Harmonic dimension of covering surfaces and minimal fine neighborhood.

*(English)*Zbl 0968.30020In this paper the authors study an aspect of harmonic dimensions of ends. An end \(\Omega\) is a compactly bordered noncompact Riemann surface \(\Omega\) having a single ideal boundary component \(\delta\Omega\) in the sense of Kerekjarto-Stoilow and having a compact border \(\partial\Omega\) consisting of a finite number of mutually disjoint simple closed analytic curves. The simplest end is the punctured unit disc \(D=\{0 <|z|<1\}\), where the border \(\partial D\) of \(D\) is the unit circle \(\{|z|=1\}\) and the single ideal boundary component \(\delta D\) of \(D\) is the origin 0. M. Heins showed in his celebrated paper [Ann. of Math., II. Ser. 55, 296-317 (1952; Zbl 0046.08702) that any bounded holomorphic function on an end \(\Omega\) has a limit at \(\delta \Omega\) like the case of \(D\) (the Riemann removability theorem) but the analogue of the principle of positive singularity for positive harmonic functions on \(\Omega\) (the Picard principle) may not hold, unlike the case of \(D\). To state this latter part more closely, let \({\mathcal P}(\Omega)\) be the convex cone of nonnegative harmonic functions on \(\Omega\) having vanishing boundary values on \(\partial \Omega\). Heins coined the term harmonic dimension of \(\Omega, \dim \Omega\) in notation, for the cardinal number of the set of extremal rays in \({\mathcal P}(\Omega)\), which is nothing but the cardinal number of the minimal boundary points lying over \(\delta\Omega\) in the Martin compactification of \(\Omega\) so that \(\dim D=1\) (the Picard principle). For each fixed integer \(1<p< \infty\) consider the class \({\mathcal E}_p\) of ends \(W\) which are represented as \(p\)-sheeted unlimited covering surfaces of \(D\). Heins noted that \(1\leq\dim W\leq p\) for each \(W\in {\mathcal E}_p\) and showed that both \(\dim W=1\) and \(\dim W=p\) can occur for a suitable choice of \(W\in {\mathcal E}_p\) by using the following particular \(W_1\in {\mathcal E}_P\) Consider two decreasing sequences \((a_n)\) and \((b_n)\) in \((0,1)\) satisfying \(b_{n+1} <a_n<b_n\) and \(a_n \downarrow 0\). Set \(G= D\setminus I\), where \(I=\bigcup_{n=1} ^\infty I_n\) with \(I_n=[a_n,b_n]\). Take \(p\) copies \(G_1,\dots, G_p\) of \(G\) and join the upper edge of \(I_n\) in \(G_j\) with the lower edge of \(I_n\) in \(G_{j+1} (j\bmod p)\) for every \(n\). The resulting \(p\)-sheeted covering surface \(W _1\) of \(D\) belongs to \({\mathcal E}_p\). Heins showed that if \(I_n\) are scattered on \((0,1)\) sparsely enough [resp. densely] toward 0, then \(\dim W_1=p\) [resp. \(\dim W_1=1]\). The latter result is the starting point of the study of the present authors. They gave in their earlier paper [Kodai Math. J. 17, No. 2, 351-359 (1994; Zbl 0811.30028)] the following precise version of the above Heins result: If \(I\) is thin [resp. not thin] in the sense of potential theory, then \(\dim W_1=p\) [resp. \(\dim W_1=1]\). The main purpose of the paper under review is to characterize \(\dim W\) for \(W\in{\mathcal E}_p\) as the maximum of the numbers of connected components of the counter images \(\pi^{-1}(M)\) of \(M\) under the projection \(\pi\) of \(W\) onto \(D\) when \(M\) vary among subdomains \(M\) of \(D\) such that \(M\cup\{0\}\) are fine neighborhoods of \(0\). This main result is then used to derive certain generalizations of the result of the authors on the above-mentioned \(W_1\). As another application of the main result it is also shown that for any \(1\leq q\leq p\) there is a \(W\in {\mathcal E}_p\) such that \(\dim W=q\). Two more results are appended. The first is that \(\dim W\) for any \(W\in {\mathcal E}_p\) divides \(p\) when the cover transformation group of \(W\) is transitive on each fiber. The end \(W_1\) is cyclic in the sense that the cover transformation group of \(W_1\) is transitive on each fiber and moreover cyclic. A natural question occurs of whether \(\dim W\) for cyclic \(W\in {\mathcal E}_p\) is either 1 or \(p\) as it is for \(\dim W_1\). The second result is to resolve this question in the negative: for each divisor \(q\) of \(p\) there is a cyclic \(W\in {\mathcal E}_p\) with \(\dim W=q\).

Reviewer: M.Nakai (MR 99g:30054)