Surfaces carrying no singular functions. (English) Zbl 1193.30058

Let \(X\) and \(Y\) be two Riemann surfaces, and let \(\gamma\) be a slit commonly contained in both. A Riemann surface \(Z= X\uplus Y\) is obtained by pasting \(X\setminus\gamma\) to \(Y\setminus\gamma\) crosswise along \(\gamma\). For \(j\in J= \{1,2,3,\dots,m\}\), let \(W_j\) be an open Riemann surface. If \(J'= \{j_1,j_2,\dots, j_m\}\) is a permutation of \(J\) and \(Z_1= W_{j_1}\uplus W_{j_2}\) for a common slit \(\gamma_{j_1}\) in \(W_{j_1}\), set \(Z_2= Z_1\uplus W_{j_3}\) for a common slit \(\gamma_{j_2}\) in \(Z_1\) and \(W_{j_3}\). Continuing, the surface \(Z_{m-1}= Z_{m-2}\uplus W_{j_m}\) is obtained.
Neglecting how the permutation and the sequence of pasting slits are chosen, a resulting surface, called the united surface of the \(W_j\), is formed. The authors show that there is a canonical isomorphism between the space of harmonic functions on the united surface and the space of harmonic functions on the bunched surface \(\bigcup_{j\in J} W_j\), assuming all the \(W_j\) are hyperbolic surfaces. An application of this result gives a sufficient condition for an afforested surface to belong to the class \(O_s\) of hyperbolic Riemann surfaces which have no nonzero singular harmonic functions for which its plantation and trees on it are all in \(O_s\).


30F20 Classification theory of Riemann surfaces
30F15 Harmonic functions on Riemann surfaces
30F25 Ideal boundary theory for Riemann surfaces
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