調和距離とケーベの定理

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(1)Title. 調和距離とケーベの定理. Author(s). 長田, 正幸. Citation. 北海道教育大学紀要. 第二部. A, 数学・物理学・化学・工学編, 30(2) : 97-100. Issue Date. 1980-03. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/6050. Rights. Hokkaido University of Education.

(2) Journal of Hokkaido University of Education (Section II A) Vol. 30, No. 2 March, 1980. »iiair^'Ê® (H 2 Â) ^ 30 ^ ^ 2 -?- Bgffi 55 ^ 3 ^. Harmonic Metric and Koebe' s Theorem. Masayuki OSADA Mathematics Laboratory, Sapporo College, Hokkaido University of Education, Sapporo 064. : eg IE^ : imît^—^î MMH:^'!NL?^MTOM. Abstract The purpose of this paper is to extend Koebe' s theorem to an analytic mapping of a hyperbolic Riemann surface into another hyperbolic Riemann suface as follows: Let R and R' be two hyperbolic Riemann surfaces, {an}nî and {bn}nî be two sequences on R tending to the ideal boundary of R such that Hz{an,bn)> 5> 0 for a fixed point z on R and for each positive integer n, where Hz denotes the harmonic metric on R—[z} introduced by N. Boboc and G. Mocanu ([!]), ffw/ \n be a Jordan arc which joins an and bn in R—[z}.Let R'* be a metrizable compactification of R' and ^'=R'*—R'. For each b' £ R'*, we set. Vn(b')={w\w^Rf,d'(w,b')^}{n=l,2,---), where d' is a metric on R'*. Set. As(7?/*)={^|&'eA; lim lvnw(w)< 0 for each w e7?'}. n-ôo. Let cf) be an analytic mapping of R into R'. If {^(/ln)}nSi converges uniformly to a point b' e R'*— A s( R'*}, then b' e R' and (f> is a constant mapping.. Introduction N. Boboc and G. Mocanu ([!]) introduced the notion of the hamonic metric on a hyperbolic Riemann surface R as follows. Let z be a fixed point on 7?. For a closed subest F of 7?—{^}, we difine lF{z}=mf{s(z)\s>0, superharmonic on R and s^l quasi everywhere on F}. For any a, b e 7?— {z}, we set. (97).

(3) Masayuki OSADA. HAa, b)=mt{lr(z)\r ^F a,^R-{z}}}, where F a,b(R—{ z}} denotes the family of all curves in R—[z} joining a and b. They showed that Hz is a metric on R— {z} and the topology induced by Hz is compatible with the original one on R—[z}. Furthermore they extended Koebe's theorem as follows.. THEOREM 1 (Theorem of N. Boboc and G. Mocanu). Let R and Rr be two hyperbolic Riemann surfaces, {an}nî and {bn}nî be two sequences on R tending to the ideal boundary of R such that Hz(an,bn) > <5'> 0 for a fixed point z on R and for each positive integer n, and \n be a Jordan arc which joins an and bn in R—[z}. Let <f) be an analytic mapping of R into R'. If {^(An)}nSi converges uniformly to a point on R', then ^ is a constant mapping. On the other hand, Y. Nagasaka ([2]) obtained the following very interesting result. THEOREM 2 (Theorem of Y. Nagasaka). Let ^ be an analytic mapping of a hyperbolic Riemann surface R into another hyperbolic Riemann surface R'. Let r:z=z(t), 0 ^t< 1 be an arc such that z(t) tends to the ideal boundary of R as t-'l and suppose lim 1 rn(R-Rn) ^0 n-»oo. on R, zuhere {Rn}nî is a normal exhaustion of R. Let R'* be a metrizable compactification of R' and /\'=R'*—R'. For each b' e R'*, we set. Vn{b')={w\w^R',df(w,bf)^}(n=l,2,-), where d' is a metric on R * Set. As=As(^7*)={6lyeA;lim ly^')W >0for each w e 7?/}. n-oo. //lim (/){z(t))=V e R'*-^s(R'*), then V e R' and ^ is a constant mapping. (-1. In this article, we shall extend Koebe' s theorem as follows.. THEOREM 3. Let R and R' be two kyperbolic Riemann surfaces, {an}nî and {bn}nî be two sequences on R tending to the ideal boundary of R such that Hân,bn}>8> 0 for a fixed point z on R and for each positive integer n, and \n be a Jordan arc zuhich joins an and bn in R-[z}. Let Rr* be a metrizable compactification of R' and ^'=R'*-R'. For each b' <= R'* we set. Vn(b'}={w\w^R',df(w,b')^}(n=l,2,-), where d' is a metric on R'* Set ^'s(R'*)={b'\b' eA;lim lvn(b')W > 0 for each w e 7?/}. n- oo. Let <f) be an analytic mapping of R into R'. //{^(An)}nî converges uniformly to a point b' <= R'*—^s(R'*), then b' <= R' and (f> is a constant mapping.. Preliminaries Let R be a hyperbolic Riemann surface. For a closed subset F of R, we denote by Ip. the lower envelope of the family of all positive superharmonic functions s on R with the property that s^l quasi everywhere on F. Let {Fn}nî be a sequence of closed subsets of. (98).

(4) Harmonic Metric and Koebe's Theorem. R such that Fn~=)Fn+^n =1, 2, •••). Set F=UFn. n=l. For a fixed ^(0< 5-<1), we set Qn{F:S}={z\zeR, lFn{z)^S}. If liml^Ô on R, then p| Q,n{F: S)^Q. Let^be a continuous mapping of R into a compact n-oo. n=l. metric space X. We define the following cluster sets:. ^(7^)=n ^Q^F-.SJFand <^^F)= U ^F:5). 0<S<1. We shall prove. LEMMA 1 (see LEMMA 1 in [2]). If limlÔ on R, then ^(7?)c^(F). n-*oo. PROOF. We set u=lim Ipn. Then u is a positive superharmonic function, 0<u^l on n^ oo. R and IF,, ^M on R for each n. Suppose z is an arbitrary point of R. If 0<u{z)<l, then z e Qn{F:u(z)) for each n. Then ^(z) 6 ^(Qn(.F:u(z))) for each n and therefore. (/,{z) e0 ^>{Qn{F:u{z)))=^{F:u{2))^ U ^(.F:^)=^(F).If u(z)=l, then z ^Qn {F:S) n=l. 0<5<l. co. and therefore ^{z} <= (f){Q.n{F:S}} for each n and each <5'(0< <5'<1). Hence ^(z) eQ ^(^». {F:S))=^F:^ d_U^F:S)=^^F). 0<5<1. Let R be a hyperbolic Riemann surface, 7?* be a metrizable compactification of R and A =R*- R. For each b e R* we set. Vn{b}={z\z eR, d(z,b)^^}{n=l,2,-), where d is a metric on R* Following Y. Nagasaka([2]) we consider the following subsets of 7?*: E(b}=E(b: R^=[a\a e R*\im Urn lv,w(z)>0}, Ei{b)=Ei{b:R*)={a\a e R* Any positive superharmonic function s on R with Ums(z) =+oo has always the poroperty lim s(z)=:+00}, z-a. E=E(R*)={a\a e A, There is not a barrier function at a} and A,=A,(7?*)={6|6£A, Hml>4,(,)(^)>0 for each ^ 6 7?}. The following lemma is due to Y. Nagasaka.. LEMMA 2 (see LEMMA 2 in [2]). (1) b Ê(b) for each b ^ R* (2) If b e7?*-As,then E(b)Êz{b). (3) If b e R, then £i(6)={&}. If 6 e A -As, then £:i(6)cA. If & e As, then. RÊ{b}. We shall prove LEMMA 3 (see PROPOSITION 2 in [3]). Let ^ be an analytic mapping of a hyperbolic Riemann surface R into another hyperbolic Riemann surface R' and let R'* be a metrizable compactification of R'. Let {Fn}n^\ be a sequence of closed subsets of R such that Fn^Fn+\ (n =1,2, •••). If {(f){Fn)}n^l converges uniformly to a point b' e Rf* where the closure is taken. in Rr* then cf>^F)Ê{b':R'*}.. (99).

(5) Masayuki OSADA. PROOF. We fix <5'(0<<5'<1). We have only to show that for each a eR'*—E(b') there exists a positive integer m=m(a') such that ^(Qm(F:S)) ~â. Suppose a <= R'*—E. (&'). Then there exists a positive integer no and a closed neighbourhood V(a') of d such that li^(y)<-j-on V{a')HR'. Since {^(Fn)}n2i converges uniformly to a point 6; there exists a positive integer m such that (/>(Fm) <^Vn,{b'). As l^)^lt/^(6/) on R', it follows that I^F^^- on V(a')HR'. Noting that IFÎ^F^^ on 7?, we see that (l^F^(/>){z)^S for each z ^Qm(F:9), so that 1^)^<5~ on ^(Qm(F:S)}. Hence (V(a)C~}Rf)n(/>(Qm(F:B))=Q So we see that a ^ <^>{Q m{F:S)), so that Q ^(Q,n{F:S))Ê{b') for each ^(0<<?<1). Thus we have ^^F)(ZE(br).. n=l. Proof of THEOREM Set Fn=U Ak and F=U An. Then Fn~^Fn+i{n=l, 2--) and F=U Fn. By Lemma 1. ^(7?)C^(.F). By Lemma 3 (/>^F)c.E{br, R'*). By Lemma 2 (1) E(b', R'*}(Êi(b', R'*). Hence ^(R)czEi{b', 7?'*). Assume 6/ e A/-As. By Lemma 2 (3) £i(6/)cA;so that ^(7?)CA: This is a contradiction. Hence b' e R'. By Lemma 2 (3) £i(6/)={6/}, so that ^>(R)={b'}. This completes the proof.. References [1] Boboc, N. and Mocanu, G. (1961), Sur la notion de metrique harmonique sur une surface riemannienne hyperbolique. Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine (N. S.) 4 (52), No. 1-2, p. 3-21. [2] Nagasaka, Y. (1977), A Lindelof type theorem on a Riemann surface. Hokkaido Math. J. Vol. 6, p.161-168.. (100).

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