1965/01/01 by Joji Kajiwara · 1 citation
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Quantum chaos and dynamical systems #Advanced Algebra and Geometry
paper · pdf · doi:10.2996/kmj/1138845123
openalex publication_date 1965/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2025/11/06
Oka On the other hand from Cartan [5]-Behnke-Stein [2]'s theorem, a Cousin-I domain in C 2 is a domain of holomorphy. In this way any domain of holomorphy in C 2 can be completely characterized by additive Cousin's problems. For n3, however, Cartan [6] showed that a Cousin-I domain in C n is not necessarily a domain of holomorphy. In the previous paper [10] we tried to characterize a domain of holomorphy in a Stein manifold by additive Cousin's problems. An open set G in C 71 is called regular if G\P is a Cousin-I open set for any relatively compact poly cylinder P in C n . We proved that a domain in C n is a domain of holomorphy if and only if it can be exhausted by regular domains. Moreover, we proved that a regular open set is pseudoconvex in the Cartan's sense at its continuous boundary point. Making use of the results of Oka [13] or Docquier-Grauert [7] respectively, we proved that a domain in C n or more generally in a Stein manifold with a smooth boundary is a domain of holomorphy if and only if it is locally regular at its each boundary point.