Analytic functions in a lacunary end of a Riemann surface
Annales de l'Institut Fourier, Tome 25 (1975) no. 3-4, pp. 353-379.

Soient G un “end” d’une surface de Riemann parabolique R et G ′ un “end” lacunaire à l’ensemble fermé F=G-G ′ . On étudie des fonctions minimales dans G et G ′ , et on montre que G et G ′ jouissent des propriétés similaires lorsque F est distribué effilément sur la frontière idéale. On discute le comportement de fonctions analytiques dans G ′ , la relation entre l’existence de fonctions analytiques de certaines classes dans G ′ , et la structure des points de la frontière de R.S. Martin sur l’“end” G. On montre aussi que l’existence des points compliqués de la frontière de R.S. Martin ne permet que des fonctions analytiques violentes à exister dans G ′ si F est très effilé à la frontière idéale de R.

Let G be an end of a Riemann surface with null boundary and let G ′ be a lacunary end with a closed set F=G-G ′ . We study minimal functions in G and G ′ to show that G and G ′ have similar properties if F is thinly distributed on the ideal boundary. We discuss the behaviour of analytic functions in G ′ and relation between the existence of analytic functions of some classes in G ′ and the structure of Martin’s boundary points over the end G. Also we show that the existence of complicated Martin’s boundary points allows only violent analytic functions to exist in G ′ , if F is very thin at the ideal boundary of R.

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     title = {Analytic functions in a lacunary end of a {Riemann} surface},
     journal = {Annales de l'Institut Fourier},
     pages = {353--379},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {25},
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Kuramochi, Zenjiro. Analytic functions in a lacunary end of a Riemann surface. Annales de l'Institut Fourier, Tome 25 (1975) no. 3-4, pp. 353-379. doi : 10.5802/aif.589. http://www.numdam.org/articles/10.5802/aif.589/

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