Decomposition theorem of normalized biholomorphic convex mappings

In this paper, we prove the decomposition theorem of biholomorphic convex mappings on the product domains of bounded convex circular domains. Namely, let Omega(i) be bounded convex and circular domains in C-ni, i = 1, 2. Then every biholomorphic convex mapping on the product domain Omega(1), x Omega(2), is the direct product of the biholomorphic convex mappings on Omega(i), i=1, 2. This generalizes the result of [Su] from polydiscs to more general product domains about the decomposition theorem of biholomorphic convex mappings. On the other hand, this also extends the result of [MT) from irreducible to reducible bounded symmetric domains about the characterization of biholomorphic convex mappings. As an application, we exhibit that balls and their product domains are the only bounded symmetric domains on which there exists a convex radius for the class of locally uniform bounded normalized biholomorphic mappings, which generalizes the result of [Sh]. The point of our decomposition theorem is that this along with the result of [MT] shows, the theory of biholomorphic convex mappings on the bounded symmetric domains is clear for rank greater than or equal to 2 and remains meaningful only on the domains of balls and their products.