Kuramochi Boundaries of Riemann Surfaces: A Symposium held by Fumi-Yuki Maeda, Makoto Ohtsuka

By Fumi-Yuki Maeda, Makoto Ohtsuka

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Extra resources for Kuramochi Boundaries of Riemann Surfaces: A Symposium held at the Research Institute for Mathematical Sciences, Kyoto University, October 1965

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Then { p E B ~ ; analytic p ~ G} Is a G 6 subset of B a. (li) S = {pEB~, Proof. a diam M(f(p)) > 0} is a Gsc subset of B a. We shall prove our lemma In case a = N. for a = K is quite analogous. The proof For any fixed z E R, N(z, p) Is a continuous function of p on B N and R_RI_GN(Z , p) Is lower semi- continuous on B N. a G~ set. Since N Therefore {pEB~; {p~BN; N(z, p) = R_RI_GN(Z , p)} Is p ~ G} Is the intersection N of the above set wlth B 1 and since B 1 is a G 6 set, we have assertion choose a sequence (I).

P). compact set whose interior contains Q, - then (NA( -, Q))~(P) w 4. = NA(p, Definition Q) for P ~ R - K. of ideal boundary Let D be a non relatively boundary continuous satisfies function M(P, Q ) o n compact M(P, Q) is a positive {P}U ~DCK Let ~(D) element of nowhere in R. be the set of all kernels If M(P, Qj) converges as J § ~, then at P = Q. compact set K in R of points to a harmonic in D clustering function {Qj} will be called a fundamental If the limiting harmonic functions locally sequence of two {M(P, Qj)} and {M(P, Qj)} are equal to each other, {Qj} and {Q~} are equivalent equivalence point of D.

Norm among the functions If r ~ 1 on ~K, by ~(P; K, 2). properties K, ~) ~ ~ and I) Then there exists K, ~) such that can be proved easily (I = i, 2) be given continuous ~(r K, 2) which vanish We shall denote h(P) by r The following r ~(r 1 In [3]) We see that h has the smallest r of functions by set In 2. ~. Then we can prove In compact ~(r functions (Cf. [3]): Let on 3K such that K, ~) ~ ~ (I = I, 2). K = al(r K + a2(02) K for any real numbers aI , a2 9 then 2) 0 < ~(P; K, ~) ~ i for P ~ ~ - K.

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