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.
© Copyright 2024 Paperzz