DECOMPOSITION THEOREM OF NORMALIZED BIHOLOMORPHIC CONVEX MAPPINGS

In this paper, we prove the decomposition theorem of biholomorphic con vex 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 doma in Omega(1), x Omega(2), is the direct product of the biholomorphic co nvex mappings on Omega(i), i=1, 2. This generalizes the result of [Su] from polydiscs to more general product domains about the decompositio n theorem of biholomorphic convex mappings. On the other hand, this al so extends the result of [MT) from irreducible to reducible bounded sy mmetric domains about the characterization of biholomorphic convex map pings. As an application, we exhibit that balls and their product doma ins are the only bounded symmetric domains on which there exists a con vex radius for the class of locally uniform bounded normalized biholom orphic mappings, which generalizes the result of [Sh]. The point of ou r decomposition theorem is that this along with the result of [MT] sho ws, the theory of biholomorphic convex mappings on the bounded symmetr ic domains is clear for rank greater than or equal to 2 and remains me aningful only on the domains of balls and their products.