# Analytic functions in a lacunary end of a Riemann surface

Annales de l'institut Fourier (1975)

- Volume: 25, Issue: 3-4, page 353-379
- ISSN: 0373-0956

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topKuramochi, Zenjiro. "Analytic functions in a lacunary end of a Riemann surface." Annales de l'institut Fourier 25.3-4 (1975): 353-379. <http://eudml.org/doc/74253>.

@article{Kuramochi1975,

abstract = {Let $G$ be an end of a Riemann surface with null boundary and let $G^\{\prime \}$ be a lacunary end with a closed set $F = G-G^\{\prime \}$. We study minimal functions in $G$ and $G^\{\prime \}$ to show that $G$ and $G^\{\prime \}$ have similar properties if $F$ is thinly distributed on the ideal boundary. We discuss the behaviour of analytic functions in $G^\{\prime \}$ and relation between the existence of analytic functions of some classes in $G^\{\prime \}$ 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 $G^\{\prime \}$, if $F$ is very thin at the ideal boundary of $R$.},

author = {Kuramochi, Zenjiro},

journal = {Annales de l'institut Fourier},

language = {eng},

number = {3-4},

pages = {353-379},

publisher = {Association des Annales de l'Institut Fourier},

title = {Analytic functions in a lacunary end of a Riemann surface},

url = {http://eudml.org/doc/74253},

volume = {25},

year = {1975},

}

TY - JOUR

AU - Kuramochi, Zenjiro

TI - Analytic functions in a lacunary end of a Riemann surface

JO - Annales de l'institut Fourier

PY - 1975

PB - Association des Annales de l'Institut Fourier

VL - 25

IS - 3-4

SP - 353

EP - 379

AB - Let $G$ be an end of a Riemann surface with null boundary and let $G^{\prime }$ be a lacunary end with a closed set $F = G-G^{\prime }$. We study minimal functions in $G$ and $G^{\prime }$ to show that $G$ and $G^{\prime }$ have similar properties if $F$ is thinly distributed on the ideal boundary. We discuss the behaviour of analytic functions in $G^{\prime }$ and relation between the existence of analytic functions of some classes in $G^{\prime }$ 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 $G^{\prime }$, if $F$ is very thin at the ideal boundary of $R$.

LA - eng

UR - http://eudml.org/doc/74253

ER -

## References

top- [1] Z. KURAMOCHI, On minimal points of Riemann surfaces, II, Hokkaido Math. J., II (1973), 139-175. Zbl0277.30015MR51 #8408
- [2] M. BRELOT, Sur le principe des singularités positives et la topologie de R. S. Martin., Ann. Univ. Grenoble, 23 (1948), 113-138. Zbl0030.25601MR10,192b
- [3] Z. KURAMOCHI, Mass distributions on the ideal boundaries of abstract Riemann surfaces, 1., Osaka Math. J., 8 (1956), 119-137. Zbl0071.07303MR18,120f
- [4] M. NAKAI, Green potential of Evans type on Royden's compactification of a Riemann surface, Nagoya Math. J., 24 (1964), 205-239. Zbl0148.09801MR31 #336
- [5] Z. KURAMOCHI, On the existence of functions of Evans's type, J. Fac. Sci. Hokkaido Univ., 19 (1965), 1-27. Zbl0168.09101MR41 #461
- [6] Z. KURAMOCHI. Analytic functions in a neighbourhood of boundary, Proc. Japan Acad., 51 (1975), 320-327. Zbl0318.30019MR51 #8409
- [7] Z. KURAMOCHI. Analytic fonctions in a neighbourhood of irregular boundary points, Hokkado Math. J., V (1976). Zbl0319.30020

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