On boundedness of circular domains
Akio Kodama · Proceedings of the Japan Academy Series A Mathematical Sciences · 1982
Introduction.The main purpose of this note is to prove the fol- lowing assertions" ( I ) The classification problem for generalized Siegel domains in C C in the sense of Kaup, Matsushima and Ochiai [3] can be com- pletely reduced to that for bounded circular domains in C , where N m-t-1 (Theorem 1) (II) Let D be a starlike circular domain in Cn.Then D is Kobayashi hyperbolic if and only if it is a bounded domain in C (Theorem 2).(I) is a supplement to our previous papers [5], [7].(II) gives a partial affirmative answer to the following fundamental problem in the theory of hyperbolic manifolds" If D is a domain in C and it is hyperbolic in the sense of Kobayashi [4], then is it true that D is holo- morphically equivalent to a bounded domain in C? Recently, Barth [1] obtained an affirmative answer to this problem in the case where D is a geometrically convex domain..The author would like to express his thanks to his colleague K. Azukawa who presents him an interesting example in 3.