A Kahler potential on the unit ball with constant differential norm

초록

Let B-n be the unit ball in C-n and H-n be the homogeneous Siegel domain of the second kind which is biholomorphic to B-n. We show that the Kahler potential of H-n is unique up to the automorphisms among Kahler potentials whose differentials have constant norms. As an application, we consider a domain Omega in C-n, which is biholomorphic to B-n. We show that if Omega is affine homogeneous, then it is affine equivalent to H-n. Assume next that its canonical potential with respect to the Kahler-Einstein metric has a differential with a constant norm. If the biholomorphism between Omega and B-n is a restriction of a Mobius transformation, then the map is affine equivalent to a Cayley transform.