The role of compactification theory in the type problem (Q934308)

A parabolic Riemann surface \(R\) is characterized by the nonexistence of Green functions on \(R\), and denoted by \(R\in O_G\). A Riemann surface not in \(O_G\) is said to be hyperbolic. For a covering surface \((R,S,\pi)\), where \(R\) and \(S\) are Riemann surfaces and \(\pi:R\to S\) denotes the projection, the (generalized) type problem is to determine whether \(R\in O_G\) or not. This problem is meaningful when the base surface \(S\) is parabolic and the covering surface is infinitely sheeted. In this paper the author constructs an infinitely sheeted covering Riemann surface \(S_{\Gamma}\) of a parabolic Riemann surface \(S\) as follows. Let \(S_n\) \((n\in \mathbb{N})\) be a replica of \(S\), and \(\Gamma\) the family of simple arcs \(\gamma_n\subset S\) with \(\gamma_{n-1}\cap \gamma_n=\emptyset\), where \(\gamma_0\) denotes a simple arc in \(S\) disjoint from all other \(\gamma_n\) for sufficiently large \(n\in \mathbb{N}\) and the end points of \(\gamma_n\) tend to \(\infty\), the Alexandroff point of \(S\). The surface \(S_{\Gamma}\) is then obtained by joining \(S_n\setminus(\gamma_{n-1}\cup\gamma_n)\) to \(S_{n+1}\setminus(\gamma_n\cup\gamma_{n+1})\) crosswise along \(\gamma_n\) for each \(n\in\mathbb{N}\). The author shows that the parabolicity of \(S_{\Gamma}\) implies that the sequence of capacities of \(\gamma_n\) relative to \(S\setminus \gamma_0\) converges to zero provided that \(\Gamma\) is monotonically disposed toward \(\infty\). The main result of this paper is to show that \(S_{\Gamma}\in O_G\) if the sequence of capacities of \(\gamma_n\) relative to \(S\setminus \gamma_0\) converges to zero so rapidly as to satisfy the sum of the sequence is finite for one and hence for every choice of admissible arc \(\gamma_0\) in \(S\). The proof is done by contradiction. If \(S_{\Gamma}\) is not in \(O_G\), it is proved that the projection \(\pi_{\Gamma}\) is not a Fatou mapping as a result of the more general situation that if an infinitely sheeted covering surface \((X,Y,\pi)\) is complete and sparsely branched, where \(X\) is hyperbolic and \(Y\) is parabolic, then \(\pi\) is not a Fatou mapping. If \(S_{\Gamma}\) is not in \(O_G\), it is also proved that \(\pi_{\Gamma}\) is a Fatou mapping by finding a compact nonpolar subset \(E\) of \(S\) such that the balayage of 1 in \(S_{\Gamma}\) with respect to \(\pi_{\Gamma}^{-1}(E)\) is a potential on \(S_{\Gamma}\).