A K\"ahler 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 K\"ahler potential of $\mathbb H^n$ is unique up to the automorphisms among K\"ahler potentials whose differentials have constant norms. As an application, we consider a domain $\Omega$ in $\mathbb C^n$, which is biholomorphic to $\mathbb B^n$. We show that if $\Omega$ is affine homogeneous, then it is affine equivalent to $\mathbb H^n$. Assume next that its canonical potential with respect to the K\"ahler--Einstein metric has a differential with a constant norm. If the biholomorphism between $\Omega$ and $\mathbb B^n$ is a restriction of a M\"obius transformation, then the map is affine equivalent to a Cayley transform.