## 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 $$O_s$$.

### MSC:

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

### Keywords:

Riemann surface; afforested surface; hyperbolic surface
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### References:

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