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Title An application of harmonic mapping to complex analytic

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Title
An application of harmonic mapping to complex analytic
geometry
Author(s)
SHIGA, Kiyoshi
Citation
[岐阜大学教養部研究報告] vol.[17] p.[57]-[60]
Issue Date
1981
Rights
Version
岐阜大学教養部 (Dept. of Math., Fac. of Gen. Educ., Gifu
Univ.)
URL
http://repository.lib.gifu-u.ac.jp/handle/123456789/47506
※この資料の著作権は、各資料の著者・学協会・出版社等に帰属します。
57
A n application of harm onic mapping
to com plex analytic geom etry
Dedicated to P rofessor S. T omatsu on his 60th birthday
Kiyoshi SH I GA
Dept.
of M ath. , F ac.
of Gen.
E du. , Gifu U niversity
(Received Oct. 5, 1981)
l ntroduction
1
Recently the theory of harmonic mapping is applied in several directions.
plex analytic geometry some applications are know n.
ty theorem
ture.
ln com
Y . T . Siu 〔8〕 showed the rigidi-
of com pleχ strud ures of K iihler m anifolds w ith adequate neg ative curva- .
T he key step of the proof of the theorem is to show that a horm onic m apping is
holom orphic or antih010morphic.
H e obtained a B ochner type equality for a harmonic
mapping and integrate on the souse m anifold。
S. N ishikawa and the author applied Siu s method to K iihler maniflods with boundary
and obtained
THEOREM ( M shikawa-Shiga 〔4〕) . Ld M 皿 d N becom折deKa沁 ylu n面 ldsof com一
j)leχdim四 si皿
≧ 2 . Let 仙 ⊂ M and 励 ⊂ N be ydd 加dycom加 d dom4泌s 組 M 皿 d
N 面 侑 (グ bo皿 dd es. S呻1)osd k ぺ 1) N k sadeq皿 temgatiw cuymtMye緬 the seuceof
S加 〔6〕, ( 2) theboMu面巧 ∂Dぶs畑mdocom a α11d ( 3) 匝e粍 a istsα(ンR di蕉 omo砂吊sm
斤 ∂£) 1→ ∂D2uJhicheχtGldstoa homo朗)ye叫i叱lmceof DU oD2.
bih㎡omo砂hicj い 岸加山 吐
TheuD 1皿 d D2aye
秘 紅 け d a dstoαbiholomo砂hicd派 omoゆhism of D 1M
D 2.
ln this paper we consider the case that the dimensions of 討 and N are not the same.
T HEOREM . Ld M n d N be coml)ld e K 晶 len u l可 olds of d面 ension m 皿 d n ( 2 ≦ g ≦
痢 yesl)ec面 dy.
Weasst4mel¥Hsof 筧昭d佃ecMymt匹eof oydey琲 泌 tk smceof S伍.
f) & α g 加だ回yy cas 加 d ゐ n z加 丿 討 面 肋 C゛ か ,2㎡ 叱回 ua; 加 凹 励 ひ
Ld
-
が 八 1) → N iSα
d而2口 lti岫lem司)折昭 such tk t八 ∂D isaC-R 泌tod珊eomo砂旨sm, thm theye isa smooth
大節i何程D→NSUd tk 八∂
D= 八aD, パsholom叫)Mc皿 D皿d几shomo幼id(汀副・
d加e ∂八
Rem ark.
T he assumption of pseudoconveχity of ∂7) can be weakened to somewhat
weaker convexity condition ; the trace of L evi form is non-negative everywhere on ∂且
K iyoshi Shiga
58
T he author w ould like to explT
ess his thanks to his colleague S. N ishikaw a for pleasant discussions w ith him .
2.
P reliminaries
ln this section we summarize definitions and known results that we need later.
Let yぼ and yV be Riemannian manifolds and μ M → N a differentiable mapping. T he
energy functional £ (/ ) isdefined by
£(/)=-y几tra
c
e
い/*ぺ)
VVe ca11y is harmonic if y is a critical mapping with respect to the above energy func-
tional.
Concem ing the Dirichlet problem of the hormonic mapping, the following
theorem holds.
THEOREM ( Hamilton 〔2〕 Schonen 〔5〕). Ld M 皿 d N becom瞬 teRiemamli皿 x m面 lds
a抑
⊂ M bea 副 d w ly com釦 d doma細 戚 th smooth bouM血 巧.
Weassume N is of m)ll-
-
知j ti碗 cun屈 ut
Tha a di加 ra 面 b19 回 卸i昭 穴 D → N is homotol)ic to α k ymo戒c
刑司)折昭 副 at加e ∂p.
F or further properties of hormonic mappings w e refere to Eells-L emaire 〔1〕.
N ow we define the notion of negative curvature of order ヵ in the sence of Siu.
DEFI NITION・ L et Af be a K ahler m anifold and R 命営 curvature tensor. yぼ isof strongly
negative (semi-negative) curvatureat 夕E yぼif
Σ R命営 ( A aBβ- C. p β)( A &B 7- Ca
ア) > O( ≧ O)
゜ β 7 ∂
_
_
E
lt g, w here ノ1α, 召 。, C 。, £) 。 are com pleχ num bers such that yl a召 β- C aD β十 O for at
least one pair of indices ( α, β ) 。
,
D
E
F
I
N
I
T
I
O
N
.
V
V
e
s
a
y
t
h
e
c
u
r
v
a
t
u
r
e
t
e
n
s
o
r
t
i
?
・
β
μ
i
s
n
e
g
a
t
v
e
o
f
o
r
d
e
r
ん
a
t
/
・
i
f
i
t
i
s
strongly sem i-negative at 力 and it enj oys the f0110w ing properties.
lf y j び → yぼ is a sm ooth m apping from an open neighbourhood び of θin C たto yぼ
4 j
R頑弧G(シ万G
一ア ー
・
ノ
V乙
with / ( ○ 二 /) and rank が = 2ん at a, and if at 夕
= O
for i ≦ も j
≦ ん
a, β, ア, δ
where
ぐ 言
二 ( aぬ ) ( (爪 石 訂 ( O) - ( aぷ ) ( O) ( び j ) ( O)
then either ∂/ = O at a or 訂 = O at a.
T he property of strong semi-negative curvature is stronger the property of non-positive curvature.
F or examples of manifolds of negative curvature of order ん see Siu
〔6〕 and Mostow-Siu 〔3〕.
W e define C-R mappings.
dimension s ≧ 2 .
Let S be a real hypersurface of a compleχ manifold 肛 of
H ( S ) 二 T 1・o ( 訂 ) IS n T ( S ) ⑧ C is a vector bundle of rank 肖-1,
where 7`1・o ( 訂 ) is the holomorphic tangent bundle of M
A n application of harmonic mopping to compleχ analytic geometry
59
DEFI NITION. L et yぼ an(1 N be comqleχ manifolds and S ⊂ yぼ be a real hypersurface
of M W e say a differentiable mapping; f : S・→ yVis a C-R mapping if が ( 耳 ( S ) ) ⊂ 7`1. 0( yV ) .
Let ♪be a point of S and { 0 , { w j be local coordinate systems at ♪and / ( /) ) .
represent y by the above coordinate system
as / =
( 石 , . . …. , 几 ). lf w e choose { 副
that ∂/∂
z1 (飴 ぷ …, ∂/∂
4 -1 (列 area baseof 亙 ( S 泌
We
so
and yis a C-R mapping then
∂/∂乱几 = O at ♪for ブニ1, ……, 撰- i , α= 1, ……, 爪
3.
P roof of the theorem
L et yぼ and yV be com plete K iihler m anifold of dim ension 聊 and 刄,
be local coordinate ststems of yぼ and Ⅳ.
y.
and g = ( g μ )
{ z, }
and { z4/ . }
be the K ahler metric on
W e consider a ( 1, 1) from 〈 島 ∂八 ∂/ 〉 on 訂 defined in terms of local local coor-
dinates
エ
〈
g ,
∂ 良
町
〉
=
W e denote by ω the K 油 ler from of な i
fundamental in the sequel ( c. f. Siu 〔6〕 ) .
倒
∂∂〈£ ∂八∂/ 〉 へωm-2=
where
c
ア
C
7ω ゛ -
Σ
ぬ
j
砿
μ
几
α, β
T he f0110wing Bochner type equality is
lf 八 訂 → N is a harmonic mapping, then
χ ω 771
な古よΣ瓦い 昌グ可
ぐげ =∂
げ。∂
むか ー∂
ソ
¯九
9∂
jT
X
9
and χ is sorne nonpositive function on A£
H ere 7? μ μ is the curvature tensor oI N 。
-
Let £) ⊂ yぼ be a bounded domain with C° boundary and ソT
: D→ N
ing such that y l∂£) is a into C-R diffeom orphism .
a smooth mapp-
B y H amilton-Schoen s theorem w e
may assume y is a harmonic mapping・
N ow we integrate (1) on 八
逞∂
ぷ
〈島
ノノ△
∂
/〉△
゜J゛¯2¯ffD(7(t
Since y is of negative curvature of order 琲 , we have
限∂
jく£j八∂
かA. -2≧O
T ake a point /・ ∈ ∂皿
and we choose a local coordinate system
{ら }
at 夕 so
that ∂/ ∂z1,……, ∂/∂4 -l span the holomorphic tangent space 耳 ( ∂7) )。 of £) at 九 and
{ ∂/∂ろ } isorthonormal at か Wedenoteby φa defining function of 7) , i. e. φisa
sm ooth function on D , Such that
フ:) = { φく O} and grad φ 中 O on ∂Z) .
φ so that ∂/ ∂Zi φ ( 列 = Oう こ 1, 。。…・, 衿1- 1, and a/ ∂4
I Ve choose
φ( 列 = 1 for brevity. W e also
choose a coordinate system 佃 。} at / り ) such that { ∂/ ∂ち 卜 s orthonorm al, and represent
y by this coordinate。
Since / ¦ ∂£) is a C-R mapping, by dired culculation we obtain ( c. f. 〔4〕 )
K iyoshi Shiga
60
一
一
(y
∂冷 φ
α= 1
Z= 1
yT
几∂
j〈島丿
広げ〉八
゜J
ご几1
2)p
o
s
itiv
e(2g-1)fo
rma
t九
)(Σ∂
瑶
J< g, ∂
八∂
/ > Aωl-2
。 j 〈£ ∂広 ∂/ 〉A°
J゛¯2
m- 2 -
by Stokes theorem.
y4
T he left hand side is non-negative, on the hand the right hand side is nonpositive if
∂i ∂・ φ is nonnegative, i. e. the trace of the Levi form of φ is nonnegative.
T hen c
y= O ori Af .
From the assumption of negative curvature of order 謂 oI N ,
we have 5f = Oor ∂/ = O at points of rank が = 2t
omorpfism, rank が = 2聊 on a neighbourhood of ∂刀.
= O or
∂ノ エ 0 .
have Jy = O.
Since / ¦ ∂D is a C-R diffe-
By thecontinuity, it hold ∂/
Sine / ¦ ∂7) satisfies the tangential C auchy
R iem ann
equation ,
we
T his com pletes the proof of the theorem .
R eferenCeS
〔1〕 J. Ee11s and L. Lemaire A report on harmonic maps.
Bu11. London Math. Soc., 10 ( 1978), 1-68.
〔2〕 R. S. Hamilton Harmonic maps of manif01ds with boundary.
Lecture note in Mathematics
N 0. 471, Springer, 1975.
〔3〕 G. D. M ostow and Y. T. Siu A compact Khhler surface of negative curvature not covered by the
baII.
A nn. of M ath.
112 ( 1980 ) 321- 360・
〔4〕 S. Nishikawa and K. Shiga On the holomorphic equivalence of bounded domains in complete
K iihler manifolds of nonpositive curvature.
〔5〕
preprint
R. M. Schoen Eχistence and regularity for some geometric variational problems. Thesis, Stan-
ford U niv. 1977.
〔6〕 Y. T. Siu The complex analyticity of harmonic maps and the strong rigidity of compact Kiihler
manifolds.
A nn. of M ath. 112 ( 1980 ) , 73- 111.
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