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83: １３２人目の素数さん [] 2024/01/24(水)11:55 ID:ZkPFgyVs(1)
Let $O_G$ denote the set of Riemann surfaces which do not have Green functions. By generalizing Evans' theorem, it was proved by by Z. Kuramochi [K-1,2,3] that $R\in O_G$ if and only if $R$ admits an Evans potential, which Kuramochi calls Evans-Selberg potential in view of a work of H. Selberg [Sb] as well as [E]\footnote{[Sb] appeared a little later than [E].}. As a background of Kuramochi's work, one can mention [Nv] which initiated complex analysis on open Riemann surfaces by establishing that, given a plane domain $D$, $D\in O_G$ if and only if the Bergman kernel of $D$ is trivial. M. Ohtsuka [Oht] proved that $D\in O_G$ if and only if $D$ admits a nonconstant bounded superharmonic function.

\begin{definition}If $R\notin O_G$, a divergent sequence of point $p_n$ in $R$ is called \textbf{irregular} if there exists $p'\in R$ such that $\liminf_{n\to\infty}{g(p_n,p')}>0$. \end{definition}

Extensions of Picard's theorem have been obtained in order to describe the essential singularities of meromorphic functions near the sets of null logarithmic capacity (cf. [Km]).

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