Geometry of Bounded Domains

In this paper, we shall study differential geometric properties of bounded domains in Cn. Here is the summary of our results. We consider an w-dimensional complex manifold M and the Hilbert space of square integrable holomorphic re-forms on M. After Bergman [3; 4; 5], we define the kernel form on M (instead of the kernel function) and, under certain assumptions, we define the invariant metric of Bergman. This method of generalizing the theory of S. Bergman (although the generalization is not essential) allows us to define the Bergman metric on certain compact complex manifolds. Some elementary properties of the kernel form and the Bergman metric (mostly already classical) are studied for the sake of completeness. Then we reexamine the theorem of H. Cartan on the group of holomorphic transformations of a bounded domain from the differential geometric point of view. We study also differential geometric properties of a manifold M which admits a discontinuous group D of holomorphic transformations such that M/D is compact. It should be noted that such a manifold possesses properties similar to those of a homogeneous manifold. Bremermann has studied the bounded domains with the following property (P): the kernel function goes to infinity at every boundary point [8]. He has shown that a bounded domain with the property (P) is a domain of holomorphy and that the converse is not true. (The same result has been obtained also by Sommer and Mehring [22].) Making use of this result, he has proved that if a bounded domain is complete with respect to the Bergman metric, then it is a domain of holomorphy. Since the kernel function is not intrinsically defined, the property (P) is not intrinsic. We consider, therefore, a condition which is stronger than (P) but which is intrinsic. This condition can be roughly stated as follows: if K is the kernel form and / is a square integrable holomorphic re-form, then (f/\J)/K goes to zero at every boundary point. We shall prove that this condition implies the completeness of the Bergman metric. From this result it can be proved, for instance, that every bounded analytic polyhedron is complete with respect to the Bergman metric. As a preparation for the above study of "completeness," we shall prove that a manifold with the Bergman metric can be isometrically imbedded, in a natural way, into a complex projective space (of infinite dimension, in general). Finally, we shall give examples of manifolds (beside the classical bounded