Hyperbolicity of circular domains
Kazuo Azukawa · Tohoku Mathematical Journal · 1983
If a domain D in C n is hyperbolic in the sense of Kobayashi [5], then every holomorphic mapping from C into D is constant.In general the converse is not true.In this paper, we show that the converse holds if D is a strictly starlike circular domain in C n .More strongly, if D is a starlike circular domain in C n with D c \D for any real λ > 1, and if every C-linear mapping from C into D is zero, then D is bounded (Proposition 4.4).Geometrically convex, circular domains or complete Reinhardt domains in C n are strictly starlike (Propositions 4.2 and 4.3).Next we obtain equivalent conditions for a starlike circular domain in C n to be pseudoconvex (Proposition 5.1 and Theorem 5.4).Finally, modifying the example in Earth [2], we construct a nonhyperbolic pseudoconvex circular domain in C 2 into which every holomorphic mapping from C is constant (Proposition 6.5).In subsequent sections, we call a subset X of C n or R n convex, for brevity, when X is geometrically convex, i.e., {\x + (1 -λ)#; 0 < λ < l}c X for any x,yeX.The author would like to express his thanks to Professors T. Kuroda and M. Suzuki for valuable discussions during the preparation of the present paper.The author would also like to thank the referee for helpful suggestions, which led to the improvement of the original manuscript. Hyperbolicity of domains in Cn .Throughout this paper we consider the following conditions on a domain D in C n :(H. 1) D is bounded.(H. 2) D is biholomorphic to a bounded domain in C n .(H. 3) D is C-hyperbolic, i.e., the Caratheodory pseudodistance of D is a distance.(H. 4) D is Jf-hyperbolic, i.e., the Kobayashi pseudodistance of D is a distance.(H. 5) D contains no entire holomorphic curve, i.e., there does not