Title A geometric characterization of open balls Author(s) SHIGA, Kiyoshi Citation [岐阜大学教養部研究報告] vol. p.- 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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