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 $Ω$ in $\mathbb C^n$, which is biholomorphic to $\mathbb B^n$. We show that if $Ω$ 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 $Ω$ and $\mathbb B^n$ is a restriction of a Möbius transformation, then the map is affine equivalent to a Cayley transform.