×

On continuable Riemann surfaces. (English) Zbl 0921.30031

A Riemann surface \(R\) is called continuable if there exists a conformal mapping of \(R\) onto a proper subregion of some other Riemann surface. \(R\) is call maximal, if it is not continuable. The author proves sufficient conditions for a Riemann surface to be continuable. Let \(R\) be a Riemann surface with Green function \(g_{p_0}(p)\) and let \(B(p_0,\alpha), \alpha >0,\) be the Betti number of \(\{p \in R: g_{p_0}(p)>\alpha \}\). If \(B(p_0,\alpha)\) satisfies some growth restriction as \(\alpha\) tends to zero, then \(R\) is continuable. As a corollary the author proves the continuability of an \(n\)-sheeted unlimited covering surface of the unit disc under certain growth restrictions on the branch points of this covering.

MSC:

30F20 Classification theory of Riemann surfaces
30F25 Ideal boundary theory for Riemann surfaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] HASUMI, M., Hardy Classes on Infinitely Connected Riemann Surfaces, Lecture Notes in Math., 1027, Springer, 1983. · Zbl 0523.30028
[2] JIN, N., On maximality of two-sheeted unlimited covering surfaces of the unit disc, to appea in J. Math. Kyoto Univ. · Zbl 0934.30031
[3] SAKAI, M., Continuations of Riemann surfaces, Canad. J. Math., 44 (1992), 357-36 · Zbl 0759.30019
[4] SARIO, L. AND M. NAKAI, Classification Theory of Riemann Surfaces, Springer, 1970 · Zbl 0199.40603
[5] SARIO, L. AND K. OIKAWA, Capacity Functions, Springer, 1969 · Zbl 0184.10503
[6] WiDOM, H., JVPsections of vector bundles over Riemann surfaces, Ann. of Math., 94 (1971), 304-324 · Zbl 0238.32014
[7] YOSHIDA, M., The method of orthogonal decomposition for differentials on open Rieman surfaces, J. Sci. Hiroshima Univ. Ser. A-I Math., 32 (1968), 181-210. · Zbl 0175.08102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.