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Title
A geometric characterization of open balls
Author(s)
SHIGA, Kiyoshi
Citation
[岐阜大学教養部研究報告] vol.[16] p.[137]-[139]
Issue Date
1980
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/123456789/47495
※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。
137
A
geometric characterization
of open balls
K iyoshi Shiga
Dept.
0f M ath. , Fac.
of Gen.
E duc., Gifu U niv.
( Received Oct. 13, 1980)
R. E . Greene and H 八Vu proposed the problems to characterize C
and bounded
dom ains of C by means of curvature of a K ahler metric in [ 1 1 . Y . T . Siu and S. T .
Yau [ 4 ] and Greene and べ
Vu [ 2 ] gave some answer to the problem of the characterization of C .
T hey constructed coordinate functions by £ 2-m ethod.
ln contrast to the case of C , w e have no method to produce non constant bounded
holomorphic functions as yet.
K . Shiga [ 3 ] gave an answer under a condition that a
K ahler lT
netric has a very strong sym metry at som e pointL
ぺ
Ve call these K ahler
manifolds K ahlerian models.
ln this note, we consider the case 巾 at the com pleχ structure has a very strong
sym m etry at som e point.
1.
D ennitions and know n results.
L et 訂 be a non compact K iihler manifold and θ be a point of 訂 . W e call ( M , 0 ) a
K ahler m anifold with a pole θ, if the eχponential mapping exp : μ 。→ 訂
is a diff-
eom orphism , where M oi s the tangent space of 訂 at θ・
べ
Ve consider the hermitian inner product on 訂。 induced from the K ahler metric on
訂.
W e denote by び ( 訂。) the unitary transform ation group of 訂。 with resped to this
inner product.
DEFI NIT ION. A K ahler m anifold w ith a pole( 皿 θ) is a 尺涌 /雨 a
g ㎡ d iff every
φ6び( 訂。) isrealized asthedifferential of an isometry φof 皿 i.e. φ0 ) = aand φ。。= φ
Let ( M,0) be a Kiihlerian model and y bethedistancefunction from θ。 ∂= gmd y
is a vecter neld on 訂 ― 佃 } . W e call a sed ional curvature of the com plex plane spanned
by ∂ the h010morphic radial curvature. Since ( 訂 / ) is a K ahlerian model the h010m orphic radial curvature is a function of , ・, and we denote it by 尺 ( 杓 .
THEOREM (K. Shiga [ 3 ]) .
Ld ( M,oう be α K池leyian 枇d d.
Tha M is bi-
holomo呻 hic to C゛ oγ tke ol)m bd l. FMytk ymoye
mo呻hic toC1.
(1
) び£(rハマこjT
7粕γla塚e y, ty)k R≦M is biholoかy
( 2) び尺( y) is 厭)石″卸si面e n d K (、
la噌ey, M is biholomor
〃
-
j
w
メ
・
●
1+ ε
戸lOg γ
油ic to the ol)m bd , uJk ye E is some 卸si面 e 印 7XSだZが .
ln the following we need a theorem of Greene and χVu.
138
K iyoshi Shiga
THEOREM( Greene and W u [ 2 ] ) . Let ( 肌 a) & α1-d面 e面 ou l K哉 ley lu n面 ld
面th a l)ole, alld uJe de, lote by r tk distallce 血れd 011 斤om o. が the ct4nJature is 710れ
t 二 _ ?j r ..
7
y7
, y
如siti叱皿 d smd ey tk 肴
1
+
£
戸
γ210g y
寸
・
〃
●
¶
●4
4
. 4
●
-
-・
●-
ミー
ミ
ニ
粕川 a曙ey, M is biholom砂hicto tk 皿 it disk, wkeye
E is some 知 si面 e coMst皿 t.
2.
T heorem and its proof.
Let ( 肌 a) be a ,x-dimensional Kiihler manifold with a pole a. W edenoteby び佃 )
the unitary group of degree 筧. N ow we assume the isotropy subgroup /1M ( 訂 ) 。 at θ of
the holomorphic autom orphism group /1耐 ( 訂 ) of 訂 contains び佃 ) . M ore pred sely, w e
assume that there is a faithfull continuous representation び佃 ) to /1耐 ( 訂 ) 。.
W e denote by 面 the H aar measure on び佃 ) , and 冶 2 the K iihler metric on 訂 . W e
define a new K iihler m etric・on 訂 by 虜 2= ん (。) ( g * 冶 2) 面 .
T hen び佃 ) operates on な
isometrically with respect to 必 2, so ( 肌 O is a K iihlerian model with respect to the new
metric 虜 2.
THEOREM.
£d ( 肌 a) be α n-d加 e面 o回 I K晶 ley m四面 ld 面 tk a 図 e.
仔服
isotyo防 s油 gyo呻 d o of tke holom叫 )hic a tomo印池 m 訂 o呻 of M co戒a泌s U ( 、
n) 、the11
M is biholom砂hic to C゛l oy tk
FI㎡ keγm oye if tk
1+ ε
tk 11
y210g r
ol)ell bd .
holomo砂 hic sed oud
clu m h傀
is 歓)11 卸 s伍詑 α11d smぶ ley
知y lα塚e y, M is biholom叫 )hic to 服 ol)a bdL
PROOF. A s is m entioned above, ( 叱 a) is a K ahlerian model with respect to the new
metric j g2.
So 訂 is bih010m orphic to C
or the open ba11. F urtherm ore for a l -dim en-
tional complex subspace H。 0f j も , /7= expぷ /7. 1s a complex submanif01d and bih010m orphic to C or the open disk ( c.f. [ 3 ] ) . A nd if /Z is biholom orphic to C ( resp. 0pen disk) ,
then 訂 is biholomorphic to C ( resp. the open ba11) .
N ow we consider yyin 訂 with the original metric ゐ 2. XVe denote by y the distance
function from θin μ , and F the distance function from θin /7 with resped to the induced
metric. Clearly 7 ≧ r on /7. 0 n the other hand the Gaussian curvature on /7with resped
to the induced m etric is sm aller than the corespQnding holom orphic sed ional curvature
oI M .
T hen
the Gaussian Curvature on // ≦ the holom orphic sed ional curvature of thetangent
plane of H.
K
-
く
ー
1+ ε
-
-
戸IOg γ
1+ ε
戸IOg 戸
for large y
for large jモ
So by a theorem of Greene and W u yy is biholomorphic to the open disk.
T hen 訂 is
biholomorphic to the open ba11, and the proof is completed.
R eferences
[ 1]
R.E. Greeneand H.χ
Vu
Analysisonnoncompact Kahler manifolds, Proc. Symp. PureMath.
A geometric charad erization of open balls.
139
V ol. 30, A . M .S. Providence R .I . ( 1977) , 69-100.
[ 2]
R.E. Greeneand H.χ
Vu
M ath.
[ 3]
699.
K. Shiga
Function theory on manifoldswhichpossessapole. Lecturenotein
Springer 1979.
A geometriccharaderizationof C andopenbaIIs. NagoyaMath. J. Vol. 75( 1979) ,
145-150.
[ 4]
Y.T. Siu and S.T. Yau
than quadratic decay, A nn.
Complete Kahler manifolds with non positive curvature of faster
of M ath.
105 ( 1977) , 225-264.

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