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A geometric characterization of Cn and open balls

Published online by Cambridge University Press:  22 January 2016

Kiyoshi Shiga
Kiyoshi Shiga*
Affiliation:
Gifu University
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The purpose of this paper is to give a result concerning the problem of geometric characterizations of the Euclidean n-space Cn and bounded domains. It is well known that a simply connected Riemann surface is biholomorphic to one of the Riemann sphere, the complex plane and the unit disc. And there are several results concerning the geometric characterization of these spaces. To show that some simply connected open Riemann surface is biholomorphic to the complex plane or the unit disc, it is sufficient to see that there exist non constant bounded sub-harmonic functions or not. But in the higher dimensional case, there is no uniformization theorem. By this reason to show that some complex manifold is biholomorphic to Cn or an open ball, we must construct a biholomorphic mapping directly.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 75 , September 1979 , pp. 145 - 150
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1979

References

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[3] Kobayashi, S., Transformation groups in differential geometry, Erg. der Math. 70, Springer, 1977.Google Scholar
[4] Milnor, J., On deciding whether a surface is parabolic or hyperbolic, Amer. Math. Monthly, 84 (1977), 43–46.Google Scholar