all
map
variables
a3<
2a<
1<i<n.
atO,
the
<1 a, <
and any 1<i1
Thenforanyl<a1<a2<a3<?n
fa2,
(fal,
whose
fa3)
domain
variables
are
wil
wi2,
<i2<i3<n,
wi3
(the
other
being fixed at 0) is locally diffeomorphic at 0. For the conclusion
< a3 < n
clearly it suffices to show that for any 1 < a1 <2
of the proposition,
and anyl
<
il
<
i2 <
i3 <
n either a,
Ct2 =
0 for 1
se, \ ? 3 or aa --fa
0
,, X < 3. Hence in order to derive the conclusion of the proposition
from the second equation of (9.3), it suffices to consider the case n = 3. We
for 1 <
assume now n = 3.
below
in full
from the second equation
of (3.9):
We
write
out
nine
equations
$8^
=
_ f 1lf2 = a~f2_2f'_ a-f2al
f2f a-
(9.4)
al
(9.5) )
a fls
(9.6)
(9.6)
aEjRf~~lpf2
(9.7)
the
af2
a:+-
__f,~af
f2f =
a~~~~~~~~~~~~f283fl
f2 _
-~
2
-3 a -fla~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~f2
a f?3fa~~~~~~~~~~~~~
~~
2
a
-_j
laf2
_-aEif31
= a-Rf2Ejalf _
_ a-Rf23
1f2 f
if3 2f2 _ af3 a1f2 = a-f2 f3
_-
-f2 1f3,
i
j
109
COMPLEX-ANALYTICITY OF HARMONIC MAPS
(9.8)
a-f3
(9.9)
a8f3a1f2
-
3f2
8f3
_
=
2f2
a-f3a3f2
J2a 3f3
f3
=af2a
-fa2
-
_
f3 ,
f3
a-f2
I
3J3a1f
&-f'af3= a-f3_2f'
3
f3 -3fa2f3 = aJ3a3f'-_3a-2f
a-fla
_aifa3fa
f3 a-f3a=fl
falf3
f3 hfla
(9.10)
(9. 1)
(9.12)
Let p be the rank of the 3 x 3 matrix (8fa) ita,3 and q be the rank of
We claim that one of p and q is at least 2.
the 3 x 3 matrix (afj)1<i?3.
Suppose the contrary. Then we can choose 1 <? x, ft 3 such that for
1 _ a <!~ 3,
af
=
jaf
at 0 for some X, g, e C (which may be zero). It follows that at 0 the 6-form
Aa==(df' A dfa)
Any
af', and hence
is a linear combination of exterior products of af'1,
must be 0, contradicting the assumption that f is locally diffeomorphicat
0. By applying the transformation z -*,z if necessary, we can assume without
loss of generality
af,,
that p > 2.
Case 1. p = 3. By Lemma
aThf=
From (9.4)-(9.12)
change in wi,
3 a), after a linear coordinate
1 < i < 3, we can assume without loss of generality
that
for 1 < i, a < 3 .
bi
it follows that
a-fl -=-f2
(9.13)
0
=-f2
=-f3= 0
- a3f3 =
(9.14)
0
0
0=
2
f
f2
3
=a-
~31 0,
_a f
(9.15)
=
af3
From (9.13), (9.14), and (9.15) it follows that
1=
Hence
jfa = 0 for 1 <i
Case 2. p
- 8_f2 = a-f3 = _a-3fI
, a < 3.
2. At 0, two
After renumbering
fl,
f2, f3
of af1, 3f2, 3f3 are
if necessary,
linearly
we can assume
independent.
without loss of
YUM-TONGSIU
110
at 0. Afterapplyinga
are linearlyindependent
linear coordinatechange to w_,1 < i < 3, we have
3f'= dw,
af = dw.
generalitythat afl,
af2
for 1 < j ? 3, 1 ? a < 2. Since afI is a linear combinaat 0; i.e., ajfa =j
tion of 3f' and af2 at 0, we have a33f3 0. From (9.5), (9.6), and (9.12) it
followsthat 3f = 0 for 1 < a ? 3. Combiningthis with a3fa 0O for
1 < a < 3, we concludethat dfO,df", 1 < a < 3, are linear combinationsof
div-2at 0. Hence
dw1,dw2, div-1,
A a=l (dfa A dfa)
vanishes at 0, contradictingthe assumptionthat f is locally diffeomorphic
at 0. Thus Case 2 cannot occur and we have &fT = 0 for 1 < i, a < 3.
10. Strong rigidity and other applications
THEOREM 6. k > 2. Suppose
g: N
If the curvature
Kdhler manifolds.
if the map H1(N, R)
->
-->
map of compact
tensor of M is negative of order k and
H1(M, R) induced
then g is homotopic to a holomorphic
M is a continuous
by f is nonzero for some I ? 2k,
or conjugate
holomorphic
map from
N to M.
Proof. There exists a harmonic map f: N -> M which is homotopicto g.
Since the map H1(N, R) -->HI(M, R) induced by f is nonzero, it follows that
rankRdf> 2k at some point of N. By Theorem5, f is either holomorphic
Q.E.D.
or conjugate holomorphic.
The followingtheoremfollowsfromTheorem6 and Lemma 2.
THEOREM 7. Suppose
M is a compact Kdhler manifold
tensor is strongly negative.
Then for k > 2 an element of H2k(M, Z) can be
represented by a complex-analytic
by the continuous
subvariety
of M if it can be represented
image of a compact Kdhler manifold.
THEOREM 8. Let g: N-> M be a continuous
manifolds,
both of complex dimension
of M is adequately
H2,-2(N, R)
-> H2-2(M,
a biholomorphic
whose curvature
negative.
n
Suppose
map
g is
of degree
R) induced by g is injective.
or conjugate
of compact
Kdhler
> 2. Suppose the curvature tensor
biholomorphic
1 and
the map
Then g is homotopic to
map from N to M.
Proof. By Theorem 6, g is homotopicto a holomorphicor conjugate
holomorphicmap f: N -> M. Let V be the set of points of N where f is not
locally homeomorphic. Since f is of degree 1, V # N. Suppose V is nonempty.
COMPLEX-ANALYTICITY
OF HARMONIC MAPS
111
We want to derive a contradiction. V is a complex-analyticsubvariety of
purecomplexcodimension1 in N, because locally V is definedby det(aza/awz)
whenf is conjugate holomorphic,
whenf is holomorphicand by det(dzc/dwi)
where (z") (respectively(wZ)) is a holomorphiclocal coordinatesystemof M
(respectivelyN). f( V) is a complex-analyticsubvarietyin M. Since f is of
degree 1, f maps N - f'- (f( V)) bijectively onto M - f( V). The complex
codimensionof f( V) in M is at least two. For otherwisethereexists v C V
such that v is an isolated pointof f-(f(v)) and, by using a local coordinate
chart (wz) of N at v and by applying the Riemannremovablesingularity
theoremto wLof l or w&o f ' on U - f( V) for some open neighborhoodof
at v, contradicting
f(v) in M, we conclude that f is locally diffeomorphic
v E V. Hence the elementin H2ff_2(N,
R) definedby V is mappedby f to the
zero element in H22, (M, R), contradicting the injectivity of the map
Q.E.D.
H2-2 (N, R) -> H2n-2(M,R) inducedby f.
Now Theorems2 and 4 followfromTheorem8, Lemma 2, and Propositions 5-8.
STANFORD UNIVERSITY, STANFORD, CALIFORNIA
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(ReceivedMarch 2, 1979)