A Kähler potential on the unit ball with constant differential norm

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