A Kähler potential on the unit ball with constant differential norm

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