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(1)On a Harmonic Function with Respect to the Bergman Metric By. Teiichi HIGUCHI* and Hiroshi SAKAMOTO"*. gO. Introduction. ･. In this paper, we find a harmonic function in a open unit ball except the origin with respect to the Bergman metric and using this function we count the number of zero points for non-degenerate holomorphic mapping having only isolated zero points.. THEoREM 1. Let M be a ztnit ball in C" and d denote the LaPlacian oPerator on M zoith resP2ct to the Bergman metric. Then there is a fzanction f with af==O.. THEoREM 2. Let zu be a non-degenerate holomorPhic maPPing ofaunit ball mto zt. Mle assum that w has only isolated 2ero Points. I?Ve Put f= 2n+i((".lii).!-i.n (iog]212+ ZII + ･･･ +('feile)l-i (.(-nil)i!)! [i2k. + ... + (n-.lln IN.121(n-i) )'. Then we have Si,i =i Clyf--7i. and S],T...ZV*(*df)==nzaMber of 2ero Points in {l2I<r}.. gl. Theorem. Let M be a complex manifold of dimension n, AP,q the sheaf of COO(p, q)forms on･M and T(AP,q) all sections of AP'q over Ml We introduce an Hermitian metric 2Zgj.pLd2,at･d2-P,･ on M. Associated to this metric we have forms. to=iZg.,,gd2evAd2t9 and to"=2"n!gdxiA･･･Adx2n, where 2a==x2a-i+ix2ec and g=det(g.p-). We denote the inverse (g.p')-i of (g.p-) by (gaP)=(g.p-)-i, that is,. 7gdv`9gB7=65 and ¥g.RgRr==6}. VXre denote as follows: * Department of Mathematics, Faculty of Education, Yokohama National University.. ** Department of Mathematics, Tokyo University of Education..

(2) 16 T. HiGucHi and H. SAKAMoTo Ap=(ai,･･･,crp),ai<cr2<･･･<crp, 1;Sa,:;ln, An-p==(crp+i,''',cr.), ap+i<･･･<a., ISai:Sn. and (cri,･･･,apu,crp+i,･･･,crn) is a permutation of (1,2,･･･,n). Then with this notation we write a (P, q)-form. {b = Z 2 9)ApRqd2AP A dzBq, where dzAp==dza'iA ･･･Adzevp and d2B-q==dz-PiA A d2-t3q. We denote. r. gbApBq =ZgttiZi ･･･ gctP2PgptiP-i ･'' g;tqPq￠2r.2pptr/tq･. Then. ,. S5 =ZZ9b-ApB-,d2""Ad2Bq ==]2]Z(-1)Pqs5ApB-,d2BqAd2"p. ==22(sb)B,z,d2BqAdg"p.. Thus (95)Bqzpu=(-1)Pq￠ApBq' We define the operator * from T(AP,9) to J'(An-q,n-p) by *sb=(i)"(-1)-l;"("-i)+PnZ2gAqA...qB-pB-.-p￠E."Aqd2An'qAdzBMn-p, Where gApAn"pgqB-n-q=ga･i･･･a･.p-i･･･p'.=det(gcr,p-j･). In particular. *1 == ing(- 1)Sn(n-i). **==(-1)pqJ on T(AP,q). Let put. 5= -*d*. ' a == dS+Sd. For a Coo function f we have 6f =O. Hence we have Af == 6df= -*d*df= -*d*( 60.f. dza'+ oO.-fp d2-P) .,, -*d{in(-1)en(n-i)+n+P-iggRa oO.f. d2Adz--iA ･･･ AdA.-PA ･･･ Ad2-n ,. +in(-1)Sn(n-')'a-'ggRev dOzi dgiA ･･･ Adt?aA ･･･ Adz"Ad2-} .. -,{in(-1)S"("-i' oO...p (ggp'ev dO.f. )d2A d2-. ,. +in(-1)S"("-i' 60.. (ggRa' oO.i )dzA d2d"} - -{22gpcr o.dO,26fm. +2( Oog.-P,cr + g3a' oO.g.,) oO.f. +z( Oog.R.S + gScr dO.g.) aO.-f,. where dz =d2iA ･･･ Adz".. }･.

(3) OnaHarmonicFunctionwithRespecttotheBergmanMetric 17 In this paper, in particular, we take M to be a unit ball in C", that is,. M:=: {zE CnlIzI2: =1ziI2+ ･- +I2fn12< 1} . ufic:otnrOi?UCoef tMheisBergMan metric in M as foilows: As the Bergman kernei fW. n! 1 K= zn a-l2;tl2)n+i' we have gcrp-= o.2S.-p iogK= (iII"S,), ((i-E2E2)s.,3+z-cr2p),. where S.p is Kronecker's delta. Then theassociatedtwo form with respect to. this metric is ' co ==i2 g.Rdzcr A dzP. and w"/n! is the volume element of ML We choose the Bergman metric as the. metric (g.ph). We have the inverse matrix of (g.-p-) . (gPcr) == ( iZi.is, (s.p-z-p2cr)) '. and. g=det(g.p-)=(1(m"i,1,)[,").det(I+lgai.Pl,)) . = a(-ni2i)t 2n)n 2 sign (i 111 ni.)(5i,,+ il iil2) ･･･ (6ni.+ ilnC:2). ' ''' 6ini,･. = (i(-nl;)i 2n). 2 sign (i 111 l.)tr., (i-lxi2)k 2"'i2"i ''' 2-"j'fef'-k6i･k.,i,-k+, = (1(-"S.1)l ,"). tr., (1-l.l,), i! z sig. (li ill ]i.2)z-ii.j'i･･･ .-ile.j'le. = (1(."i.1)1 2")n tl., (1.l.s2)k kl! 2 sign (l･i, IIiii.:)z-iiz"i ･･･ z-ikzj'le '. = (1(-"F?S)l #)n {1+ 1-l.l2 2z-`z`}. m (n+1)n. (1-l2[2)n+i. and :}]( Oog.-Pia + {a oO.-gp )== :i)( .-+2Pi (s.p-2-pza)- .li (i-lz[2)2a. ' ' +(1(7,t21i)2.).","2(s.p-2-p2a)(n+1)n(1(lllli)2Z)Pn+2> ' == n'l1 (-zcr+I2[22a-n(1-1zI2)2ev+(2ak]2za) == o. Then we have.

(4) 18 '･ T.HiGucHiandH.SAKAMoTo Af= -22gpcr o.-Opgf.. ., A function f with Af==O on M is called a harmonic function on M. Mo is a compact complex manifold without its boundary, then harmonic functions on Mo are only constant functions.. 02log i2l2 Aloglzl2=ZgPa o2-po7a-. '. 1-[z12 = -2 (n+1)1214 2(Sa･p-2-`9za)(S.,g k12-2crz-t3) .. = -2 (nlill)i L214 :(Sapl2i2-Sa`3z"2-P'5a,3z-PzcrIzl2+1zcrl2l2Pl2). = -2 (.lill)i L214 (n[2i2-Izt2-[z1`+12l`) '. '. ' 1-[212 n-1. ==-2 n+1 l212' a kiL2le = T2i2]gp-a o.-pOii.a l.it,le. == -2 (ff4, (i)iilg,l2.),, ￡(6.,,-2-pza)(6.,,I212-(fe+1)2cr2p). '. ' =-2 (;i±llll)f2i(L:)2) z(6cr,gI212-(le+1)Sa･p2cr2-P -S.plz12z-t9za+(le+1)12a]2lzP12) = -'2 (;IItilil)fii,I2.),, (nlzl2-(k+1)sz12-lzl`+(fe+1)1z1`). '. '. -le2 -le(n-k-1) +1-I212 =-2(1-1212 n+1 I2I2(le'i' n+1 l2]2k)' Hence A(loglzl2+ ZIEII ,lm+ ･'' + (-lele1)!le-' (.(-"lilli)iT)!l.l2le + ''' + (.--1)ln 1.121(n-i) ). = o. ,. Therefore log [2L2+ 4I fi 21' t ''' + (.'.1)ln 1.121(n-i). '. is a harmonic function with respect to the Bergman metric in ML. Secon'dlyweculculate ' *d(iog l'zl2+ k-1 2i + ･'･･ + (Mlelei)!le-" (.(lliZll{ll)i!)! kl2k' + ''' + (.--il" l.I2i(n-i) ). ==(Ei2" ?2'l 41 + '" + (il)k (.(-"i!)iT)!1.121ik+i) +(-1)nviLtl2. )*d1z12..

(5) OnaHarmonicFunctionwithRespecttotheBergmanMetric 19 Since. *diz[2. =*(2(z-"d.7. a'+2crdz-ev)) . t =in(ne1)S'i`"-i""z{((ln.+1lz)l",ff).i(s..,3-z'P2cr)2-nyd2ad.7iA･･･Ad12crA･･･Ad?'crA･･･Ad2-n '. . -((in-+Ei.)i",-)i.(6.,3-zPz-a)zcrd2-a'd2iA･･･AdA2evA･･･AdAz-evA...Ad2-n}. =((ln-+11.)]",-)i.in(-1)nz((s.,p-2-Pzev)2-avdxa' .. , -(S.,,g-zP2-a)zad2-cr)(dZiAd2Mi+(ii/li)di"Ad2-n)n-i == (i(ZSi),")M.i-, in(('.l)in)"l-i (z*dx-d2*･z)(d2*･dz)n-i,. we have. '. .. '. *d(iog12[2+ '7sl 2i + ''' + (Pkife)l-i (.(-"il)i!)!182le + ''' + (.-.il" l.i2i(n-o ). ==(li2' 72-1 4i + ''' + (i l,)k (n(llXli)i')! li 2le +(-i)nL-iI.l2n). '(i(4i.l)2gli f".(ii))! (z"dz-dz*2)(dz*d2)n-i = (ii),".Mi(i-Ez]2)n-i (i(4;2),n)-.i-, (Zin!-S)./ (2*dx-d2*2)(dz*dz)n-i. = ?'nn(-Mll))n! (n+1)n-i ]S,. (2*dz-dz*z)(dz*d2)n-i.. On the other hand. '. '. Sl,1=,1212n(2*d2'd2*2)(d2*dz)n-'==S1,ts.,2(dz*d2)n == 2S(.b2+ +(.2.)2sn!2nindxiA ･･･ Adx2n .. 2n+iinzn. F. Let us put ･. '. f= 2n+i((nnii))n!-iTn (iog[zl2+ kii-+ ･･･ + (n--ii" E212i(.ei) ).. Therefore we have 'S Tz, =," df == 1. and S,,E=,ZV*("df)=the numbers of zero points of w in {;21<r} ..

(6) 20 T. HiGucHi and H. SAKAMoTo ･ References [1] BERGMAN, S., The kernel function and confofal mapping, American Math. Society, Math. Surveys number V, Second (Revised) Edition, 1970. [2] BoTT, R. and CHERN, S,S., Hermitian vector btmdles and the equidistribution of the zeroes of their holomorphic sections, Acta Math,, 114, 71-112 (1965).. [3] HiGucHi, T. and TsuBoi, M., On the exactness of a Kodaira-Spencer sequences on complex spaces, Math. Zeitsch., 116, 331-337 (1970).. [4] KoDAiRA, K., Holomorphic mappings of polydiscs into compact complex mani-. v. folds, J. Differential Geometry, 6, 33-46 (1970).. [5] LEviNE, H., A theorem on holomorphic mappingsinto complex projective space, Ann. Math., 71, 529-535 (1960). [6] MATsusHiMA, Y., Differentiable manifold, New York:Marcel Dekker, Inc., 1972.. .. [7] MoRRow, J. and KoDAiRA, K., Complex manifolds, New York: Holt, Rinehart and Winston, Inc., 1971. [8] STEiN, E.M., Boundary behavior of holomorphic functions of several complex variables, Princeton Mathematical Notes, Princeton, N.J.: Princeton University Press, 1972.. `. ).

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